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Unformatted text preview: 5E-03(pp 126-135) 1/17/06 1:49 PM Page 126 CHAPTER 3 By measuring slopes at points on the sine curve, we get strong visual evidence that the derivative of the sine function is the cosine function. Derivatives 5E-03(pp 126-135) 1/17/06 1:49 PM Page 127 In this chapter we begin our study of differential calculus, which is concerned with how one quantity changes in relation to another quantity. The central concept of differential calculus is the derivative, which is an outgrowth of the velocities and slopes of tangents that we considered in Chapter 2. After learning how to calculate derivatives, we use them to solve problems involving rates of change and the approximation of functions. |||| 3.1 Derivatives In Section 2.6 we defined the slope of the tangent to a curve with equation y ෇ f ͑x͒ at the point where x ෇ a to be 1 m ෇ lim h l0 f ͑a ϩ h͒ Ϫ f ͑a͒ h We also saw that the velocity of an object with position function s ෇ f ͑t͒ at time t ෇ a is v ͑a͒ ෇ lim h l0 f ͑a ϩ h͒ Ϫ f ͑a͒ h In fact, limits of the form lim h l0 f ͑a ϩ h͒ Ϫ f ͑a͒ h arise whenever we calculate a rate of change in any of the sciences or engineering, such as a rate of reaction in chemistry or a marginal cost in economics. Since this type of limit occurs so widely, it is given a special name and notation. 2 Definition The derivative of a function f at a number a, denoted by f Ј͑a͒, is f Ј͑a͒ ෇ lim |||| f Ј͑a͒ is read “f prime of a.” h l0 f ͑a ϩ h͒ Ϫ f ͑a͒ h if this limit exists. If we write x ෇ a ϩ h, then h ෇ x Ϫ a and h approaches 0 if and only if x approaches a. Therefore, an equivalent way of stating the definition of the derivative, as we saw in finding tangent lines, is 3 f Ј͑a͒ ෇ lim xla f ͑x͒ Ϫ f ͑a͒ xϪa 127 5E-03(pp 126-135) 128 ❙❙❙❙ 1/17/06 1:49 PM Page 128 CHAPTER 3 DERIVATIVES EXAMPLE 1 Find the derivative of the function f ͑x͒ ෇ x 2 Ϫ 8x ϩ 9 at the number a. SOLUTION From Definition 2 we have f Ј͑a͒ ෇ lim Try problems like this one. Resources / Module 3 / Derivative at a Point / Problem Wizard h l0 f ͑a ϩ h͒ Ϫ f ͑a͒ h ෇ lim ͓͑a ϩ h͒2 Ϫ 8͑a ϩ h͒ ϩ 9͔ Ϫ ͓a 2 Ϫ 8a ϩ 9͔ h ෇ lim a 2 ϩ 2ah ϩ h 2 Ϫ 8a Ϫ 8h ϩ 9 Ϫ a 2 ϩ 8a Ϫ 9 h ෇ lim 2ah ϩ h 2 Ϫ 8h ෇ lim ͑2a ϩ h Ϫ 8͒ h l0 h h l0 h l0 h l0 ෇ 2a Ϫ 8 Interpretation of the Derivative as the Slope of a Tangent In Section 2.6 we defined the tangent line to the curve y ෇ f ͑x͒ at the point P͑a, f ͑a͒͒ to be the line that passes through P and has slope m given by Equation 1. Since, by Definition 2, this is the same as the derivative f Ј͑a͒, we can now say the following. The tangent line to y ෇ f ͑x͒ at ͑a, f ͑a͒͒ is the line through ͑a, f ͑a͒͒ whose slope is equal to f Ј͑a͒, the derivative of f at a. Thus, the geometric interpretation of a derivative [as defined by either (2) or (3)] is as shown in Figure 1. y y y=ƒ f(a+h)-f(a) P y=ƒ h x-a 0 0 a FIGURE 1 x a+h f(a+h)-f(a) h h=0 =slope of tangent at P =slope of curve at P (a) f ª(a)=lim Geometric interpretation of the derivative ƒ-f(a) P a x x ƒ-f(a) x-a x=a =slope of tangent at P =slope of curve at P (b) f ª(a)=lim If we use the point-slope form of the equation of a line, we can write an equation of the tangent line to the curve y ෇ f ͑x͒ at the point ͑a, f ͑a͒͒: y Ϫ f ͑a͒ ෇ f Ј͑a͒͑x Ϫ a͒ EXAMPLE 2 Find an equation of the tangent line to the parabola y ෇ x 2 Ϫ 8x ϩ 9 at the point ͑3, Ϫ6͒. SOLUTION From Example 1 we know that the derivative of f ͑x͒ ෇ x 2 Ϫ 8x ϩ 9 at the number a is f Ј͑a͒ ෇ 2a Ϫ 8. Therefore, the slope of the tangent line at ͑3, Ϫ6͒ is 5E-03(pp 126-135) 1/17/06 1:49 PM Page 129 SECTION 3.1 DERIVATIVES ❙❙❙❙ 129 f Ј͑3͒ ෇ 2͑3͒ Ϫ 8 ෇ Ϫ2. Thus, an equation of the tangent line, shown in Figure 2, is y y=≈-8x+9 y Ϫ ͑Ϫ6͒ ෇ ͑Ϫ2͒͑x Ϫ 3͒ x 0 (3, _6) y=_2x y ෇ Ϫ2x or EXAMPLE 3 Let f ͑x͒ ෇ 2 x. Estimate the value of f Ј͑0͒ in two ways: (a) By using Definition 2 and taking successively smaller values of h. (b) By interpreting f Ј͑0͒ as the slope of a tangent and using a graphing calculator to zoom in on the graph of y ෇ 2 x. SOLUTION (a) From Definition 2 we have FIGURE 2 f Ј͑0͒ ෇ lim h l0 h 2 Ϫ1 h 0.1 0.01 0.001 0.0001 Ϫ0.1 Ϫ0.01 Ϫ0.001 Ϫ0.0001 0.718 0.696 0.693 0.693 0.670 0.691 0.693 0.693 h f ͑h͒ Ϫ f ͑0͒ 2h Ϫ 1 ෇ lim h l0 h h Since we are not yet able to evaluate this limit exactly, we use a calculator to approximate the values of ͑2 h Ϫ 1͒͞h. From the numerical evidence in the table at the left we see that as h approaches 0, these values appear to approach a number near 0.69. So our estimate is f Ј͑0͒ Ϸ 0.69 (b) In Figure 3 we graph the curve y ෇ 2 x and zoom in toward the point ͑0, 1͒. We see that the closer we get to ͑0, 1͒, the more the curve looks like a straight line. In fact, in Figure 3(c) the curve is practically indistinguishable from its tangent line at ͑0, 1͒. Since the x-scale and the y-scale are both 0.01, we estimate that the slope of this line is 0.14 ෇ 0.7 0.20 So our estimate of the derivative is f Ј͑0͒ Ϸ 0.7. In Chapter 7 we will show that, correct to six decimal places, f Ј͑0͒ Ϸ 0.693147. (0, 1) (a) ͓_1, 1͔ by ͓0, 2͔ FIGURE 3 (0, 1) (0, 1) (b) ͓_0.5, 0.5͔ by ͓0.5, 1.5͔ (c) ͓_0.1, 0.1͔ by ͓0.9, 1.1͔ Zooming in on the graph of y=2® near (0, 1) Interpretation of the Derivative as a Rate of Change In Section 2.6 we defined the instantaneous rate of change of y ෇ f ͑x͒ with respect to x at x ෇ x 1 as the limit of the average rates of change over smaller and smaller intervals. If the interval is ͓x 1, x 2 ͔, then the change in x is ⌬x ෇ x 2 Ϫ x 1, the corresponding change in y is ⌬y ෇ f ͑x 2 ͒ Ϫ f ͑x 1͒ and 4 instantaneous rate of change ෇ lim ⌬x l 0 ⌬y f ͑x 2 ͒ Ϫ f ͑x1͒ ෇ lim x lx ⌬x x 2 Ϫ x1 2 1 5E-03(pp 126-135) 130 ❙❙❙❙ 1/17/06 1:49 PM Page 130 CHAPTER 3 DERIVATIVES From Equation 3 we recognize this limit as being the derivative of f at x 1, that is, f Ј͑x 1͒. This gives a second interpretation of the derivative: The derivative f Ј͑a͒ is the instantaneous rate of change of y ෇ f ͑x͒ with respect to x when x ෇ a. y Q P x FIGURE 4 The y-values are changing rapidly at P and slowly at Q. The connection with the first interpretation is that if we sketch the curve y ෇ f ͑x͒, then the instantaneous rate of change is the slope of the tangent to this curve at the point where x ෇ a. This means that when the derivative is large (and therefore the curve is steep, as at the point P in Figure 4), the y-values change rapidly. When the derivative is small, the curve is relatively flat and the y-values change slowly. In particular, if s ෇ f ͑t͒ is the position function of a particle that moves along a straight line, then f Ј͑a͒ is the rate of change of the displacement s with respect to the time t. In other words, f Ј͑a͒ is the velocity of the particle at time t ෇ a. (See Section 2.6.) The speed of the particle is the absolute value of the velocity, that is, f Ј͑a͒ . Խ Խ EXAMPLE 4 The position of a particle is given by the equation of motion s ෇ f ͑t͒ ෇ 1͑͞1 ϩ t͒, where t is measured in seconds and s in meters. Find the velocity and the speed after 2 seconds. SOLUTION The derivative of f when t ෇ 2 is 1 1 Ϫ f ͑2 ϩ h͒ Ϫ f ͑2͒ 1 ϩ ͑2 ϩ h͒ 1ϩ2 f Ј͑2͒ ෇ lim ෇ lim h l0 h l0 h h In Module 3.1 you are asked to compare and order the slopes of tangent and secant lines at several points on a curve. 1 1 3 Ϫ ͑3 ϩ h͒ Ϫ 3ϩh 3 3͑3 ϩ h͒ ෇ lim ෇ lim h l0 h l0 h h ෇ lim h l0 Ϫh Ϫ1 1 ෇ lim ෇Ϫ h l 0 3͑3 ϩ h͒ 3͑3 ϩ h͒h 9 Thus, the velocity after 2 seconds is f Ј͑2͒ ෇ Ϫ 1 m͞s, and the speed is 9 f Ј͑2͒ ෇ Ϫ 1 ෇ 1 m͞s. 9 9 Խ Խ Խ Խ EXAMPLE 5 A manufacturer produces bolts of a fabric with a fixed width. The cost of producing x yards of this fabric is C ෇ f ͑x͒ dollars. (a) What is the meaning of the derivative f Ј͑x͒? What are its units? (b) In practical terms, what does it mean to say that f Ј͑1000͒ ෇ 9 ? (c) Which do you think is greater, f Ј͑50͒ or f Ј͑500͒? What about f Ј͑5000͒? SOLUTION (a) The derivative f Ј͑x͒ is the instantaneous rate of change of C with respect to x; that is, f Ј͑x͒ means the rate of change of the production cost with respect to the number of yards produced. (Economists call this rate of change the marginal cost. This idea is discussed in more detail in Sections 3.4 and 4.8.) Because ⌬C f Ј͑x͒ ෇ lim ⌬x l 0 ⌬x the units for f Ј͑x͒ are the same as the units for the difference quotient ⌬C͞⌬x. Since ⌬C is measured in dollars and ⌬x in yards, it follows that the units for f Ј͑x͒ are dollars per yard. 5E-03(pp 126-135) 1/17/06 1:49 PM Page 131 SECTION 3.1 DERIVATIVES ❙❙❙❙ 131 (b) The statement that f Ј͑1000͒ ෇ 9 means that, after 1000 yards of fabric have been manufactured, the rate at which the production cost is increasing is $9͞yard. (When x ෇ 1000, C is increasing 9 times as fast as x.) Since ⌬x ෇ 1 is small compared with x ෇ 1000, we could use the approximation |||| Here we are assuming that the cost function is well behaved; in other words, C͑x͒ doesn’t oscillate rapidly near x ෇ 1000. f Ј͑1000͒ Ϸ ⌬C ⌬C ෇ ෇ ⌬C ⌬x 1 and say that the cost of manufacturing the 1000th yard (or the 1001st) is about $9. (c) The rate at which the production cost is increasing (per yard) is probably lower when x ෇ 500 than when x ෇ 50 (the cost of making the 500th yard is less than the cost of the 50th yard) because of economies of scale. (The manufacturer makes more efficient use of the fixed costs of production.) So f Ј͑50͒ Ͼ f Ј͑500͒ But, as production expands, the resulting large-scale operation might become inefficient and there might be overtime costs. Thus, it is possible that the rate of increase of costs will eventually start to rise. So it may happen that f Ј͑5000͒ Ͼ f Ј͑500͒ The following example shows how to estimate the derivative of a tabular function, that is, a function defined not by a formula but by a table of values. t D͑t͒ 1980 1985 1990 1995 2000 930.2 1945.9 3233.3 4974.0 5674.2 EXAMPLE 6 Let D͑t͒ be the U.S. national debt at time t. The table in the margin gives approximate values of this function by providing end of year estimates, in billions of dollars, from 1980 to 2000. Interpret and estimate the value of DЈ͑1990͒. SOLUTION The derivative DЈ͑1990͒ means the rate of change of D with respect to t when t ෇ 1990, that is, the rate of increase of the national debt in 1990. According to Equation 3, DЈ͑1990͒ ෇ lim t l1990 D͑t͒ Ϫ D͑1990͒ t Ϫ 1990 So we compute and tabulate values of the difference quotient (the average rates of change) as follows. t 1980 1985 1995 2000 |||| Another method is to plot the debt function and estimate the slope of the tangent line when t ෇ 1990. (See Example 5 in Section 2.6.) D͑t͒ Ϫ D͑1990͒ t Ϫ 1990 230.31 257.48 348.14 244.09 From this table we see that DЈ͑1990͒ lies somewhere between 257.48 and 348.14 billion dollars per year. [Here we are making the reasonable assumption that the debt didn’t fluctuate wildly between 1980 and 2000.] We estimate that the rate of increase of the national debt of the United States in 1990 was the average of these two numbers, namely DЈ͑1990͒ Ϸ 303 billion dollars per year 5E-03(pp 126-135) 132 ❙❙❙❙ 1/17/06 1:50 PM Page 132 CHAPTER 3 DERIVATIVES |||| 3.1 Exercises 1. On the given graph of f, mark lengths that represent f ͑2͒, f ͑2 ϩ h͒, f ͑2 ϩ h͒ Ϫ f ͑2͒, and h. (Choose h Ͼ 0.) What f ͑2 ϩ h͒ Ϫ f ͑2͒ line has slope ? h 10. (a) If G͑x͒ ෇ x͑͞1 ϩ 2x͒, find GЈ͑a͒ and use it to find an equation of the tangent line to the curve y ෇ x͑͞1 ϩ 2x͒ at 1 the point (Ϫ 4 , Ϫ 1 ). 2 (b) Illustrate part (a) by graphing the curve and the tangent line on the same screen. ; y 11. Let f ͑x͒ ෇ 3 x. Estimate the value of f Ј͑1͒ in two ways: y=ƒ (a) By using Definition 2 and taking successively smaller values of h. (b) By zooming in on the graph of y ෇ 3 x and estimating the slope. ; 0 12. Let t͑x͒ ෇ tan x. Estimate the value of tЈ͑␲͞4͒ in two ways: x 2 2. For the function f whose graph is shown in Exercise 1, arrange the following numbers in increasing order and explain your reasoning: f Ј͑2͒ 0 f ͑3͒ Ϫ f ͑2͒ 1 2 ͓ f ͑4͒ Ϫ f ͑2͔͒ numbers in increasing order and explain your reasoning: tЈ͑Ϫ2͒ tЈ͑0͒ tЈ͑2͒ ; 13–18 |||| Find f Ј͑a͒. 13. f ͑x͒ ෇ 3 Ϫ 2x ϩ 4x 2 14. f ͑t͒ ෇ t 4 Ϫ 5t tЈ͑4͒ y 15. f ͑t͒ ෇ 2t ϩ 1 tϩ3 16. f ͑x͒ ෇ 17. f ͑x͒ ෇ 3. For the function t whose graph is given, arrange the following 0 (a) By using Definition 2 and taking successively smaller values of h. (b) By zooming in on the graph of y ෇ tan x and estimating the slope. 1 sx ϩ 2 18. f ͑x͒ ෇ s3x ϩ 1 ■ ■ ■ ■ ■ ■ ■ ■ x2 ϩ 1 xϪ2 ■ ■ ■ ■ y=© 19–24 |||| Each limit represents the derivative of some function f at some number a. State such an f and a in each case. 0 1 2 3 4 x 19. lim ͑1 ϩ h͒10 Ϫ 1 h 20. lim 21. lim _1 2 x Ϫ 32 xϪ5 22. lim 23. lim cos͑␲ ϩ h͒ ϩ 1 h 24. lim h l0 x l5 4. If the tangent line to y ෇ f ͑x͒ at (4, 3) passes through the point (0, 2), find f ͑4͒ and f Ј͑4͒. 5. Sketch the graph of a function f for which f ͑0͒ ෇ 0, f Ј͑0͒ ෇ 3, h l0 ■ ■ ■ h l0 4 s16 ϩ h Ϫ 2 h x l ␲͞4 ■ ■ t l1 ■ ■ ■ tan x Ϫ 1 x Ϫ ␲͞4 t4 ϩ t Ϫ 2 tϪ1 ■ ■ ■ ■ f Ј͑1͒ ෇ 0, and f Ј͑2͒ ෇ Ϫ1. 6. Sketch the graph of a function t for which t͑0͒ ෇ 0, tЈ͑0͒ ෇ 3, tЈ͑1͒ ෇ 0, and tЈ͑2͒ ෇ 1. 7. If f ͑x͒ ෇ 3x 2 Ϫ 5x, find f Ј͑2͒ and use it to find an equation of the tangent line to the parabola y ෇ 3x 2 Ϫ 5x at the point ͑2, 2͒. 8. If t͑x͒ ෇ 1 Ϫ x 3, find tЈ͑0͒ and use it to find an equation of the tangent line to the curve y ෇ 1 Ϫ x 3 at the point ͑0, 1͒. 9. (a) If F͑x͒ ෇ x 3 Ϫ 5x ϩ 1, find FЈ͑1͒ and use it to find an ; equation of the tangent line to the curve y ෇ x 3 Ϫ 5x ϩ 1 at the point ͑1, Ϫ3͒. (b) Illustrate part (a) by graphing the curve and the tangent line on the same screen. 25–26 |||| A particle moves along a straight line with equation of motion s ෇ f ͑t͒, where s is measured in meters and t in seconds. Find the velocity when t ෇ 2. 25. f ͑t͒ ෇ t 2 Ϫ 6t Ϫ 5 ■ ■ ■ ■ 26. f ͑t͒ ෇ 2t 3 Ϫ t ϩ 1 ■ ■ ■ ■ ■ ■ ■ 27. The cost of producing x ounces of gold from a new gold mine is C ෇ f ͑x͒ dollars. (a) What is the meaning of the derivative f Ј͑x͒? What are its units? (b) What does the statement f Ј͑800͒ ෇ 17 mean? (c) Do you think the values of f Ј͑x͒ will increase or decrease in the short term? What about the long term? Explain. ■ 5E-03(pp 126-135) 1/17/06 1:50 PM Page 133 WRITING PROJECT EARLY METHODS FOR FINDING TANGENTS ❙❙❙❙ 133 28. The number of bacteria after t hours in a controlled laboratory 33. The quantity of oxygen that can dissolve in water depends on experiment is n ෇ f ͑t͒. (a) What is the meaning of the derivative f Ј͑5͒? What are its units? (b) Suppose there is an unlimited amount of space and nutrients for the bacteria. Which do you think is larger, f Ј͑5͒ or f Ј͑10͒? If the supply of nutrients is limited, would that affect your conclusion? Explain. the temperature of the water. (So thermal pollution influences the oxygen content of water.) The graph shows how oxygen solubility S varies as a function of the water temperature T . (a) What is the meaning of the derivative SЈ͑T ͒? What are its units? (b) Estimate the value of SЈ͑16͒ and interpret it. S (mg/L) 16 29. The fuel consumption (measured in gallons per hour) of a car traveling at a speed of v miles per hour is c ෇ f ͑v. (a) What is the meaning of the derivative f Ј͑v͒? What are its 12 units? (b) Write a sentence (in layman’s terms) that explains the meaning of the equation f Ј͑20͒ ෇ Ϫ0.05. 8 4 30. The quantity (in pounds) of a gourmet ground coffee that is sold by a coffee company at a price of p dollars per pound is Q ෇ f ͑ p͒. (a) What is the meaning of the derivative f Ј͑8͒? What are its units? (b) Is f Ј͑8͒ positive or negative? Explain. 0 2 4 6 8 10 12 73 73 70 69 72 81 88 40 T (°C) 32 S (cm /s) 20 14 T 24 maximum sustainable swimming speed S of Coho salmon. (a) What is the meaning of the derivative SЈ͑T ͒? What are its units? (b) Estimate the values of SЈ͑15͒ and SЈ͑25͒ and interpret them. night on June 2, 2001. The table shows values of this function recorded every two hours. What is the meaning of T Ј͑10͒? Estimate its value. 0 16 34. The graph shows the influence of the temperature T on the 31. Let T͑t͒ be the temperature (in Њ F ) in Dallas t hours after mid- t 8 91 32. Life expectancy improved dramatically in the 20th century. The table gives values of E͑t͒, the life expectancy at birth (in years) of a male born in the year t in the United States. Interpret and estimate the values of EЈ͑1910͒ and EЈ͑1950͒. t E͑t͒ t E͑t͒ 1900 1910 1920 1930 1940 1950 48.3 51.1 55.2 57.4 62.5 65.6 1960 1970 1980 1990 2000 66.6 67.1 70.0 71.8 74.1 0 35–36 |||| 10 Determine whether f Ј͑0͒ exists. 35. f ͑x͒ ෇ ͭ ͭ x sin 1 x if x 36. f ͑x͒ ෇ x 2 sin 1 x if x ■ 0 if x ෇ 0 0 ■ 0 if x ෇ 0 0 ■ T (°C) 20 ■ ■ ■ ■ ■ ■ ■ ■ WRITING PROJECT Early Methods for Finding Tangents The first person to formulate explicitly the ideas of limits and derivatives was Sir Isaac Newton in the 1660s. But Newton acknowledged that “If I have seen further than other men, it is because I have stood on the shoulders of giants.” Two of those giants were Pierre Fermat (1601–1665) and Newton’s teacher at Cambridge, Isaac Barrow (1630–1677). Newton was familiar with the methods that these men used to find tangent lines, and their methods played a role in Newton’s eventual formulation of calculus. ■ 5E-03(pp 126-135) 134 ❙❙❙❙ 1/17/06 1:50 PM Page 134 CHAPTER 3 DERIVATIVES The following references contain explanations of these methods. Read one or more of the references and write a report comparing the methods of either Fermat or Barrow to modern methods. In particular, use the method of Section 3.1 to find an equation of the tangent line to the curve y ෇ x 3 ϩ 2x at the point (1, 3) and show how either Fermat or Barrow would have solved the same problem. Although you used derivatives and they did not, point out similarities between the methods. 1. Carl Boyer and Uta Merzbach, A History of Mathematics (New York: Wiley, 1989), pp. 389, 432. 2. C. H. Edwards, The Historical Development of the Calculus (New York: Springer-Verlag, 1979), pp. 124, 132. 3. Howard Eves, An Introduction to the History of Mathematics, 6th ed. (New York: Saunders, 1990), pp. 391, 395. 4. Morris Kline, Mathematical Thought from Ancient to Modern Times (New York: Oxford University Press, 1972), pp. 344, 346. |||| 3.2 The Derivative as a Function In the preceding section we considered the derivative of a function f at a fixed number a: 1 f Ј͑a͒ ෇ lim hl0 f ͑a ϩ h͒ Ϫ f ͑a͒ h Here we change our point of view and let the number a vary. If we replace a in Equation 1 by a variable x, we obtain 2 f Ј͑x͒ ෇ lim hl0 f ͑x ϩ h͒ Ϫ f ͑x͒ h Given any number x for which this limit exists, we assign to x the number f Ј͑x͒. So we can regard f Ј as a new function, called the derivative of f and defined by Equation 2. We know that the value of f Ј at x, f Ј͑x͒, can be interpreted geometrically as the slope of the tangent line to the graph of f at the point ͑x, f ͑x͒͒. The function f Ј is called the derivative of f because it has been “derived” from f by the limiting operation in Equation 2. The domain of f Ј is the set ͕x f Ј͑x͒ exists͖ and may be smaller than the domain of f . Խ EXAMPLE 1 The graph of a function f is given in Figure 1. Use it to sketch the graph of the derivative f Ј. y y=ƒ 1 0 FIGURE 1 1 x 5E-03(pp 126-135) 1/17/06 1:50 PM Page 135 SECTION 3.2 THE DERIVATIVE AS A FUNCTION Watch an animation of the relation between a function and its derivative. Resources / Module 3 / Derivatives as Functions / Mars Rover Resources / Module 3 / Slope-a-Scope / Derivative of a Cubic ❙❙❙❙ 135 SOLUTION We can estimate the value of the derivative at any value of x by drawing the tangent at the point ͑x, f ͑x͒͒ and estimating its slope. For instance, for x ෇ 5 we draw the tangent at P in Figure 2(a) and estimate its slope to be about 3 , so f Ј͑5͒ Ϸ 1.5. This 2 allows us to plot the point PЈ͑5, 1.5͒ on the graph of f Ј directly beneath P. Repeating this procedure at several points, we get the graph shown in Figure 2(b). Notice that the tangents at A, B, and C are horizontal, so the derivative is 0 there and the graph of f Ј crosses the x-axis at the points AЈ, BЈ, and CЈ, directly beneath A, B, and C. Between A and B the tangents have positive slope, so f Ј͑x͒ is positive there. But between B and C the tangents have negative slope, so f Ј͑x͒ is negative there. y B y=ƒ 1 P A 0 5 1 x C |||| Notice that where the derivative is positive (to the right of C and between A and B), the function f is increasing. Where f Ј͑x͒ is negative (to the left of A and between B and C ), f is decreasing. In Section 4.3 we will prove that this is true for all functions. (a) y P ª (5, 1.5) y=fª(x) 1 Bª 0 FIGURE 2 Aª Cª 1 5 x (b) If a function is defined by a table of values, then we can construct a table of approximate values of its derivative, as in the next example. 5E-03(pp 136-145) 136 ❙❙❙❙ 1/17/06 2:33 PM Page 136 CHAPTER 3 DERIVATIVES t 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 EXAMPLE 2 Let B͑t͒ be the population of Belgium at time t. The table at the left gives midyear values of B͑t͒, in thousands, from 1980 to 2000. Construct a table of values for the derivative of this function. B͑t͒ 9,847 9,856 9,855 9,862 9,884 9,962 10,036 10,109 10,152 10,175 10,186 SOLUTION We assume that there were no wild fluctuations in the population between the stated values. Let’s start by approximating BЈ͑1988͒, the rate of increase of the population of Belgium in mid-1988. Since BЈ͑1988͒ ෇ lim h l0 B͑1988 ϩ h͒ Ϫ B͑1988͒ h we have BЈ͑1988͒ Ϸ B͑1988 ϩ h͒ Ϫ B͑1988͒ h for small values of h. For h ෇ 2, we get BЈ͑1988͒ Ϸ B͑1990͒ Ϫ B͑1988͒ 9962 Ϫ 9884 ෇ ෇ 39 2 2 (This is the average rate of increase between 1988 and 1990.) For h ෇ Ϫ2, we have BЈ͑1988͒ Ϸ t which is the average rate of increase between 1986 and 1988. We get a more accurate approximation if we take the average of these rates of change: BЈ͑t͒ 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 B͑1986͒ Ϫ B͑1988͒ 9862 Ϫ 9884 ෇ ෇ 11 Ϫ2 Ϫ2 4.5 2.0 1.5 7.3 25.0 38.0 36.8 29.0 16.5 8.5 5.5 BЈ͑1988͒ Ϸ 1͑39 ϩ 11͒ ෇ 25 2 This means that in 1988 the population was increasing at a rate of about 25,000 people per year. Making similar calculations for the other values (except at the endpoints), we get the table at the left, which shows the approximate values for the derivative. y 10,200 10,100 y=B(t) 10,000 9,900 9,800 |||| Figure 3 illustrates Example 2 by showing graphs of the population function B͑t͒ and its derivative BЈ͑t͒. Notice how the rate of population growth increases to a maximum in 1990 and decreases thereafter. 1980 1984 1988 1992 1996 2000 t 1988 1992 1996 2000 t y 30 20 y=Bª(t) 10 FIGURE 3 1980 1984 5E-03(pp 136-145) 1/17/06 2:33 PM Page 137 SECTION 3.2 THE DERIVATIVE AS A FUNCTION ❙❙❙❙ 137 EXAMPLE 3 (a) If f ͑x͒ ෇ x 3 Ϫ x, find a formula for f Ј͑x͒. (b) Illustrate by comparing the graphs of f and f . SOLUTION (a) When using Equation 2 to compute a derivative, we must remember that the variable is h and that x is temporarily regarded as a constant during the calculation of the limit. f ͑x ϩ h͒ Ϫ f ͑x͒ ͓͑x ϩ h͒3 Ϫ ͑x ϩ h͔͒ Ϫ ͓x 3 Ϫ x͔ ෇ lim hl0 h h f Ј͑x͒ ෇ lim hl0 ෇ lim x 3 ϩ 3x 2h ϩ 3xh 2 ϩ h 3 Ϫ x Ϫ h Ϫ x 3 ϩ x h ෇ lim 3x 2h ϩ 3xh 2 ϩ h 3 Ϫ h h hl0 hl0 ෇ lim ͑3x 2 ϩ 3xh ϩ h 2 Ϫ 1͒ ෇ 3x 2 Ϫ 1 hl0 (b) We use a graphing device to graph f and f Ј in Figure 4. Notice that f Ј͑x͒ ෇ 0 when f has horizontal tangents and f Ј͑x͒ is positive when the tangents have positive slope. So these graphs serve as a check on our work in part (a). 2 2 fª f _2 2 FIGURE 4 See more problems like these. Resources / Module 3 / How to Calculate / The Essential Examples _2 _2 2 _2 EXAMPLE 4 If f ͑x͒ ෇ sx Ϫ 1, find the derivative of f . State the domain of f Ј. SOLUTION f Ј͑x͒ ෇ lim hl0 f ͑x ϩ h͒ Ϫ f ͑x͒ h ෇ lim sx ϩ h Ϫ 1 Ϫ sx Ϫ 1 h ෇ lim sx ϩ h Ϫ 1 Ϫ sx Ϫ 1 sx ϩ h Ϫ 1 ϩ sx Ϫ 1 ؒ h sx ϩ h Ϫ 1 ϩ sx Ϫ 1 ෇ lim ͑x ϩ h Ϫ 1͒ Ϫ ͑x Ϫ 1͒ h(sx ϩ h Ϫ 1 ϩ sx Ϫ 1 ) ෇ lim 1 sx ϩ h Ϫ 1 ϩ sx Ϫ 1 hl0 Here we rationalize the numerator. hl0 hl0 hl0 ෇ 1 1 ෇ 2sx Ϫ 1 sx Ϫ 1 ϩ sx Ϫ 1 We see that f Ј͑x͒ exists if x Ͼ 1, so the domain of f Ј is ͑1, ϱ͒. This is smaller than the domain of f , which is ͓1, ϱ͒. 5E-03(pp 136-145) 138 ❙❙❙❙ 1/17/06 2:33 PM Page 138 CHAPTER 3 DERIVATIVES Let’s check to see that the result of Example 4 is reasonable by looking at the graphs of f and f Ј in Figure 5. When x is close to 1, sx Ϫ 1 is close to 0, so f Ј͑x͒ ෇ 1͞(2sx Ϫ 1 ) is very large; this corresponds to the steep tangent lines near ͑1, 0͒ in Figure 5(a) and the large values of f Ј͑x͒ just to the right of 1 in Figure 5(b). When x is large, f Ј͑x͒ is very small; this corresponds to the flatter tangent lines at the far right of the graph of f and the horizontal asymptote of the graph of f Ј. y y 1 1 0 FIGURE 5 x 1 (a) ƒ=œ„„„„ x-1 EXAMPLE 5 Find f Ј if f ͑x͒ ෇ SOLUTION f Ј͑x͒ ෇ lim hl0 0 x 1 (b) f ª(x)= 1 2œ„„„„ x-1 1Ϫx . 2ϩx f ͑x ϩ h͒ Ϫ f ͑x͒ h 1 Ϫ ͑x ϩ h͒ 1Ϫx Ϫ 2 ϩ ͑x ϩ h͒ 2ϩx ෇ lim hl0 h a c Ϫ b d ad Ϫ bc 1 ෇ ؒ e bd e ෇ lim ͑1 Ϫ x Ϫ h͒͑2 ϩ x͒ Ϫ ͑1 Ϫ x͒͑2 ϩ x ϩ h͒ h͑2 ϩ x ϩ h͒͑2 ϩ x͒ ෇ lim ͑2 Ϫ x Ϫ 2h Ϫ x 2 Ϫ xh͒ Ϫ ͑2 Ϫ x ϩ h Ϫ x 2 Ϫ xh͒ h͑2 ϩ x ϩ h͒͑2 ϩ x͒ ෇ lim Ϫ3h h͑2 ϩ x ϩ h͒͑2 ϩ x͒ ෇ lim Ϫ3 3 ෇Ϫ ͑2 ϩ x ϩ h͒͑2 ϩ x͒ ͑2 ϩ x͒2 hl0 hl0 hl0 hl0 Other Notations If we use the traditional notation y ෇ f ͑x͒ to indicate that the independent variable is x and the dependent variable is y, then some common alternative notations for the derivative are as follows: f Ј͑x͒ ෇ yЈ ෇ dy df d ෇ ෇ f ͑x͒ ෇ Df ͑x͒ ෇ Dx f ͑x͒ dx dx dx The symbols D and d͞dx are called differentiation operators because they indicate the operation of differentiation, which is the process of calculating a derivative. 5E-03(pp 136-145) 1/17/06 2:33 PM Page 139 SECTION 3.2 THE DERIVATIVE AS A FUNCTION |||| Gottfried Wilhelm Leibniz was born in Leipzig in 1646 and studied law, theology, philosophy, and mathematics at the university there, graduating with a bachelor’s degree at age 17. After earning his doctorate in law at age 20, Leibniz entered the diplomatic service and spent most of his life traveling to the capitals of Europe on political missions. In particular, he worked to avert a French military threat against Germany and attempted to reconcile the Catholic and Protestant churches. His serious study of mathematics did not begin until 1672 while he was on a diplomatic mission in Paris. There he built a calculating machine and met scientists, like Huygens, who directed his attention to the latest developments in mathematics and science. Leibniz sought to develop a symbolic logic and system of notation that would simplify logical reasoning. In particular, the version of calculus that he published in 1684 established the notation and the rules for finding derivatives that we use today. Unfortunately, a dreadful priority dispute arose in the 1690s between the followers of Newton and those of Leibniz as to who had invented calculus first. Leibniz was even accused of plagiarism by members of the Royal Society in England. The truth is that each man invented calculus independently. Newton arrived at his version of calculus first but, because of his fear of controversy, did not publish it immediately. So Leibniz’s 1684 account of calculus was the first to be published. ❙❙❙❙ 139 The symbol dy͞dx, which was introduced by Leibniz, should not be regarded as a ratio (for the time being); it is simply a synonym for f Ј͑x͒. Nonetheless, it is a very useful and suggestive notation, especially when used in conjunction with increment notation. Referring to Equation 3.1.4, we can rewrite the definition of derivative in Leibniz notation in the form dy ⌬y ෇ lim ⌬x l 0 ⌬x dx If we want to indicate the value of a derivative dy͞dx in Leibniz notation at a specific number a, we use the notation dy dx Ϳ or x෇a dy dx ͬ x෇a which is a synonym for f Ј͑a͒. 3 Definition A function f is differentiable at a if f Ј͑a͒ exists. It is differentiable on an open interval ͑a, b͒ [or ͑a, ϱ͒ or ͑Ϫϱ, a͒ or ͑Ϫϱ, ϱ͒] if it is differentiable at every number in the interval. Խ Խ EXAMPLE 6 Where is the function f ͑x͒ ෇ x differentiable? Խ Խ SOLUTION If x Ͼ 0, then x ෇ x and we can choose h small enough that x ϩ h Ͼ 0 and Խ Խ hence x ϩ h ෇ x ϩ h. Therefore, for x Ͼ 0 we have f Ј͑x͒ ෇ lim Խx ϩ hԽ Ϫ ԽxԽ h hl0 ෇ lim hl0 ͑x ϩ h͒ Ϫ x h ෇ lim ෇ lim 1 ෇ 1 hl0 h hl0 h and so f is differentiable for any x Ͼ 0. Similarly, for x Ͻ 0 we have x ෇ Ϫx and h can be chosen small enough that x ϩ h Ͻ 0 and so x ϩ h ෇ Ϫ͑x ϩ h͒. Therefore, for x Ͻ 0, Խ Խ f Ј͑x͒ ෇ lim hl0 ෇ lim hl0 Խ Խ Խx ϩ hԽ Ϫ ԽxԽ h Ϫ͑x ϩ h͒ Ϫ ͑Ϫx͒ Ϫh ෇ lim ෇ lim ͑Ϫ1͒ ෇ Ϫ1 hl0 h hl0 h and so f is differentiable for any x Ͻ 0. For x ෇ 0 we have to investigate f Ј͑0͒ ෇ lim hl0 ෇ lim hl0 f ͑0 ϩ h͒ Ϫ f ͑0͒ h Խ0 ϩ hԽ Ϫ Խ0Խ h ͑if it exists͒ 5E-03(pp 136-145) 140 ❙❙❙❙ 1/17/06 2:34 PM Page 140 CHAPTER 3 DERIVATIVES Let’s compute the left and right limits separately: limϩ Խ0 ϩ hԽ Ϫ Խ0Խ ෇ limϪ Խ0 ϩ hԽ Ϫ Խ0Խ ෇ hl0 and hl0 h h limϩ ԽhԽ ෇ limϪ ԽhԽ ෇ h hl0 h hl0 limϩ h ෇ limϩ 1 ෇ 1 hl0 h limϪ Ϫh ෇ limϪ ͑Ϫ1͒ ෇ Ϫ1 hl0 h hl0 hl0 Since these limits are different, f Ј͑0͒ does not exist. Thus, f is differentiable at all x except 0. A formula for f Ј is given by f Ј͑x͒ ෇ ͭ 1 Ϫ1 if x Ͼ 0 if x Ͻ 0 and its graph is shown in Figure 6(b). The fact that f Ј͑0͒ does not exist is reflected geometrically in the fact that the curve y ෇ x does not have a tangent line at ͑0, 0͒. [See Figure 6(a).] Խ Խ y y 1 x 0 x 0 FIGURE 6 _1 (a) y=ƒ=| x | (b) y=fª(x) Both continuity and differentiability are desirable properties for a function to have. The following theorem shows how these properties are related. 4 Theorem If f is differentiable at a, then f is continuous at a. Proof To prove that f is continuous at a, we have to show that lim x l a f ͑x͒ ෇ f ͑a͒. We do this by showing that the difference f ͑x͒ Ϫ f ͑a͒ approaches 0. The given information is that f is differentiable at a, that is, f Ј͑a͒ ෇ lim xla f ͑x͒ Ϫ f ͑a͒ xϪa exists (see Equation 3.1.3). To connect the given and the unknown, we divide and multiply f ͑x͒ Ϫ f ͑a͒ by x Ϫ a (which we can do when x a): f ͑x͒ Ϫ f ͑a͒ ෇ f ͑x͒ Ϫ f ͑a͒ ͑x Ϫ a͒ xϪa Thus, using the Product Law and (3.1.3), we can write lim ͓ f ͑x͒ Ϫ f ͑a͔͒ ෇ lim xla xla f ͑x͒ Ϫ f ͑a͒ ͑x Ϫ a͒ xϪa 5E-03(pp 136-145) 1/17/06 2:34 PM Page 141 SECTION 3.2 THE DERIVATIVE AS A FUNCTION ෇ lim xla ❙❙❙❙ 141 f ͑x͒ Ϫ f ͑a͒ lim ͑x Ϫ a͒ xla xϪa ෇ f Ј͑a͒ ؒ 0 ෇ 0 To use what we have just proved, we start with f ͑x͒ and add and subtract f ͑a͒: lim f ͑x͒ ෇ lim ͓ f ͑a͒ ϩ ͑ f ͑x͒ Ϫ f ͑a͔͒͒ xla xla ෇ lim f ͑a͒ ϩ lim ͓ f ͑x͒ Ϫ f ͑a͔͒ xla xla ෇ f ͑a͒ ϩ 0 ෇ f ͑a͒ Therefore, f is continuous at a. | NOTE The converse of Theorem 4 is false; that is, there are functions that are continuous but not differentiable. For instance, the function f ͑x͒ ෇ x is continuous at 0 because ■ Խ Խ Խ Խ lim f ͑x͒ ෇ lim x ෇ 0 ෇ f ͑0͒ xl0 xl0 (See Example 7 in Section 2.3.) But in Example 6 we showed that f is not differentiable at 0. How Can a Function Fail to Be Differentiable? Խ Խ We saw that the function y ෇ x in Example 6 is not differentiable at 0 and Figure 6(a) shows that its graph changes direction abruptly when x ෇ 0. In general, if the graph of a function f has a “corner” or “kink” in it, then the graph of f has no tangent at this point and f is not differentiable there. [In trying to compute f Ј͑a͒, we find that the left and right limits are different.] Theorem 4 gives another way for a function not to have a derivative. It says that if f is not continuous at a, then f is not differentiable at a. So at any discontinuity (for instance, a jump discontinuity) f fails to be differentiable. A third possibility is that the curve has a vertical tangent line when x ෇ a; that is, f is continuous at a and y vertical tangent line 0 a x FIGURE 7 Խ Խ lim f Ј͑x͒ ෇ ϱ xla This means that the tangent lines become steeper and steeper as x l a. Figure 7 shows one way that this can happen; Figure 8(c) shows another. Figure 8 illustrates the three possibilities that we have discussed. y 0 y a x 0 y a x 0 a FIGURE 8 Three ways for ƒ not to be differentiable at a (a) A corner (b) A discontinuity (c) A vertical tangent x 5E-03(pp 136-145) 142 ❙❙❙❙ 1/17/06 2:34 PM Page 142 CHAPTER 3 DERIVATIVES A graphing calculator or computer provides another way of looking at differentiability. If f is differentiable at a, then when we zoom in toward the point ͑a, f ͑a͒͒ the graph straightens out and appears more and more like a line. (See Figure 9. We saw a specific example of this in Figure 3 in Section 3.1.) But no matter how much we zoom in toward a point like the ones in Figures 7 and 8(a), we can’t eliminate the sharp point or corner (see Figure 10). y 0 y a x 0 x a FIGURE 9 |||| 3.2 FIGURE 10 ƒ is differentiable at a. ƒ is not differentiable at a. Exercises 3. (a) f Ј͑Ϫ3͒ 1–3 |||| Use the given graph to estimate the value of each derivative. Then sketch the graph of f Ј. 1. (a) f Ј͑1͒ (c) f Ј͑3͒ (b) f Ј͑Ϫ2͒ (b) f Ј͑2͒ (d) f Ј͑4͒ 0 (e) f Ј͑1͒ x 1 (f ) f Ј͑2͒ (g) f Ј͑3͒ ■ 1 ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ 4. Match the graph of each function in (a)–(d) with the graph of 1 x its derivative in I–IV. Give reasons for your choices. (a) (c) f Ј͑2͒ (e) f Ј͑4͒ 1 (d) f Ј͑0͒ y=ƒ 2. (a) f Ј͑0͒ y=f(x) (c) f Ј͑Ϫ1͒ y 0 y y (b) f Ј͑1͒ (d) f Ј͑3͒ (f ) f Ј͑5͒ (b) 0 x y 0 x y y=f(x) (c) 0 1 0 y 1 x (d) x y 0 x 5E-03(pp 136-145) 1/17/06 2:34 PM Page 143 SECTION 3.2 THE DERIVATIVE AS A FUNCTION y I II 0 x 13. y 0 y IV 143 y x 0 III ❙❙❙❙ y ■ ■ ■ x ■ ■ ■ ■ ■ ■ ■ ■ ■ 14. Shown is the graph of the population function P͑t͒ for yeast 0 x 0 cells in a laboratory culture. Use the method of Example 1 to graph the derivative PЈ͑t͒. What does the graph of PЈ tell us about the yeast population? x P (yeast cells) 5–13 Trace or copy the graph of the given function f . (Assume that the axes have equal scales.) Then use the method of Example 1 to sketch the graph of f Ј below it. |||| 5. 6. y 0 500 y 0 x 0 5 10 15 t (hours) x 15. The graph shows how the average age of first marriage of 7. 8. y Japanese men varied in the last half of the 20th century. Sketch the graph of the derivative function MЈ͑t͒. During which years was the derivative negative? y M 0 0 x x 27 25 9. 10. y 1960 y 1970 1980 1990 t 16. Make a careful sketch of the graph of the sine function and 0 x 0 x below it sketch the graph of its derivative in the same manner as in Exercises 5–13. Can you guess what the derivative of the sine function is from its graph? 2 ; 17. Let f ͑x͒ ෇ x . 11. 12. y 0 x y 0 x (a) Estimate the values of f Ј͑0͒, f Ј( 1 ), f Ј͑1͒, and f Ј͑2͒ by 2 using a graphing device to zoom in on the graph of f. (b) Use symmetry to deduce the values of f Ј(Ϫ 1 ), f Ј͑Ϫ1͒, 2 and f Ј͑Ϫ2͒. (c) Use the results from parts (a) and (b) to guess a formula for f Ј͑x͒. (d) Use the definition of a derivative to prove that your guess in part (c) is correct. 5E-03(pp 136-145) 144 ❙❙❙❙ 1/17/06 2:34 PM Page 144 CHAPTER 3 DERIVATIVES 34. Let P͑t͒ be the percentage of Americans under the age of 18 at 3 ; 18. Let f ͑x͒ ෇ x . (a) Estimate the values of f Ј͑0͒, f Ј( 1 ), f Ј͑1͒, f Ј͑2͒, and f Ј͑3͒ 2 by using a graphing device to zoom in on the graph of f. (b) Use symmetry to deduce the values of f Ј(Ϫ 1 ), f Ј͑Ϫ1͒, 2 f Ј͑Ϫ2͒, and f Ј͑Ϫ3͒. (c) Use the values from parts (a) and (b) to graph f Ј. (d) Guess a formula for f Ј͑x͒. (e) Use the definition of a derivative to prove that your guess in part (d) is correct. 19–29 |||| Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative. 19. f ͑x͒ ෇ 37 20. f ͑x͒ ෇ 12 ϩ 7x 21. f ͑x͒ ෇ 1 Ϫ 3x 22. f ͑x͒ ෇ 5x 2 ϩ 3x Ϫ 2 2 23. f ͑x͒ ෇ x 3 Ϫ 3x ϩ 5 24. f ͑x͒ ෇ x ϩ sx 25. t͑x͒ ෇ s1 ϩ 2x 26. f ͑x͒ ෇ 3ϩx 1 Ϫ 3x 28. t͑x͒ ෇ 1 x2 ■ ■ 27. G͑t͒ ෇ 4t tϩ1 29. f ͑x͒ ෇ x ■ ■ ■ time t. The table gives values of this function in census years from 1950 to 2000. t t P͑t͒ 1950 1960 1970 (a) (b) (c) (d) P͑t͒ 31.1 35.7 34.0 1980 1990 2000 28.0 25.7 25.7 What is the meaning of PЈ͑t͒? What are its units? Construct a table of values for PЈ͑t͒. Graph P and PЈ. How would it be possible to get more accurate values for PЈ͑t͒? 35. The graph of f is given. State, with reasons, the numbers at which f is not differentiable. y 4 ■ ■ ■ ■ ■ ■ ■ 30. (a) Sketch the graph of f ͑x͒ ෇ s6 Ϫ x by starting with the ; graph of y ෇ sx and using the transformations of Section 1.3. (b) Use the graph from part (a) to sketch the graph of f Ј. (c) Use the definition of a derivative to find f Ј͑x͒. What are the domains of f and f Ј? (d) Use a graphing device to graph f Ј and compare with your sketch in part (b). 31. (a) If f ͑x͒ ෇ x Ϫ ͑2͞x͒, find f Ј͑x͒. ; (b) Check to see that your answer to part (a) is reasonable by comparing the graphs of f and f Ј. 2 4 6 8 12 x 10 36. The graph of t is given. (a) At what numbers is t discontinuous? Why? (b) At what numbers is t not differentiable? Why? y 32. (a) If f ͑t͒ ෇ 6͑͞1 ϩ t 2 ͒, find f Ј͑t͒. ; (b) Check to see that your answer to part (a) is reasonable by comparing the graphs of f and f Ј. 33. The unemployment rate U͑t͒ varies with time. The table (from the Bureau of Labor Statistics) gives the percentage of unemployed in the U.S. labor force from 1991 to 2000. t U͑t͒ t 6.8 7.5 6.9 6.1 5.6 1996 1997 1998 1999 2000 5.4 4.9 4.5 4.2 4.0 x U͑t͒ 1991 1992 1993 1994 1995 0 1 (a) What is the meaning of UЈ͑t͒? What are its units? (b) Construct a table of values for UЈ͑t͒. ; 37. Graph the function f ͑x͒ ෇ x ϩ sԽ x Խ . Zoom in repeatedly, first toward the point (Ϫ1, 0) and then toward the origin. What is different about the behavior of f in the vicinity of these two points? What do you conclude about the differentiability of f ? ; 38. Zoom in toward the points (1, 0), (0, 1), and (Ϫ1, 0) on the graph of the function t͑x͒ ෇ ͑x 2 Ϫ 1͒2͞3. What do you notice? Account for what you see in terms of the differentiability of t. 5E-03(pp 136-145) 1/17/06 2:34 PM Page 145 SECTION 3.3 DIFFERENTIATION FORMULAS (a) If a 0, use Equation 3.1.3 to find f Ј͑a͒. (b) Show that f Ј͑0͒ does not exist. 3 (c) Show that y ෇ sx has a vertical tangent line at ͑0, 0͒. (Recall the shape of the graph of f . See Figure 13 in Section 1.2.) 0 5Ϫx f ͑x͒ ෇ 1 5Ϫx 40. (a) If t͑x͒ ෇ x 2͞3, show that tЈ͑0͒ does not exist. (b) If a 0, find tЈ͑a͒. (c) Show that y ෇ x 2͞3 has a vertical tangent line at ͑0, 0͒. (d) Illustrate part (c) by graphing y ෇ x 2͞3. Խ Խ 41. Show that the function f ͑x͒ ෇ x Ϫ 6 is not differentiable at 6. Find a formula for f Ј and sketch its graph. Խ Խ in its domain and odd if f ͑Ϫx͒ ෇ Ϫf ͑x͒ for all such x. Prove each of the following. (a) The derivative of an even function is an odd function. (b) The derivative of an odd function is an even function. 46. When you turn on a hot-water faucet, the temperature T of the water depends on how long the water has been running. (a) Sketch a possible graph of T as a function of the time t that has elapsed since the faucet was turned on. (b) Describe how the rate of change of T with respect to t varies as t increases. (c) Sketch a graph of the derivative of T . 44. The left-hand and right-hand derivatives of f at a are defined by f ͑a ϩ h͒ Ϫ f ͑a͒ h f Ј ͑a͒ ෇ limϪ Ϫ hl0 |||| 47. Let ᐍ be the tangent line to the parabola y ෇ x 2 at the point ͑1, 1͒. The angle of inclination of ᐍ is the angle ␾ that ᐍ makes with the positive direction of the x-axis. Calculate ␾ correct to the nearest degree. f ͑a ϩ h͒ Ϫ f ͑a͒ f Ј ͑a͒ ෇ limϩ ϩ hl0 h and if x ജ 4 45. Recall that a function f is called even if f ͑Ϫx͒ ෇ f ͑x͒ for all x tiable? Find a formula for f Ј and sketch its graph. (b) For what values of x is f differentiable? (c) Find a formula for f Ј. if x ഛ 0 if 0 Ͻ x Ͻ 4 (b) Sketch the graph of f . (c) Where is f discontinuous? (d) Where is f not differentiable? 42. Where is the greatest integer function f ͑x͒ ෇ ͠ x͡ not differen43. (a) Sketch the graph of the function f ͑x͒ ෇ x x . 145 if these limits exist. Then f Ј͑a͒ exists if and only if these onesided derivatives exist and are equal. (a) Find f Ј ͑4͒ and f Ј ͑4͒ for the function Ϫ ϩ 3 39. Let f ͑x͒ ෇ sx. ; ❙❙❙❙ 3.3 Differentiation Formulas If it were always necessary to compute derivatives directly from the definition, as we did in the preceding section, such computations would be tedious and the evaluation of some limits would require ingenuity. Fortunately, several rules have been developed for finding derivatives without having to use the definition directly. These formulas greatly simplify the task of differentiation. Let’s start with the simplest of all functions, the constant function f ͑x͒ ෇ c. The graph of this function is the horizontal line y ෇ c, which has slope 0, so we must have f Ј͑x͒ ෇ 0. (See Figure 1.) A formal proof, from the definition of a derivative, is also easy: y c y=c f Ј͑x͒ ෇ lim hl0 slope=0 f ͑x ϩ h͒ Ϫ f ͑x͒ cϪc ෇ lim hl0 h h ෇ lim 0 ෇ 0 hl0 0 FIGURE 1 The graph of ƒ=c is the line y=c, so f ª(x)=0. x In Leibniz notation, we write this rule as follows. Derivative of a Constant Function d ͑c͒ ෇ 0 dx 5E-03(pp 146-155) 146 ❙❙❙❙ 1/17/06 2:14 PM Page 146 CHAPTER 3 DERIVATIVES Power Functions y We next look at the functions f ͑x͒ ෇ x n, where n is a positive integer. If n ෇ 1, the graph of f ͑x͒ ෇ x is the line y ෇ x, which has slope 1 (see Figure 2). So y=x slope=1 d ͑x͒ ෇ 1 dx 1 0 x FIGURE 2 The graph of ƒ=x is the line y=x, so f ª(x)=1. (You can also verify Equation 1 from the definition of a derivative.) We have already investigated the cases n ෇ 2 and n ෇ 3. In fact, in Section 3.2 (Exercises 17 and 18) we found that d d ͑x 2 ͒ ෇ 2x ͑x 3 ͒ ෇ 3x 2 2 dx dx For n ෇ 4 we find the derivative of f ͑x͒ ෇ x 4 as follows: f Ј͑x͒ ෇ lim f ͑x ϩ h͒ Ϫ f ͑x͒ ͑x ϩ h͒4 Ϫ x 4 ෇ lim hl0 h h ෇ lim x 4 ϩ 4x 3h ϩ 6x 2h 2 ϩ 4xh 3 ϩ h 4 Ϫ x 4 h ෇ lim 4x 3h ϩ 6x 2h 2 ϩ 4xh 3 ϩ h 4 h hl0 hl0 hl0 ෇ lim ͑4x 3 ϩ 6x 2h ϩ 4xh 2 ϩ h 3 ͒ ෇ 4x 3 hl0 Thus d ͑x 4 ͒ ෇ 4x 3 dx 3 Comparing the equations in (1), (2), and (3), we see a pattern emerging. It seems to be a reasonable guess that, when n is a positive integer, ͑d͞dx͒͑x n ͒ ෇ nx nϪ1. This turns out to be true. We prove it in two ways; the second proof uses the Binomial Theorem. The Power Rule If n is a positive integer, then d ͑x n ͒ ෇ nx nϪ1 dx First Proof The formula x n Ϫ a n ෇ ͑x Ϫ a͒͑x nϪ1 ϩ x nϪ2a ϩ и и и ϩ xa nϪ2 ϩ a nϪ1 ͒ can be verified simply by multiplying out the right-hand side (or by summing the second factor as a geometric series). If f ͑x͒ ෇ x n, we can use Equation 3.1.3 for f Ј͑a͒ and the equation above to write f Ј͑a͒ ෇ lim xla f ͑x͒ Ϫ f ͑a͒ xn Ϫ an ෇ lim xla xϪa xϪa 5E-03(pp 146-155) 1/17/06 2:15 PM Page 147 SECTION 3.3 DIFFERENTIATION FORMULAS ❙❙❙❙ 147 ෇ lim ͑x nϪ1 ϩ x nϪ2a ϩ и и и ϩ xa nϪ2 ϩ a nϪ1 ͒ xla ෇ a nϪ1 ϩ a nϪ2a ϩ и и и ϩ aa nϪ2 ϩ a nϪ1 ෇ na nϪ1 Second Proof f Ј͑x͒ ෇ lim hl0 |||| The Binomial Theorem is given on Reference Page 1. f ͑x ϩ h͒ Ϫ f ͑x͒ ͑x ϩ h͒n Ϫ x n ෇ lim hl0 h h In finding the derivative of x 4 we had to expand ͑x ϩ h͒4. Here we need to expand ͑x ϩ h͒n and we use the Binomial Theorem to do so: ͫ x n ϩ nx nϪ1h ϩ f Ј͑x͒ ෇ lim hl0 nx nϪ1h ϩ ෇ lim hl0 ͫ n͑n Ϫ 1͒ nϪ2 2 x h ϩ и и и ϩ nxh nϪ1 ϩ h n 2 h ෇ lim nx nϪ1 ϩ hl0 ͬ n͑n Ϫ 1͒ nϪ2 2 x h ϩ и и и ϩ nxh nϪ1 ϩ h n Ϫ x n 2 h n͑n Ϫ 1͒ nϪ2 x h ϩ и и и ϩ nxh nϪ2 ϩ h nϪ1 2 ͬ ෇ nx nϪ1 because every term except the first has h as a factor and therefore approaches 0. We illustrate the Power Rule using various notations in Example 1. EXAMPLE 1 (a) If f ͑x͒ ෇ x 6, then f Ј͑x͒ ෇ 6x 5. (c) If y ෇ t 4, then dy ෇ 4t 3. dt (b) If y ෇ x 1000, then yЈ ෇ 1000x 999. (d) d 3 ͑r ͒ ෇ 3r 2 dr (e) Du͑u m ͒ ෇ mu mϪ1 New Derivatives from Old When new functions are formed from old functions by addition, subtraction, multiplication, or division, their derivatives can be calculated in terms of derivatives of the old functions. In particular, the following formula says that the derivative of a constant times a function is the constant times the derivative of the function. The Constant Multiple Rule If c is a constant and f is a differentiable function, then d d ͓cf ͑x͔͒ ෇ c f ͑x͒ dx dx 5E-03(pp 146-155) 148 ❙❙❙❙ 1/17/06 2:15 PM Page 148 CHAPTER 3 DERIVATIVES Proof Let t͑x͒ ෇ cf ͑x͒. Then |||| GEOMETRIC INTERPRETATION OF THE CONSTANT MULTIPLE RULE tЈ͑x͒ ෇ lim y hl0 t͑x ϩ h͒ Ϫ t͑x͒ cf ͑x ϩ h͒ Ϫ cf ͑x͒ ෇ lim hl0 h h y=2ƒ ͫ ෇ lim c hl0 y=ƒ 0 ෇ c lim x Multiplying by c ෇ 2 stretches the graph vertically by a factor of 2. All the rises have been doubled but the runs stay the same. So the slopes are doubled, too. hl0 f ͑x ϩ h͒ Ϫ f ͑x͒ h f ͑x ϩ h͒ Ϫ f ͑x͒ h ͬ (by Law 3 of limits) ෇ cf Ј͑x͒ EXAMPLE 2 (a) d d ͑3x 4 ͒ ෇ 3 ͑x 4 ͒ ෇ 3͑4x 3 ͒ ෇ 12x 3 dx dx (b) d d d ͑Ϫx͒ ෇ ͓͑Ϫ1͒x͔ ෇ ͑Ϫ1͒ ͑x͒ ෇ Ϫ1͑1͒ ෇ Ϫ1 dx dx dx The next rule tells us that the derivative of a sum of functions is the sum of the derivatives. The Sum Rule If f and t are both differentiable, then |||| Using prime notation, we can write the Sum Rule as ͑ f ϩ t͒Ј ෇ f Ј ϩ tЈ d d d ͓ f ͑x͒ ϩ t͑x͔͒ ෇ f ͑x͒ ϩ t͑x͒ dx dx dx Proof Let F͑x͒ ෇ f ͑x͒ ϩ t͑x͒. Then FЈ͑x͒ ෇ lim hl0 ෇ lim hl0 ෇ lim hl0 ෇ lim hl0 F͑x ϩ h͒ Ϫ F͑x͒ h ͓ f ͑x ϩ h͒ ϩ t͑x ϩ h͔͒ Ϫ ͓ f ͑x͒ ϩ t͑x͔ h ͫ f ͑x ϩ h͒ Ϫ f ͑x͒ t͑x ϩ h͒ Ϫ t͑x͒ ϩ h h ͬ f ͑x ϩ h͒ Ϫ f ͑x͒ t͑x ϩ h͒ Ϫ t͑x͒ ϩ lim hl0 h h (by Law 1) ෇ f Ј͑x͒ ϩ tЈ͑x͒ The Sum Rule can be extended to the sum of any number of functions. For instance, using this theorem twice, we get ͑ f ϩ t ϩ h͒Ј ෇ ͓͑ f ϩ t͒ ϩ h͔Ј ෇ ͑ f ϩ t͒Ј ϩ hЈ ෇ f Ј ϩ tЈ ϩ hЈ By writing f Ϫ t as f ϩ ͑Ϫ1͒t and applying the Sum Rule and the Constant Multiple Rule, we get the following formula. 5E-03(pp 146-155) 1/17/06 2:16 PM Page 149 SECTION 3.3 DIFFERENTIATION FORMULAS ❙❙❙❙ 149 The Difference Rule If f and t are both differentiable, then d d d ͓ f ͑x͒ Ϫ t͑x͔͒ ෇ f ͑x͒ Ϫ t͑x͒ dx dx dx The Constant Multiple Rule, the Sum Rule, and the Difference Rule can be combined with the Power Rule to differentiate any polynomial, as the following examples demonstrate. EXAMPLE 3 Try more problems like this one. Resources / Module 4 / Polynomial Models / Basic Differentiation Rules and Quiz d ͑x 8 ϩ 12x 5 Ϫ 4x 4 ϩ 10x 3 Ϫ 6x ϩ 5͒ dx d d d d d d ෇ ͑x 8 ͒ ϩ 12 ͑x 5 ͒ Ϫ 4 ͑x 4 ͒ ϩ 10 ͑x 3 ͒ Ϫ 6 ͑x͒ ϩ ͑5͒ dx dx dx dx dx dx ෇ 8x 7 ϩ 12͑5x 4 ͒ Ϫ 4͑4x 3 ͒ ϩ 10͑3x 2 ͒ Ϫ 6͑1͒ ϩ 0 ෇ 8x 7 ϩ 60x 4 Ϫ 16x 3 ϩ 30x 2 Ϫ 6 EXAMPLE 4 Find the points on the curve y ෇ x 4 Ϫ 6x 2 ϩ 4 where the tangent line is horizontal. SOLUTION Horizontal tangents occur where the derivative is zero. We have dy d d d ෇ ͑x 4 ͒ Ϫ 6 ͑x 2 ͒ ϩ ͑4͒ dx dx dx dx ෇ 4x 3 Ϫ 12x ϩ 0 ෇ 4x͑x 2 Ϫ 3͒ Thus, dy͞dx ෇ 0 if x ෇ 0 or x 2 Ϫ 3 ෇ 0, that is, x ෇ Ϯs3. So the given curve has horizontal tangents when x ෇ 0, s3, and Ϫs3. The corresponding points are ͑0, 4͒, (s3, Ϫ5), and (Ϫs3, Ϫ5). (See Figure 3.) y (0, 4) 0 x FIGURE 3 The curve y=x$-6x@+4 and its horizontal tangents Resources / Module 4 / Polynomial Models / Product and Quotient Rules {_ œ„, _5} 3 {œ„, _5} 3 Next we need a formula for the derivative of a product of two functions. By analogy with the Sum and Difference Rules, one might be tempted to guess, as Leibniz did three centuries ago, that the derivative of a product is the product of the derivatives. We can see, however, that this guess is wrong by looking at a particular example. Let f ͑x͒ ෇ x and t͑x͒ ෇ x 2. Then the Power Rule gives f Ј͑x͒ ෇ 1 and tЈ͑x͒ ෇ 2x. But ͑ ft͒͑x͒ ෇ x 3, so | ͑ ft͒Ј͑x͒ ෇ 3x 2. Thus, ͑ ft͒Ј f ЈtЈ. The correct formula was discovered by Leibniz (soon after his false start) and is called the Product Rule. 5E-03(pp 146-155) 150 ❙❙❙❙ 1/17/06 2:17 PM Page 150 CHAPTER 3 DERIVATIVES |||| We can write the Product Rule in prime notation as ͑ ft͒Ј ෇ ftЈ ϩ t f Ј The Product Rule If f and t are both differentiable, then d d d ͓ f ͑x͒t͑x͔͒ ෇ f ͑x͒ ͓t͑x͔͒ ϩ t͑x͒ ͓ f ͑x͔͒ dx dx dx Proof Let F͑x͒ ෇ f ͑x͒t͑x͒. Then FЈ͑x͒ ෇ lim hl0 ෇ lim hl0 F͑x ϩ h͒ Ϫ F͑x͒ h f ͑x ϩ h͒t͑x ϩ h͒ Ϫ f ͑x͒t͑x͒ h In order to evaluate this limit, we would like to separate the functions f and t as in the proof of the Sum Rule. We can achieve this separation by subtracting and adding the term f ͑x ϩ h͒t͑x͒ in the numerator: FЈ͑x͒ ෇ lim hl0 f ͑x ϩ h͒t͑x ϩ h͒ Ϫ f ͑x ϩ h͒t͑x͒ ϩ f ͑x ϩ h͒t͑x͒ Ϫ f ͑x͒t͑x͒ h ͫ ෇ lim f ͑x ϩ h͒ hl0 t͑x ϩ h͒ Ϫ t͑x͒ f ͑x ϩ h͒ Ϫ f ͑x͒ ϩ t͑x͒ h h ෇ lim f ͑x ϩ h͒ ؒ lim hl0 hl0 ͬ t͑x ϩ h͒ Ϫ t͑x͒ f ͑x ϩ h͒ Ϫ f ͑x͒ ϩ lim t͑x͒ ؒ lim hl0 hl0 h h ෇ f ͑x͒tЈ͑x͒ ϩ t͑x͒f Ј͑x͒ Note that lim h l 0 t͑x͒ ෇ t͑x͒ because t͑x͒ is a constant with respect to the variable h. Also, since f is differentiable at x, it is continuous at x by Theorem 3.2.4, and so lim h l 0 f ͑x ϩ h͒ ෇ f ͑x͒. (See Exercise 53 in Section 2.5.) In words, the Product Rule says that the derivative of a product of two functions is the first function times the derivative of the second function plus the second function times the derivative of the first function. EXAMPLE 5 Find FЈ͑x͒ if F͑x͒ ෇ ͑6x 3 ͒͑7x 4 ͒. SOLUTION By the Product Rule, we have FЈ͑x͒ ෇ ͑6x 3 ͒ d d ͑7x 4 ͒ ϩ ͑7x 4 ͒ ͑6x 3 ͒ dx dx ෇ ͑6x 3 ͒͑28x 3 ͒ ϩ ͑7x 4 ͒͑18x 2 ͒ ෇ 168x 6 ϩ 126x 6 ෇ 294x 6 Notice that we could verify the answer to Example 5 directly by first multiplying the factors: F͑x͒ ෇ ͑6x 3 ͒͑7x 4 ͒ ෇ 42x 7 ? FЈ͑x͒ ෇ 42͑7x 6 ͒ ෇ 294x 6 5E-03(pp 146-155) 1/17/06 2:18 PM Page 151 SECTION 3.3 DIFFERENTIATION FORMULAS ❙❙❙❙ 151 But later we will meet functions, such as y ෇ x 2 sin x, for which the Product Rule is the only possible method. EXAMPLE 6 If h͑x͒ ෇ xt͑x͒ and it is known that t͑3͒ ෇ 5 and tЈ͑3͒ ෇ 2, find hЈ͑3͒. SOLUTION Applying the Product Rule, we get hЈ͑x͒ ෇ d d d ͓xt͑x͔͒ ෇ x ͓t͑x͔͒ ϩ t͑x͒ ͓x͔ dx dx dx ෇ xtЈ͑x͒ ϩ t͑x͒ hЈ͑3͒ ෇ 3tЈ͑3͒ ϩ t͑3͒ ෇ 3 ؒ 2 ϩ 5 ෇ 11 Therefore The Quotient Rule If f and t are differentiable, then |||| In prime notation we can write the Quotient Rule as t f Ј Ϫ ftЈ f Ј ෇ t2 t ͩͪ d dx ͫ ͬ f ͑x͒ t͑x͒ t͑x͒ ෇ d d ͓ f ͑x͔͒ Ϫ f ͑x͒ ͓t͑x͔͒ dx dx ͓t͑x͔͒ 2 Proof Let F͑x͒ ෇ f ͑x͒͞t͑x͒. Then f ͑x ϩ h͒ f ͑x͒ Ϫ F͑x ϩ h͒ Ϫ F͑x͒ t͑x ϩ h͒ t͑x͒ FЈ͑x͒ ෇ lim ෇ lim hl0 hl0 h h ෇ lim hl0 f ͑x ϩ h͒t͑x͒ Ϫ f ͑x͒t͑x ϩ h͒ ht͑x ϩ h͒t͑x͒ We can separate f and t in this expression by subtracting and adding the term f ͑x͒t͑x͒ in the numerator: FЈ͑x͒ ෇ lim hl0 f ͑x ϩ h͒t͑x͒ Ϫ f ͑x͒t͑x͒ ϩ f ͑x͒t͑x͒ Ϫ f ͑x͒t͑x ϩ h͒ ht͑x ϩ h͒t͑x͒ t͑x͒ ෇ lim hl0 f ͑x ϩ h͒ Ϫ f ͑x͒ t͑x ϩ h͒ Ϫ t͑x͒ Ϫ f ͑x͒ h h t͑x ϩ h͒t͑x͒ lim t͑x͒ ؒ lim ෇ hl0 hl0 f ͑x ϩ h͒ Ϫ f ͑x͒ t͑x ϩ h͒ Ϫ t͑x͒ Ϫ lim f ͑x͒ ؒ lim hl0 hl0 h h lim t͑x ϩ h͒ ؒ lim t͑x͒ hl0 ෇ hl0 t͑x͒f Ј͑x͒ Ϫ f ͑x͒tЈ͑x͒ ͓t͑x͔͒ 2 Again t is continuous by Theorem 3.2.4, so lim h l 0 t͑x ϩ h͒ ෇ t͑x͒. In words, the Quotient Rule says that the derivative of a quotient is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the square of the denominator. 5E-03(pp 146-155) 152 ❙❙❙❙ 1/17/06 2:19 PM Page 152 CHAPTER 3 DERIVATIVES The theorems of this section show that any polynomial is differentiable on ‫ ޒ‬and any rational function is differentiable on its domain. Furthermore, the Quotient Rule and the other differentiation formulas enable us to compute the derivative of any rational function, as the next example illustrates. |||| We can use a graphing device to check that the answer to Example 7 is plausible. Figure 4 shows the graphs of the function of Example 7 and its derivative. Notice that when y grows rapidly (near Ϫ2), yЈ is large. And when y grows slowly, yЈ is near 0. EXAMPLE 7 Let y ෇ x2 ϩ x Ϫ 2 . x3 ϩ 6 Then ͑x 3 ϩ 6͒ yЈ ෇ 1.5 yª d d ͑x 2 ϩ x Ϫ 2͒ Ϫ ͑x 2 ϩ x Ϫ 2͒ ͑x 3 ϩ 6͒ dx dx ͑x 3 ϩ 6͒2 ෇ ෇ _4 ͑x 3 ϩ 6͒͑2x ϩ 1͒ Ϫ ͑x 2 ϩ x Ϫ 2͒͑3x 2 ͒ ͑x 3 ϩ 6͒2 ͑2x 4 ϩ x 3 ϩ 12x ϩ 6͒ Ϫ ͑3x 4 ϩ 3x 3 Ϫ 6x 2 ͒ ͑x 3 ϩ 6͒2 ෇ Ϫx 4 Ϫ 2x 3 ϩ 6x 2 ϩ 12x ϩ 6 ͑x 3 ϩ 6͒2 4 y _1.5 FIGURE 4 General Power Functions The Quotient Rule can also be used to extend the Power Rule to the case where the exponent is a negative integer. If n is a positive integer, then d Ϫn ͑x ͒ ෇ Ϫnx ϪnϪ1 dx d d ͑x Ϫn ͒ ෇ dx dx Proof xn ෇ ෇ ͩͪ 1 xn d d ͑1͒ Ϫ 1 ؒ ͑x n ͒ dx dx x n ؒ 0 Ϫ 1 ؒ nx nϪ1 ෇ n 2 ͑x ͒ x 2n Ϫnx nϪ1 ෇ Ϫnx nϪ1Ϫ2n ෇ Ϫnx ϪnϪ1 x 2n EXAMPLE 8 (a) If y ෇ (b) d dt 1 dy d 1 , then ෇ ͑x Ϫ1 ͒ ෇ Ϫx Ϫ2 ෇ Ϫ 2 x dx dx x ͩͪ 6 t3 ෇6 d Ϫ3 18 ͑t ͒ ෇ 6͑Ϫ3͒t Ϫ4 ෇ Ϫ 4 dt t 5E-03(pp 146-155) 1/17/06 2:19 PM Page 153 SECTION 3.3 DIFFERENTIATION FORMULAS ❙❙❙❙ 153 5E-03(pp 146-155) ❙❙❙❙ 154 1/17/06 2:20 PM Page 154 CHAPTER 3 DERIVATIVES So far we know that the Power Rule holds if the exponent n is a positive or negative integer. If n ෇ 0, then x 0 ෇ 1, which we know has a derivative of 0. Thus, the Power Rule holds for any integer n. What if the exponent is a fraction? In Example 3 in Section 2.6 we found, in effect, that 1 1 ”1, 2 ’ y= x œ„ 1+≈ d 1 sx ෇ dx 2sx 4 0 which can be written as FIGURE 5 d 1͞2 ͑x ͒ ෇ 1 xϪ1͞2 2 dx This shows that the Power Rule is true even when n ෇ 1 . In fact, it also holds for any real 2 number n, as we will prove in Chapter 7. (A proof for rational values of n is indicated in Exercise 40 in Section 3.7.) In the meantime we state the general version and use it in the examples and exercises. y (2, 6) xy=12 0 The Power Rule (General Version) If n is any real number, then x d ͑x n ͒ ෇ nx nϪ1 dx (_2, _6) 3x+y=0 FIGURE 6 EXAMPLE 9 (a) If f ͑x͒ ෇ x ␲, then f Ј͑x͒ ෇ ␲ x ␲ Ϫ1. y෇ (b) Let Then 1 sx 2 3 dy d Ϫ2͞3 ෇ ͑x ͒ ෇ Ϫ2 xϪ͑2͞3͒Ϫ1 3 dx dx ෇ Ϫ2 xϪ5͞3 3 EXAMPLE 10 Find an equation of the tangent line to the curve y ෇ sx͑͞1 ϩ x 2 ͒ at the point (1, 1 ). 2 |||| 3.3 1–20 |||| Exercises Differentiate the function. 1. f ͑x͒ ෇ 186.5 2. f ͑x͒ ෇ s30 3. f ͑x͒ ෇ 5x Ϫ 1 4. F͑x͒ ෇ Ϫ4x 10 5. f ͑x͒ ෇ x 2 ϩ 3x Ϫ 4 6. t͑x͒ ෇ 5x 8 Ϫ 2x 5 ϩ 6 7. f ͑t͒ ෇ 4 ͑t 4 ϩ 8͒ 8. f ͑t͒ ෇ 2 t 6 Ϫ 3t 4 ϩ t 1 9. V͑r͒ ෇ 3 ␲ r 3 4 1 10. R͑t͒ ෇ 5tϪ3͞5 11. Y͑t͒ ෇ 6t Ϫ9 13. F ͑x͒ ෇ ( 1 x) 5 2 12. R͑x͒ ෇ s10 x7 14. f ͑t͒ ෇ st Ϫ 1 st 15. y ෇ x Ϫ2͞5 3 16. y ෇ sx 17. y ෇ 4␲ 2 18. t͑u͒ ෇ s2u ϩ s3u 5E-03(pp 146-155) 1/17/06 2:21 PM Page 155 SECTION 3.3 DIFFERENTIATION FORMULAS 19. v ෇ t 2 Ϫ ■ ■ 1 4 st 3 ■ ■ ■ ■ ■ ■ ■ |||| Estimate the value of f Ј͑a͒ by zooming in on the graph of f . Then differentiate f to find the exact value of f Ј͑a͒ and compare with your estimate. ■ ■ 21. Find the derivative of y ෇ ͑x 2 ϩ 1͒͑x 3 ϩ 1͒ in two ways: by 47. f ͑x͒ ෇ 3x 2 Ϫ x 3, using the Product Rule and by performing the multiplication first. Do your answers agree? ■ |||| x Ϫ 3x sx sx Differentiate. ͪ 1 3 Ϫ 4 ͑ y ϩ 5y 3͒ y2 y 26. y ෇ sx ͑x Ϫ 1͒ 27. t͑x͒ ෇ 29. y ෇ 31. y ෇ 33. y ෇ 3x Ϫ 1 2x ϩ 1 28. f ͑t͒ ෇ t2 3t Ϫ 2t ϩ 1 30. y ෇ 2 v 3 Ϫ 2v sv 1 x4 ϩ x2 ϩ 1 36. y ෇ A ϩ ■ ■ ■ ■ ; u Ϫ 2u ϩ 5 u2 ■ ■ ; ax ϩ b cx ϩ d P͑x͒ ෇ a n x ϩ a nϪ1 x where a n nϪ1 ■ ■ ■ ■ 44. f ͑x͒ ෇ x͑͞x 2 Ϫ 1͒ ■ ■ ■ ■ 52. y ෇ ͑1, 2͒ Ϫ 5x ϩ 3 ■ ■ ■ ■ ■ ■ ■ ■ sx , xϩ1 ■ ͑4, 0.4͒ 54. y ෇ ͑1 ϩ 2x͒2, ■ ■ ■ ■ ■ ͑1, 9͒ ■ ■ Agnesi. Find an equation of the tangent line to this curve at the point (Ϫ1, 1 ). 2 (b) Illustrate part (a) by graphing the curve and the tangent line on the same screen. equation of the tangent line to this curve at the point ͑3, 0.3͒. (b) Illustrate part (a) by graphing the curve and the tangent line on the same screen. Find the following values. (a) ͑ ft͒Ј͑5͒ (b) ͑ f͞t͒Ј͑5͒ (c) ͑ t͞f ͒Ј͑5͒ (c) (b) ͑ ft͒Ј͑3͒ ͩͪ f Ј ͑3͒ t (d) ■ ■ ■ ͩ ͪ Ј f ͑3͒ fϪt 59. If f ͑x͒ ෇ sx t͑x͒, where t͑4͒ ෇ 8 and tЈ͑4͒ ෇ 7, find f Ј͑4͒. 60. If h͑2͒ ෇ 4 and hЈ͑2͒ ෇ Ϫ3, find 1 46. f ͑x͒ ෇ x ϩ x 3 ■ ͑1, 1͒ lowing numbers. (a) ͑ f ϩ t͒Ј͑3͒ ϩ и и и ϩ a2 x ϩ a1 x ϩ a0 |||| 45. f ͑x͒ ෇ 3x ■ 0. Find the derivative of P. 15 a෇4 ■ 58. If f ͑3͒ ෇ 4, t͑3͒ ෇ 2, f Ј͑3͒ ෇ Ϫ6, and tЈ͑3͒ ෇ 5, find the fol- 2 Find f Ј͑x͒. Compare the graphs of f and f Ј and use them to explain why your answer is reasonable. ; 44–46 ■ 57. Suppose that f ͑5͒ ෇ 1, f Ј͑5͒ ෇ 6, t͑5͒ ෇ Ϫ3, and tЈ͑5͒ ෇ 2. 43. The general polynomial of degree n has the form n ■ 56. (a) The curve y ෇ x͑͞1 ϩ x 2 ͒ is called a serpentine. Find an 3 42. f ͑x͒ ෇ ■ ■ 55. (a) The curve y ෇ 1͑͞1 ϩ x 2 ͒ is called a witch of Maria cx 1 ϩ cx 40. y ෇ c xϩ x 2x , xϩ1 53. y ෇ x ϩ sx, C B ϩ 2 x x 6 x ■ 51. y ෇ 38. y ෇ 3 39. y ෇ st ͑t 2 ϩ t ϩ t Ϫ1 ͒ 41. f ͑x͒ ෇ t3 ϩ t t4 Ϫ 2 34. y ෇ x 2 ϩ x ϩ xϪ1 ϩ xϪ2 r2 1 ϩ sr 48. f ͑x͒ ෇ 1͞sx, ■ 51–54 |||| Find an equation of the tangent line to the curve at the given point. ■ 35. y ෇ ax 2 ϩ bx ϩ c 37. y ෇ 2t 4 ϩ t2 sx Ϫ 1 32. y ෇ sx ϩ 1 v ■ t͑x͒ ෇ x 2͑͞x 2 ϩ 1͒ in the viewing rectangle ͓Ϫ4, 4͔ by ͓Ϫ1, 1.5͔. (b) Using the graph in part (a) to estimate slopes, make a rough sketch, by hand, of the graph of tЈ. (See Example 1 in Section 3.2.) (c) Calculate tЈ͑x͒ and use this expression, with a graphing device, to graph tЈ. Compare with your sketch in part (b). 24. Y͑u͒ ෇ ͑uϪ2 ϩ uϪ3 ͒͑u 5 Ϫ 2u 2 ͒ ͩ a෇1 ■ ; 50. (a) Use a graphing calculator or computer to graph the function 23. V͑x͒ ෇ ͑2x 3 ϩ 3͒͑x 4 Ϫ 2x͒ 25. F͑ y͒ ෇ ■ tion f ͑x͒ ෇ x 4 Ϫ 3x 3 Ϫ 6x 2 ϩ 7x ϩ 30 in the viewing rectangle ͓Ϫ3, 5͔ by ͓Ϫ10, 50͔. (b) Using the graph in part (a) to estimate slopes, make a rough sketch, by hand, of the graph of f Ј. (See Example 1 in Section 3.2.) (c) Calculate f Ј͑x͒ and use this expression, with a graphing device, to graph f Ј. Compare with your sketch in part (b). in two ways: by using the Quotient Rule and by simplifying first. Show that your answers are equivalent. Which method do you prefer? 23–42 ■ ; 49. (a) Use a graphing calculator or computer to graph the func- 22. Find the derivative of the function F͑x͒ ෇ 155 ; 47–48 3 20. u ෇ st 2 ϩ 2 st 3 ■ ❙❙❙❙ d dx ͩ ͪͿ h͑x͒ x x෇2 5E-03(pp 156-165) 156 ❙❙❙❙ 1/17/06 2:09 PM Page 156 CHAPTER 3 DERIVATIVES 61. If f and t are the functions whose graphs are shown, let u͑x͒ ෇ f ͑x͒t͑x͒ and v͑x͒ ෇ f ͑x͒͞t͑x͒. (b) Find vЈ͑5͒. (a) Find uЈ͑1͒. 69. How many tangent lines to the curve y ෇ x͑͞x ϩ 1) pass through the point ͑1, 2͒? At which points do these tangent lines touch the curve? 70. Draw a diagram to show that there are two tangent lines to the y parabola y ෇ x 2 that pass through the point ͑0, Ϫ4͒. Find the coordinates of the points where these tangent lines intersect the parabola. f g 71. Show that the curve y ෇ 6x 3 ϩ 5x Ϫ 3 has no tangent line 1 with slope 4. 0 x 1 72. Find equations of both lines through the point ͑2, Ϫ3͒ that are tangent to the parabola y ෇ x 2 ϩ x. 62. Let P͑x͒ ෇ F͑x͒G͑x͒ and Q͑x͒ ෇ F͑x͒͞G͑x͒, where F and G are the functions whose graphs are shown. (a) Find PЈ͑2͒. (b) Find QЈ͑7͒. 73. (a) Use the Product Rule twice to prove that if f , t, and h are differentiable, then ͑ fth͒Ј ෇ f Ј ϩ ftЈh ϩ fthЈ th y (b) Use part (a) to differentiate y ෇ sx ͑x 4 ϩ x ϩ 1͒͑2x Ϫ 3͒. F 74. (a) Taking f ෇ t ෇ h in Exercise 73, show that 0 d ͓ f ͑x͔͒ 3 ෇ 3͓ f ͑x͔͒ 2 f Ј͑x͒ dx G 1 x 1 63. If t is a differentiable function, find an expression for the deriv- ative of each of the following functions. x (a) y ෇ xt͑x͒ (b) y ෇ t͑x͒ t͑x͒ (c) y ෇ x 64. If f is a differentiable function, find an expression for the derivative of each of the following functions. (b) y ෇ (c) y ෇ x2 f ͑x͒ f ͑x͒ x2 (d) y ෇ (a) y ෇ x 2 f ͑x͒ 1 ϩ x f ͑x͒ sx 65. The normal line to a curve C at a point P is, by definition, the line that passes through P and is perpendicular to the tangent line to C at P. Find an equation of the normal line to the parabola y ෇ 1 Ϫ x 2 at the point (2, Ϫ3). Sketch the parabola and its normal line. 66. Where does the normal line to the parabola y ෇ x Ϫ x 2 at the point (1, 0) intersect the parabola a second time? Illustrate with a sketch. 67. Find the points on the curve y ෇ x 3 Ϫ x 2 Ϫ x ϩ 1 where the tangent is horizontal. (b) Use part (a) to differentiate y ෇ ͑x 4 ϩ 3x 3 ϩ 17x ϩ 82͒3. 75. Find a cubic function y ෇ ax 3 ϩ bx 2 ϩ cx ϩ d whose graph has horizontal tangents at the points ͑Ϫ2, 6͒ and ͑2, 0͒. 76. A telephone company wants to estimate the number of new residential phone lines that it will need to install during the upcoming month. At the beginning of January the company had 100,000 subscribers, each of whom had 1.2 phone lines, on average. The company estimated that its subscribership was increasing at the rate of 1000 monthly. By polling its existing subscribers, the company found that each intended to install an average of 0.01 new phone lines by the end of January. (a) Let s͑t͒ be the number of subscribers and let n͑t͒ be the number of phone lines per subscriber at time t, where t is measured in years and t ෇ 0 corresponds to the beginning of January. What are the values of s͑0͒ and n͑0͒? What are the company’s estimates for sЈ͑0͒ and nЈ͑0͒? (b) Estimate the number of new lines the company will have to install in January by using the Product Rule to calculate the rate of increase of lines at the beginning of the month. 77. In this exercise we estimate the rate at which the total personal 68. Find equations of the tangent lines to the curve y෇ xϪ1 xϩ1 that are parallel to the line x Ϫ 2y ෇ 2. income is rising in the Richmond-Petersburg, Virginia, metropolitan area. In 1999, the population of this area was 961,400, and the population was increasing at roughly 9200 people per year. The average annual income was $30,593 per capita, and this average was increasing at about $1400 per year (a little above the national average of about $1225 yearly). Use the 5E-03(pp 156-165) 1/17/06 2:10 PM Page 157 SECTION 3.4 RATES OF CHANGE IN THE NATURAL AND SOCIAL SCIENCES Product Rule and these figures to estimate the rate at which total personal income was rising in the Richmond-Petersburg area in 1999. Explain the meaning of each term in the Product Rule. 78. A manufacturer produces bolts of a fabric with a fixed width. The quantity q of this fabric (measured in yards) that is sold is a function of the selling price p (in dollars per yard), so we can write q ෇ f ͑ p͒. Then the total revenue earned with selling price p is R͑ p͒ ෇ pf ͑ p͒. (a) What does it mean to say that f ͑20͒ ෇ 10,000 and f Ј͑20͒ ෇ Ϫ350? (b) Assuming the values in part (a), find RЈ͑20͒ and interpret your answer. 79. Let f ͑x͒ ෇ ͭ 2Ϫx x 2 Ϫ 2x ϩ 2 if x ഛ 1 if x Ͼ 1 Խ 80. At what numbers is the following function t differentiable? ͭ Ϫ1 Ϫ 2x if x Ͻ Ϫ1 t͑x͒ ෇ x 2 if Ϫ1 ഛ x ഛ 1 x if x Ͼ 1 Խ 82. Where is the function h͑x͒ ෇ x Ϫ 1 ϩ x ϩ 2 differenti- able? Give a formula for hЈ and sketch the graphs of h and hЈ. 83. For what values of a and b is the line 2x ϩ y ෇ b tangent to the parabola y ෇ ax 2 when x ෇ 2? 84. Let f ͑x͒ ෇ ͭ x2 mx ϩ b if x ഛ 2 if x Ͼ 2 Find the values of m and b that make f differentiable everywhere. 85. An easy proof of the Quotient Rule can be given if we make the prior assumption that FЈ͑x͒ exists, where F ෇ f͞t. Write f ෇ Ft; then differentiate using the Product Rule and solve the resulting equation for FЈ. (a) Show that the midpoint of the line segment cut from this tangent line by the coordinate axes is P. (b) Show that the triangle formed by the tangent line and the coordinate axes always has the same area, no matter where P is located on the hyperbola. 87. Evaluate lim xl1 Give a formula for tЈ and sketch the graphs of t and tЈ. Խ 81. (a) For what values of x is the function f ͑x͒ ෇ x Ϫ 9 |||| 3.4 157 86. A tangent line is drawn to the hyperbola xy ෇ c at a point P. Is f differentiable at 1? Sketch the graphs of f and f Ј. differentiable? Find a formula for f Ј. (b) Sketch the graphs of f and f Ј. Խ Խ ❙❙❙❙ 2 Խ x 1000 Ϫ 1 . xϪ1 88. Draw a diagram showing two perpendicular lines that intersect on the y-axis and are both tangent to the parabola y ෇ x 2. Where do these lines intersect? Rates of Change in the Natural and Social Sciences Recall from Section 3.1 that if y ෇ f ͑x͒, then the derivative dy͞dx can be interpreted as the rate of change of y with respect to x. In this section we examine some of the applications of this idea to physics, chemistry, biology, economics, and other sciences. Let’s recall from Section 2.6 the basic idea behind rates of change. If x changes from x 1 to x 2, then the change in x is ⌬x ෇ x 2 Ϫ x 1 y and the corresponding change in y is Q { ¤, ‡} ⌬y ෇ f ͑x 2 ͒ Ϫ f ͑x 1 ͒ Îy P { ⁄, fl} The difference quotient Îx 0 ⁄ ¤ mPQ ϭ average rate of change m=fª(⁄)=instantaneous rate of change FIGURE 1 x f ͑x 2 ͒ Ϫ f ͑x 1 ͒ ⌬y ෇ ⌬x x2 Ϫ x1 is the average rate of change of y with respect to x over the interval ͓x 1, x 2 ͔ and can be interpreted as the slope of the secant line PQ in Figure 1. Its limit as ⌬x l 0 is the derivative f Ј͑x 1 ͒, which can therefore be interpreted as the instantaneous rate of change of y 5E-03(pp 156-165) 158 ❙❙❙❙ 1/17/06 2:10 PM Page 158 CHAPTER 3 DERIVATIVES with respect to x or the slope of the tangent line at P͑x 1, f ͑x 1 ͒͒. Using Leibniz notation, we write the process in the form dy ⌬y ෇ lim ⌬x l 0 ⌬x dx Whenever the function y ෇ f ͑x͒ has a specific interpretation in one of the sciences, its derivative will have a specific interpretation as a rate of change. (As we discussed in Section 2.6, the units for dy͞dx are the units for y divided by the units for x.) We now look at some of these interpretations in the natural and social sciences. Physics Resources / Module 4 / Polynomial Models / Start of Polynomial Models If s ෇ f ͑t͒ is the position function of a particle that is moving in a straight line, then ⌬s͞⌬t represents the average velocity over a time period ⌬t, and v ෇ ds͞dt represents the instantaneous velocity (the rate of change of displacement with respect to time). This was discussed in Sections 2.6 and 3.1, but now that we know the differentiation formulas, we are able to solve velocity problems more easily. EXAMPLE 1 The position of a particle is given by the equation s ෇ f ͑t͒ ෇ t 3 Ϫ 6t 2 ϩ 9t where t is measured in seconds and s in meters. (a) Find the velocity at time t. (b) What is the velocity after 2 s? After 4 s? (c) When is the particle at rest? (d) When is the particle moving forward (that is, in the positive direction)? (e) Draw a diagram to represent the motion of the particle. (f ) Find the total distance traveled by the particle during the first five seconds. SOLUTION (a) The velocity function is the derivative of the position function. s ෇ f ͑t͒ ෇ t 3 Ϫ 6t 2 ϩ 9t v ͑t͒ ෇ ds ෇ 3t 2 Ϫ 12t ϩ 9 dt (b) The velocity after 2 s means the instantaneous velocity when t ෇ 2, that is, v ͑2͒ ෇ ds dt Ϳ t෇2 ෇ 3͑2͒2 Ϫ 12͑2͒ ϩ 9 ෇ Ϫ3 m͞s The velocity after 4 s is v ͑4͒ ෇ 3͑4͒2 Ϫ 12͑4͒ ϩ 9 ෇ 9 m͞s (c) The particle is at rest when v ͑t͒ ෇ 0, that is, 3t 2 Ϫ 12t ϩ 9 ෇ 3͑t 2 Ϫ 4t ϩ 3͒ ෇ 3͑t Ϫ 1͒͑t Ϫ 3͒ ෇ 0 and this is true when t ෇ 1 or t ෇ 3. Thus, the particle is at rest after 1 s and after 3 s. (d) The particle moves in the positive direction when v ͑t͒ Ͼ 0, that is, 3t 2 Ϫ 12t ϩ 9 ෇ 3͑t Ϫ 1͒͑t Ϫ 3͒ Ͼ 0 5E-03(pp 156-165) 1/17/06 2:11 PM Page 159 SECTION 3.4 RATES OF CHANGE IN THE NATURAL AND SOCIAL SCIENCES t=3 s=0 t=0 s=0 t=1 s=4 s ❙❙❙❙ 159 This inequality is true when both factors are positive ͑t Ͼ 3͒ or when both factors are negative ͑t Ͻ 1͒. Thus, the particle moves in the positive direction in the time intervals t Ͻ 1 and t Ͼ 3. It moves backward (in the negative direction) when 1 Ͻ t Ͻ 3. (e) Using the information from part (d), we make a schematic sketch as shown in Figure 2 of the motion of the particle back and forth along a line (the s-axis). (f ) Because of what we learned in parts (d) and (e), we need to calculate the distances traveled during the time intervals [0, 1], [1, 3], and [3, 5] separately. The distance traveled in the first second is Խ f ͑1͒ Ϫ f ͑0͒ Խ ෇ Խ 4 Ϫ 0 Խ ෇ 4 m FIGURE 2 From t ෇ 1 to t ෇ 3 the distance traveled is Խ f ͑3͒ Ϫ f ͑1͒ Խ ෇ Խ 0 Ϫ 4 Խ ෇ 4 m From t ෇ 3 to t ෇ 5 the distance traveled is Խ f ͑5͒ Ϫ f ͑3͒ Խ ෇ Խ 20 Ϫ 0 Խ ෇ 20 m The total distance is 4 ϩ 4 ϩ 20 ෇ 28 m. EXAMPLE 2 If a rod or piece of wire is homogeneous, then its linear density is uniform and is defined as the mass per unit length ͑ ␳ ෇ m͞l͒ and measured in kilograms per meter. Suppose, however, that the rod is not homogeneous but that its mass measured from its left end to a point x is m ෇ f ͑x͒, as shown in Figure 3. x x¡ FIGURE 3 x™ This part of the rod has mass ƒ. The mass of the part of the rod that lies between x ෇ x 1 and x ෇ x 2 is given by ⌬m ෇ f ͑x 2 ͒ Ϫ f ͑x 1 ͒, so the average density of that part of the rod is average density ෇ ⌬m f ͑x 2 ͒ Ϫ f ͑x 1 ͒ ෇ ⌬x x2 Ϫ x1 If we now let ⌬x l 0 (that is, x 2 l x 1 ), we are computing the average density over smaller and smaller intervals. The linear density ␳ at x 1 is the limit of these average densities as ⌬x l 0; that is, the linear density is the rate of change of mass with respect to length. Symbolically, ⌬m dm ␳ ෇ lim ෇ ⌬x l 0 ⌬x dx Thus, the linear density of the rod is the derivative of mass with respect to length. For instance, if m ෇ f ͑x͒ ෇ sx, where x is measured in meters and m in kilograms, then the average density of the part of the rod given by 1 ഛ x ഛ 1.2 is ⌬m f ͑1.2͒ Ϫ f ͑1͒ s1.2 Ϫ 1 ෇ ෇ Ϸ 0.48 kg͞m ⌬x 1.2 Ϫ 1 0.2 while the density right at x ෇ 1 is ␳෇ dm dx Ϳ x෇1 ෇ 1 2sx Ϳ x෇1 ෇ 0.50 kg͞m 5E-03(pp 156-165) 160 ❙❙❙❙ Ϫ 1/17/06 2:11 PM Page 160 CHAPTER 3 DERIVATIVES Ϫ FIGURE 4 Ϫ Ϫ Ϫ Ϫ Ϫ EXAMPLE 3 A current exists whenever electric charges move. Figure 4 shows part of a wire and electrons moving through a shaded plane surface. If ⌬Q is the net charge that passes through this surface during a time period ⌬t, then the average current during this time interval is defined as average current ෇ ⌬Q Q2 Ϫ Q1 ෇ ⌬t t2 Ϫ t1 If we take the limit of this average current over smaller and smaller time intervals, we get what is called the current I at a given time t1 : I ෇ lim ⌬t l 0 ⌬Q dQ ෇ ⌬t dt Thus, the current is the rate at which charge flows through a surface. It is measured in units of charge per unit time (often coulombs per second, called amperes). Velocity, density, and current are not the only rates of change that are important in physics. Others include power (the rate at which work is done), the rate of heat flow, temperature gradient (the rate of change of temperature with respect to position), and the rate of decay of a radioactive substance in nuclear physics. Chemistry EXAMPLE 4 A chemical reaction results in the formation of one or more substances (called products) from one or more starting materials (called reactants). For instance, the “equation” 2H2 ϩ O2 l 2H2 O indicates that two molecules of hydrogen and one molecule of oxygen form two molecules of water. Let’s consider the reaction AϩBlC where A and B are the reactants and C is the product. The concentration of a reactant A is the number of moles (1 mole ෇ 6.022 ϫ 10 23 molecules) per liter and is denoted by ͓A͔. The concentration varies during a reaction, so ͓A͔, ͓B͔, and ͓C͔ are all functions of time ͑t͒. The average rate of reaction of the product C over a time interval t1 ഛ t ഛ t2 is ⌬͓C͔ ͓C͔͑t2 ͒ Ϫ ͓C͔͑t1 ͒ ෇ ⌬t t2 Ϫ t1 But chemists are more interested in the instantaneous rate of reaction, which is obtained by taking the limit of the average rate of reaction as the time interval ⌬t approaches 0: rate of reaction ෇ lim ⌬t l 0 ⌬͓C͔ d͓C͔ ෇ ⌬t dt Since the concentration of the product increases as the reaction proceeds, the derivative d͓C͔͞dt will be positive. (You can see intuitively that the slope of the tangent to the 5E-03(pp 156-165) 1/17/06 2:11 PM Page 161 SECTION 3.4 RATES OF CHANGE IN THE NATURAL AND SOCIAL SCIENCES ❙❙❙❙ 161 graph of an increasing function is positive.) Thus, the rate of reaction of C is positive. The concentrations of the reactants, however, decrease during the reaction, so, to make the rates of reaction of A and B positive numbers, we put minus signs in front of the derivatives d͓A͔͞dt and d͓B͔͞dt. Since ͓A͔ and ͓B͔ each decrease at the same rate that ͓C͔ increases, we have rate of reaction ෇ d͓C͔ d͓A͔ d͓B͔ ෇Ϫ ෇Ϫ dt dt dt More generally, it turns out that for a reaction of the form aA ϩ bB l cC ϩ dD we have Ϫ 1 d͓A͔ 1 d͓B͔ 1 d͓C͔ 1 d ͓D͔ ෇Ϫ ෇ ෇ a dt b dt c dt d dt The rate of reaction can be determined by graphical methods (see Exercise 22). In some cases we can use the rate of reaction to find explicit formulas for the concentrations as functions of time (see Exercises 10.3). EXAMPLE 5 One of the quantities of interest in thermodynamics is compressibility. If a given substance is kept at a constant temperature, then its volume V depends on its pressure P. We can consider the rate of change of volume with respect to pressure—namely, the derivative dV͞dP. As P increases, V decreases, so dV͞dP Ͻ 0. The compressibility is defined by introducing a minus sign and dividing this derivative by the volume V : isothermal compressibility ෇ ␤ ෇ Ϫ 1 dV V dP Thus, ␤ measures how fast, per unit volume, the volume of a substance decreases as the pressure on it increases at constant temperature. For instance, the volume V (in cubic meters) of a sample of air at 25ЊC was found to be related to the pressure P (in kilopascals) by the equation V෇ 5.3 P The rate of change of V with respect to P when P ෇ 50 kPa is dV dP Ϳ ෇Ϫ P෇50 ෇Ϫ 5.3 P2 Ϳ P෇50 5.3 ෇ Ϫ0.00212 m 3͞kPa 2500 The compressibility at that pressure is ␤෇Ϫ 1 dV V dP Ϳ P෇50 ෇ 0.00212 ෇ 0.02 ͑m 3͞kPa͒͞m 3 5.3 50 5E-03(pp 156-165) 162 ❙❙❙❙ 1/17/06 2:12 PM Page 162 CHAPTER 3 DERIVATIVES Biology EXAMPLE 6 Let n ෇ f ͑t͒ be the number of individuals in an animal or plant population at time t. The change in the population size between the times t ෇ t1 and t ෇ t2 is ⌬n ෇ f ͑t2 ͒ Ϫ f ͑t1 ͒, and so the average rate of growth during the time period t1 ഛ t ഛ t2 is ⌬n f ͑t2 ͒ Ϫ f ͑t1 ͒ average rate of growth ෇ ෇ ⌬t t2 Ϫ t1 The instantaneous rate of growth is obtained from this average rate of growth by letting the time period ⌬t approach 0: growth rate ෇ lim ⌬t l 0 ⌬n dn ෇ ⌬t dt Strictly speaking, this is not quite accurate because the actual graph of a population function n ෇ f ͑t͒ would be a step function that is discontinuous whenever a birth or death occurs and, therefore, not differentiable. However, for a large animal or plant population, we can replace the graph by a smooth approximating curve as in Figure 5. n FIGURE 5 t 0 A smooth curve approximating a growth function To be more specific, consider a population of bacteria in a homogeneous nutrient medium. Suppose that by sampling the population at certain intervals it is determined that the population doubles every hour. If the initial population is n0 and the time t is measured in hours, then f ͑1͒ ෇ 2f ͑0͒ ෇ 2n0 f ͑2͒ ෇ 2f ͑1͒ ෇ 2 2n0 f ͑3͒ ෇ 2f ͑2͒ ෇ 2 3n0 and, in general, f ͑t͒ ෇ 2 tn0 The population function is n ෇ n0 2 t. This is an example of an exponential function. In Chapter 7 we will discuss exponential functions in general; at that time we will be able to compute their derivatives and thereby determine the rate of growth of the bacteria population. 5E-03(pp 156-165) 1/17/06 2:12 PM Page 163 SECTION 3.4 RATES OF CHANGE IN THE NATURAL AND SOCIAL SCIENCES ❙❙❙❙ 163 EXAMPLE 7 When we consider the flow of blood through a blood vessel, such as a vein or artery, we can take the shape of the blood vessel to be a cylindrical tube with radius R and length l as illustrated in Figure 6. R r FIGURE 6 l Blood flow in an artery Because of friction at the walls of the tube, the velocity v of the blood is greatest along the central axis of the tube and decreases as the distance r from the axis increases until v becomes 0 at the wall. The relationship between v and r is given by the law of laminar flow discovered by the French physician Jean-Louis-Marie Poiseuille in 1840. This states that v෇ 1 P ͑R 2 Ϫ r 2 ͒ 4␩ l where ␩ is the viscosity of the blood and P is the pressure difference between the ends of the tube. If P and l are constant, then v is a function of r with domain ͓0, R͔. [For more detailed information, see W. Nichols and M. O’Rourke (eds.), McDonald’s Blood Flow in Arteries: Theoretic, Experimental, and Clinical Principles, 4th ed. (New York: Oxford University Press, 1998).] The average rate of change of the velocity as we move from r ෇ r1 outward to r ෇ r2 is given by ⌬v v͑r2 ͒ Ϫ v͑r1 ͒ ෇ ⌬r r2 Ϫ r1 and if we let ⌬r l 0, we obtain the velocity gradient, that is, the instantaneous rate of change of velocity with respect to r: velocity gradient ෇ lim ⌬r l 0 ⌬v dv ෇ ⌬r dr Using Equation 1, we obtain dv P Pr ෇ ͑0 Ϫ 2r͒ ෇ Ϫ dr 4␩l 2␩ l For one of the smaller human arteries we can take ␩ ෇ 0.027, R ෇ 0.008 cm, l ෇ 2 cm, and P ෇ 4000 dynes͞cm2, which gives v෇ 4000 ͑0.000064 Ϫ r 2 ͒ 4͑0.027͒2 Ϸ 1.85 ϫ 10 4͑6.4 ϫ 10 Ϫ5 Ϫ r 2 ͒ At r ෇ 0.002 cm the blood is flowing at a speed of v͑0.002͒ Ϸ 1.85 ϫ 10 4͑64 ϫ 10 Ϫ6 Ϫ 4 ϫ 10 Ϫ6 ͒ ෇ 1.11 cm͞s 5E-03(pp 156-165) 164 ❙❙❙❙ 1/17/06 2:12 PM Page 164 CHAPTER 3 DERIVATIVES and the velocity gradient at that point is dv dr Ϳ r෇0.002 ෇Ϫ 4000͑0.002͒ Ϸ Ϫ74 ͑cm͞s͒͞cm 2͑0.027͒2 To get a feeling for what this statement means, let’s change our units from centimeters to micrometers (1 cm ෇ 10,000 ␮m). Then the radius of the artery is 80 ␮m. The velocity at the central axis is 11,850 ␮m͞s, which decreases to 11,110 ␮m͞s at a distance of r ෇ 20 ␮m. The fact that dv͞dr ෇ Ϫ74 (␮m͞s)͞␮m means that, when r ෇ 20 ␮m, the velocity is decreasing at a rate of about 74 ␮m͞s for each micrometer that we proceed away from the center. Economics EXAMPLE 8 Suppose C͑x͒ is the total cost that a company incurs in producing x units of a certain commodity. The function C is called a cost function. If the number of items produced is increased from x 1 to x 2 , the additional cost is ⌬C ෇ C͑x 2 ͒ Ϫ C͑x 1 ͒, and the average rate of change of the cost is ⌬C C͑x 2 ͒ Ϫ C͑x 1 ͒ C͑x 1 ϩ ⌬x͒ Ϫ C͑x 1 ͒ ෇ ෇ ⌬x x2 Ϫ x1 ⌬x The limit of this quantity as ⌬x l 0, that is, the instantaneous rate of change of cost with respect to the number of items produced, is called the marginal cost by economists: marginal cost ෇ lim ⌬x l 0 ⌬C dC ෇ ⌬x dx [Since x often takes on only integer values, it may not make literal sense to let ⌬x approach 0, but we can always replace C͑x͒ by a smooth approximating function as in Example 6.] Taking ⌬x ෇ 1 and n large (so that ⌬x is small compared to n), we have CЈ͑n͒ Ϸ C͑n ϩ 1͒ Ϫ C͑n͒ Thus, the marginal cost of producing n units is approximately equal to the cost of producing one more unit [the ͑n ϩ 1͒st unit]. It is often appropriate to represent a total cost function by a polynomial C͑x͒ ෇ a ϩ bx ϩ cx 2 ϩ dx 3 where a represents the overhead cost (rent, heat, maintenance) and the other terms represent the cost of raw materials, labor, and so on. (The cost of raw materials may be proportional to x, but labor costs might depend partly on higher powers of x because of overtime costs and inefficiencies involved in large-scale operations.) For instance, suppose a company has estimated that the cost (in dollars) of producing x items is C͑x͒ ෇ 10,000 ϩ 5x ϩ 0.01x 2 Then the marginal cost function is CЈ͑x͒ ෇ 5 ϩ 0.02x 5E-03(pp 156-165) 1/17/06 2:13 PM Page 165 SECTION 3.4 RATES OF CHANGE IN THE NATURAL AND SOCIAL SCIENCES ❙❙❙❙ 165 The marginal cost at the production level of 500 items is CЈ͑500͒ ෇ 5 ϩ 0.02͑500͒ ෇ $15͞item This gives the rate at which costs are increasing with respect to the production level when x ෇ 500 and predicts the cost of the 501st item. The actual cost of producing the 501st item is C͑501͒ Ϫ C͑500͒ ෇ ͓10,000 ϩ 5͑501͒ ϩ 0.01͑501͒2 ͔ ෇ Ϫ ͓10,000 ϩ 5͑500͒ ϩ 0.01͑500͒2 ͔ ෇ $15.01 Notice that CЈ͑500͒ Ϸ C͑501͒ Ϫ C͑500͒. Economists also study marginal demand, marginal revenue, and marginal profit, which are the derivatives of the demand, revenue, and profit functions. These will be considered in Chapter 4 after we have developed techniques for finding the maximum and minimum values of functions. Other Sciences Rates of change occur in all the sciences. A geologist is interested in knowing the rate at which an intruded body of molten rock cools by conduction of heat into surrounding rocks. An engineer wants to know the rate at which water flows into or out of a reservoir. An urban geographer is interested in the rate of change of the population density in a city as the distance from the city center increases. A meteorologist is concerned with the rate of change of atmospheric pressure with respect to height (see Exercise 17 in Section 10.4). In psychology, those interested in learning theory study the so-called learning curve, which graphs the performance P͑t͒ of someone learning a skill as a function of the training time t. Of particular interest is the rate at which performance improves as time passes, that is, dP͞dt. In sociology, differential calculus is used in analyzing the spread of rumors (or innovations or fads or fashions). If p͑t͒ denotes the proportion of a population that knows a rumor by time t, then the derivative dp͞dt represents the rate of spread of the rumor (see Exercise 57 in Section 7.2). Summary Velocity, density, current, power, and temperature gradient in physics, rate of reaction and compressibility in chemistry, rate of growth and blood velocity gradient in biology, marginal cost and marginal profit in economics, rate of heat flow in geology, rate of improvement of performance in psychology, rate of spread of a rumor in sociology—these are all special cases of a single mathematical concept, the derivative. This is an illustration of the fact that part of the power of mathematics lies in its abstractness. A single abstract mathematical concept (such as the derivative) can have different interpretations in each of the sciences. When we develop the properties of the mathematical concept once and for all, we can then turn around and apply these results to all of the sciences. This is much more efficient than developing properties of special concepts in each separate science. The French mathematician Joseph Fourier (1768–1830) put it succinctly: “Mathematics compares the most diverse phenomena and discovers the secret analogies that unite them.” 5E-03(pp 166-175) 166 ❙❙❙❙ 1/17/06 2:03 PM Page 166 CHAPTER 3 DERIVATIVES |||| 3.4 Exercises |||| A particle moves according to a law of motion s ෇ f ͑t͒, t ജ 0, where t is measured in seconds and s in feet. (a) Find the velocity at time t. (b) What is the velocity after 3 s? (c) When is the particle at rest? (d) When is the particle moving in the positive direction? (e) Find the total distance traveled during the first 8 s. (f ) Draw a diagram like Figure 2 to illustrate the motion of the particle. 1–6 (b) Show that the rate of change of the volume of a cube with respect to its edge length is equal to half the surface area of the cube. Explain geometrically why this result is true by arguing by analogy with Exercise 11(b). 13. (a) Find the average rate of change of the area of a circle with 1. f ͑t͒ ෇ t 2 Ϫ 10t ϩ 12 2. f ͑t͒ ෇ t 3 Ϫ 9t 2 ϩ 15t ϩ 10 3. f ͑t͒ ෇ t 3 Ϫ 12t 2 ϩ 36t 4. f ͑t͒ ෇ t 4 Ϫ 4t ϩ 1 5. s ෇ ■ ■ t t2 ϩ 1 ■ 6. s ෇ st ͑3t 2 Ϫ 35t ϩ 90͒ ■ ■ ■ ■ ■ ■ ■ ■ ■ 7. The position function of a particle is given by s ෇ t 3 Ϫ 4.5t 2 Ϫ 7t tജ0 When does the particle reach a velocity of 5 m͞s? 8. If a ball is given a push so that it has an initial velocity of 5 m͞s down a certain inclined plane, then the distance it has rolled after t seconds is s ෇ 5t ϩ 3t 2. (a) Find the velocity after 2 s. (b) How long does it take for the velocity to reach 35 m͞s? 9. If a stone is thrown vertically upward from the surface of the moon with a velocity of 10 m͞s, its height (in meters) after t seconds is h ෇ 10t Ϫ 0.83t 2. (a) What is the velocity of the stone after 3 s? (b) What is the velocity of the stone after it has risen 25 m? 10. If a ball is thrown vertically upward with a velocity of 80 ft͞s, then its height after t seconds is s ෇ 80t Ϫ 16t 2. (a) What is the maximum height reached by the ball? (b) What is the velocity of the ball when it is 96 ft above the ground on its way up? On its way down? 11. (a) A company makes computer chips from square wafers of silicon. It wants to keep the side length of a wafer very close to 15 mm and it wants to know how the area A͑x͒ of a wafer changes when the side length x changes. Find AЈ͑15͒ and explain its meaning in this situation. (b) Show that the rate of change of the area of a square with respect to its side length is half its perimeter. Try to explain geometrically why this is true by drawing a square whose side length x is increased by an amount ⌬x. How can you approximate the resulting change in area ⌬A if ⌬x is small? 12. (a) Sodium chlorate crystals are easy to grow in the shape of cubes by allowing a solution of water and sodium chlorate to evaporate slowly. If V is the volume of such a cube with side length x, calculate dV͞dx when x ෇ 3 mm and explain its meaning. respect to its radius r as r changes from (i) 2 to 3 (ii) 2 to 2.5 (iii) 2 to 2.1 (b) Find the instantaneous rate of change when r ෇ 2. (c) Show that the rate of change of the area of a circle with respect to its radius (at any r) is equal to the circumference of the circle. Try to explain geometrically why this is true by drawing a circle whose radius is increased by an amount ⌬r. How can you approximate the resulting change in area ⌬A if ⌬r is small? 14. A stone is dropped into a lake, creating a circular ripple that travels outward at a speed of 60 cm͞s. Find the rate at which the area within the circle is increasing after (a) 1 s, (b) 3 s, and (c) 5 s. What can you conclude? 15. A spherical balloon is being inflated. Find the rate of increase of the surface area ͑S ෇ 4␲ r 2 ͒ with respect to the radius r when r is (a) 1 ft, (b) 2 ft, and (c) 3 ft. What conclusion can you make? 16. (a) The volume of a growing spherical cell is V ෇ 3 ␲ r 3, where 4 the radius r is measured in micrometers (1 ␮m ෇ 10Ϫ6 m). Find the average rate of change of V with respect to r when r changes from (i) 5 to 8 ␮m (ii) 5 to 6 ␮m (iii) 5 to 5.1 ␮m (b) Find the instantaneous rate of change of V with respect to r when r ෇ 5 ␮m. (c) Show that the rate of change of the volume of a sphere with respect to its radius is equal to its surface area. Explain geometrically why this result is true. Argue by analogy with Exercise 13(c). 17. The mass of the part of a metal rod that lies between its left end and a point x meters to the right is 3x 2 kg. Find the linear density (see Example 2) when x is (a) 1 m, (b) 2 m, and (c) 3 m. Where is the density the highest? The lowest? 18. If a tank holds 5000 gallons of water, which drains from the bottom of the tank in 40 minutes, then Torricelli’s Law gives the volume V of water remaining in the tank after t minutes as ͩ V ෇ 5000 1 Ϫ t 40 ͪ 2 0 ഛ t ഛ 40 Find the rate at which water is draining from the tank after (a) 5 min, (b) 10 min, (c) 20 min, and (d) 40 min. At what time is the water flowing out the fastest? The slowest? Summarize your findings. 19. The quantity of charge Q in coulombs (C) that has passed through a point in a wire up to time t (measured in seconds) is 5E-03(pp 166-175) 1/17/06 2:03 PM Page 167 SECTION 3.4 RATES OF CHANGE IN THE NATURAL AND SOCIAL SCIENCES given by Q͑t͒ ෇ t 3 Ϫ 2t 2 ϩ 6t ϩ 2. Find the current when (a) t ෇ 0.5 s and (b) t ෇ 1 s. [See Example 3. The unit of current is an ampere (1 A ෇ 1 C͞s).] At what time is the current lowest? 20. Newton’s Law of Gravitation says that the magnitude F of the force exerted by a body of mass m on a body of mass M is F෇ GmM r2 where G is the gravitational constant and r is the distance between the bodies. (a) Find dF͞dr and explain its meaning. What does the minus sign indicate? (b) Suppose it is known that Earth attracts an object with a force that decreases at the rate of 2 N͞km when r ෇ 20,000 km. How fast does this force change when r ෇ 10,000 km? 21. Boyle’s Law states that when a sample of gas is compressed at a constant temperature, the product of the pressure and the volume remains constant: PV ෇ C. (a) Find the rate of change of volume with respect to pressure. (b) A sample of gas is in a container at low pressure and is steadily compressed at constant temperature for 10 minutes. Is the volume decreasing more rapidly at the beginning or the end of the 10 minutes? Explain. (c) Prove that the isothermal compressibility (see Example 5) is given by ␤ ෇ 1͞P. 22. The data in the table concern the lactonization of hydroxy- valeric acid at 25ЊC. They give the concentration C͑t͒ of this acid in moles per liter after t minutes. t 0 2 4 6 8 C(t ) 0.0800 0.0570 0.0408 0.0295 0.0210 (a) Find the average rate of reaction for the following time intervals: (i) 2 ഛ t ഛ 6 (ii) 2 ഛ t ഛ 4 (iii) 0 ഛ t ഛ 2 (b) Plot the points from the table and draw a smooth curve through them as an approximation to the graph of the concentration function. Then draw the tangent at t ෇ 2 and use it to estimate the instantaneous rate of reaction when t ෇ 2. ; 23. The table gives the population of the world in the 20th century. Year Population (in millions) 1900 1910 1920 1930 1940 1950 1650 1750 1860 2070 2300 2560 Year Population (in millions) 1960 1970 1980 1990 2000 3040 3710 4450 5280 6080 ❙❙❙❙ 167 (a) Estimate the rate of population growth in 1920 and in 1980 by averaging the slopes of two secant lines. (b) Use a graphing calculator or computer to find a cubic function (a third-degree polynomial) that models the data. (See Section 1.2.) (c) Use your model in part (b) to find a model for the rate of population growth in the 20th century. (d) Use part (c) to estimate the rates of growth in 1920 and 1980. Compare with your estimates in part (a). (e) Estimate the rate of growth in 1985. ; 24. The table shows how the average age of first marriage of Japanese women varied in the last half of the 20th century. t A͑t͒ t A͑t͒ 1950 1955 1960 1965 1970 23.0 23.8 24.4 24.5 24.2 1975 1980 1985 1990 1995 24.7 25.2 25.5 25.9 26.3 (a) Use a graphing calculator or computer to model these data with a fourth-degree polynomial. (b) Use part (a) to find a model for AЈ͑t͒. (c) Estimate the rate of change of marriage age for women in 1990. (d) Graph the data points and the models for A and AЈ. 25. If, in Example 4, one molecule of the product C is formed from one molecule of the reactant A and one molecule of the reactant B, and the initial concentrations of A and B have a common value ͓A͔ ෇ ͓B͔ ෇ a moles͞L, then ͓C͔ ෇ a 2kt͑͞akt ϩ 1͒ where k is a constant. (a) Find the rate of reaction at time t. (b) Show that if x ෇ ͓C͔, then dx ෇ k͑a Ϫ x͒2 dt 26. If f is the focal length of a convex lens and an object is placed at a distance p from the lens, then its image will be at a distance q from the lens, where f , p, and q are related by the lens equation 1 1 1 ෇ ϩ f p q Find the rate of change of p with respect to q. 27. Refer to the law of laminar flow given in Example 7. Consider a blood vessel with radius 0.01 cm, length 3 cm, pressure difference 3000 dynes͞cm2, and viscosity ␩ ෇ 0.027. (a) Find the velocity of the blood along the centerline r ෇ 0, at radius r ෇ 0.005 cm, and at the wall r ෇ R ෇ 0.01 cm. 5E-03(pp 166-175) 168 ❙❙❙❙ 1/17/06 2:04 PM Page 168 CHAPTER 3 DERIVATIVES (b) Find the velocity gradient at r ෇ 0, r ෇ 0.005, and r ෇ 0.01. (c) Where is the velocity the greatest? Where is the velocity changing most? when the brightness x of a light source is increased, the eye reacts by decreasing the area R of the pupil. The experimental formula R෇ 28. The frequency of vibrations of a vibrating violin string is given by f෇ 1 2L ͱ T ␳ where L is the length of the string, T is its tension, and ␳ is its linear density. [See Chapter 11 in Donald E. Hall, Musical Acoustics, 3d ed. (Pacific Grove, CA: Brooks/Cole, 2002).] (a) Find the rate of change of the frequency with respect to (i) the length (when T and ␳ are constant), (ii) the tension (when L and ␳ are constant), and (iii) the linear density (when L and T are constant). (b) The pitch of a note (how high or low the note sounds) is determined by the frequency f . (The higher the frequency, the higher the pitch.) Use the signs of the derivatives in part (a) to determine what happens to the pitch of a note (i) when the effective length of a string is decreased by placing a finger on the string so a shorter portion of the string vibrates, (ii) when the tension is increased by turning a tuning peg, (iii) when the linear density is increased by switching to another string. 29. Suppose that the cost (in dollars) for a company to produce x pairs of a new line of jeans is C͑x͒ ෇ 2000 ϩ 3x ϩ 0.01x 2 ϩ 0.0002x 3 (a) Find the marginal cost function. (b) Find CЈ͑100͒ and explain its meaning. What does it predict? (c) Compare CЈ͑100͒ with the cost of manufacturing the 101st pair of jeans. 30. The cost function for a certain commodity is C͑x͒ ෇ 84 ϩ 0.16x Ϫ 0.0006x 2 ϩ 0.000003x 3 (a) Find and interpret CЈ͑100͒. (b) Compare CЈ͑100͒ with the cost of producing the 101st item. 31. If p͑x͒ is the total value of the production when there are x workers in a plant, then the average productivity of the workforce at the plant is A͑x͒ ෇ p͑x͒ x (a) Find AЈ͑x͒. Why does the company want to hire more workers if AЈ͑x͒ Ͼ 0 ? (b) Show that AЈ͑x͒ Ͼ 0 if pЈ͑x͒ is greater than the average productivity. 32. If R denotes the reaction of the body to some stimulus of strength x, the sensitivity S is defined to be the rate of change of the reaction with respect to x. A particular example is that ; 40 ϩ 24x 0.4 1 ϩ 4x 0.4 has been used to model the dependence of R on x when R is measured in square millimeters and x is measured in appropriate units of brightness. (a) Find the sensitivity. (b) Illustrate part (a) by graphing both R and S as functions of x. Comment on the values of R and S at low levels of brightness. Is this what you would expect? 33. The gas law for an ideal gas at absolute temperature T (in kelvins), pressure P (in atmospheres), and volume V (in liters) is PV ෇ nRT , where n is the number of moles of the gas and R ෇ 0.0821 is the gas constant. Suppose that, at a certain instant, P ෇ 8.0 atm and is increasing at a rate of 0.10 atm͞min and V ෇ 10 L and is decreasing at a rate of 0.15 L͞min. Find the rate of change of T with respect to time at that instant if n ෇ 10 mol. 34. In a fish farm, a population of fish is introduced into a pond and harvested regularly. A model for the rate of change of the fish population is given by the equation ͩ ͪ dP P͑t͒ ෇ r0 1 Ϫ P͑t͒ Ϫ ␤P͑t͒ dt Pc where r0 is the birth rate of the fish, Pc is the maximum population that the pond can sustain (called the carrying capacity), and ␤ is the percentage of the population that is harvested. (a) What value of dP͞dt corresponds to a stable population? (b) If the pond can sustain 10,000 fish, the birth rate is 5%, and the harvesting rate is 4%, find the stable population level. (c) What happens if ␤ is raised to 5%? 35. In the study of ecosystems, predator-prey models are often used to study the interaction between species. Consider populations of tundra wolves, given by W͑t͒, and caribou, given by C͑t͒, in northern Canada. The interaction has been modeled by the equations dC ෇ aC Ϫ bCW dt dW ෇ ϪcW ϩ dCW dt (a) What values of dC͞dt and dW͞dt correspond to stable populations? (b) How would the statement “The caribou go extinct” be represented mathematically? (c) Suppose that a ෇ 0.05, b ෇ 0.001, c ෇ 0.05, and d ෇ 0.0001. Find all population pairs ͑C, W ͒ that lead to stable populations. According to this model, is it possible for the two species to live in balance or will one or both species become extinct? 5E-03(pp 166-175) 1/17/06 2:04 PM Page 169 SECTION 3.5 DERIVATIVES OF TRIGONOMETRIC FUNCTIONS |||| 3.5 ❙❙❙❙ 169 Derivatives of Trigonometric Functions |||| A review of the trigonometric functions is given in Appendix D. Before starting this section, you might need to review the trigonometric functions. In particular, it is important to remember that when we talk about the function f defined for all real numbers x by f ͑x͒ ෇ sin x it is understood that sin x means the sine of the angle whose radian measure is x. A similar convention holds for the other trigonometric functions cos, tan, csc, sec, and cot. Recall from Section 2.5 that all of the trigonometric functions are continuous at every number in their domains. If we sketch the graph of the function f ͑x͒ ෇ sin x and use the interpretation of f Ј͑x͒ as the slope of the tangent to the sine curve in order to sketch the graph of f Ј (see Exercise 16 in Section 3.2), then it looks as if the graph of f Ј may be the same as the cosine curve (see Figure 1 and also page 126). See an animation of Figure 1. Resources / Module 4 / Trigonometric Models / Slope-A-Scope for Sine ƒ=sin x 0 π 2 π π 2 π 2π x fª(x) 0 x FIGURE 1 Let’s try to confirm our guess that if f ͑x͒ ෇ sin x, then f Ј͑x͒ ෇ cos x. From the definition of a derivative, we have f Ј͑x͒ ෇ lim hl0 f ͑x ϩ h͒ Ϫ f ͑x͒ h ෇ lim sin͑x ϩ h͒ Ϫ sin x h ෇ lim sin x cos h ϩ cos x sin h Ϫ sin x h hl0 |||| We have used the addition formula for sine. See Appendix D. hl0 ෇ lim hl0 ͫ ͫ ͩ ෇ lim sin x hl0 1 cos h Ϫ 1 h ෇ lim sin x ؒ lim hl0 ͬ ͩ ͪͬ cos x sin h sin x cos h Ϫ sin x ϩ h h hl0 ͪ ϩ cos x sin h h cos h Ϫ 1 sin h ϩ lim cos x ؒ lim hl0 hl0 h h 5E-03(pp 166-175) 170 ❙❙❙❙ 1/17/06 2:05 PM Page 170 CHAPTER 3 DERIVATIVES Two of these four limits are easy to evaluate. Since we regard x as a constant when computing a limit as h l 0, we have lim sin x ෇ sin x and hl0 lim cos x ෇ cos x hl0 The limit of ͑sin h͒͞h is not so obvious. In Example 3 in Section 2.2 we made the guess, on the basis of numerical and graphical evidence, that lim 2 D B ␪l0 We now use a geometric argument to prove Equation 2. Assume first that ␪ lies between 0 and ␲͞2. Figure 2(a) shows a sector of a circle with center O, central angle ␪, and radius 1. BC is drawn perpendicular to OA. By the definition of radian measure, we have arc AB ෇ ␪. Also, BC ෇ OB sin ␪ ෇ sin ␪. From the diagram we see that Խ Խ Խ Խ Խ BC Խ Ͻ Խ AB Խ Ͻ arc AB E sin ␪ Ͻ ␪ Therefore O ¨ 1 A C (a) so sin ␪ Ͻ1 ␪ Let the tangent lines at A and B intersect at E. You can see from Figure 2(b) that the circumference of a circle is smaller than the length of a circumscribed polygon, and so arc AB Ͻ AE ϩ EB . Thus Խ Խ Խ Խ B ␪ ෇ arc AB Ͻ Խ AE Խ ϩ Խ EB Խ E Խ Խ Խ Խ ෇ Խ AD Խ ෇ Խ OA Խ tan ␪ Ͻ AE ϩ ED A O sin ␪ ෇1 ␪ ෇ tan ␪ (b) FIGURE 2 (In Appendix F the inequality ␪ ഛ tan ␪ is proved directly from the definition of the length of an arc without resorting to geometric intuition as we did here.) Therefore, we have ␪Ͻ so cos ␪ Ͻ sin ␪ cos ␪ sin ␪ Ͻ1 ␪ We know that lim ␪ l 0 1 ෇ 1 and lim ␪ l 0 cos ␪ ෇ 1, so by the Squeeze Theorem, we have lim ␪ l 0ϩ sin ␪ ෇1 ␪ But the function ͑sin ␪͒͞␪ is an even function, so its right and left limits must be equal. Hence, we have lim ␪l0 so we have proved Equation 2. sin ␪ ෇1 ␪ 5E-03(pp 166-175) 1/17/06 2:05 PM Page 171 SECTION 3.5 DERIVATIVES OF TRIGONOMETRIC FUNCTIONS ❙❙❙❙ 171 We can deduce the value of the remaining limit in (1) as follows: |||| We multiply numerator and denominator by cos ␪ ϩ 1 in order to put the function in a form in which we can use the limits we know. lim ␪l0 cos ␪ Ϫ 1 ෇ lim ␪l0 ␪ ෇ lim ␪l0 ͩ cos ␪ Ϫ 1 cos ␪ ϩ 1 ؒ ␪ cos ␪ ϩ 1 ෇ lim ␪l0 cos2␪ Ϫ 1 ␪ ͑cos ␪ ϩ 1͒ Ϫsin 2␪ sin ␪ sin ␪ ෇ Ϫlim ؒ ␪l0 ␪ ͑cos ␪ ϩ 1͒ ␪ cos ␪ ϩ 1 ෇ Ϫlim ␪l0 ෇ Ϫ1 ؒ sin ␪ sin ␪ ؒ lim ␪ l 0 cos ␪ ϩ 1 ␪ ͩ ͪ 0 1ϩ1 lim 3 ͪ ␪l0 ෇0 (by Equation 2) cos ␪ Ϫ 1 ෇0 ␪ If we now put the limits (2) and (3) in (1), we get f Ј͑x͒ ෇ lim sin x ؒ lim hl0 hl0 cos h Ϫ 1 sin h ϩ lim cos x ؒ lim hl0 hl0 h h ෇ ͑sin x͒ ؒ 0 ϩ ͑cos x͒ ؒ 1 ෇ cos x So we have proved the formula for the derivative of the sine function: 4 |||| Figure 3 shows the graphs of the function of Example 1 and its derivative. Notice that yЈ ෇ 0 whenever y has a horizontal tangent. EXAMPLE 1 Differentiate y ෇ x 2 sin x. SOLUTION Using the Product Rule and Formula 4, we have 5 yª _4 d ͑sin x͒ ෇ cos x dx dy d d ෇ x2 ͑sin x͒ ϩ sin x ͑x 2 ͒ dx dx dx y ෇ x 2 cos x ϩ 2x sin x 4 Using the same methods as in the proof of Formula 4, one can prove (see Exercise 20) that _5 FIGURE 3 5 d ͑cos x͒ ෇ Ϫsin x dx The tangent function can also be differentiated by using the definition of a derivative, 5E-03(pp 166-175) 172 ❙❙❙❙ 1/17/06 2:06 PM Page 172 CHAPTER 3 DERIVATIVES but it is easier to use the Quotient Rule together with Formulas 4 and 5: d d ͑tan x͒ ෇ dx dx ͩ ͪ sin x cos x cos x ෇ d d ͑sin x͒ Ϫ sin x ͑cos x͒ dx dx cos2x ෇ cos x ؒ cos x Ϫ sin x ͑Ϫsin x͒ cos2x ෇ cos2x ϩ sin2x cos2x ෇ 1 ෇ sec2x cos2x d ͑tan x͒ ෇ sec2x dx 6 The derivatives of the remaining trigonometric functions, csc, sec, and cot , can also be found easily using the Quotient Rule (see Exercises 17–19). We collect all the differentiation formulas for trigonometric functions in the following table. Remember that they are valid only when x is measured in radians. Derivatives of Trigonometric Functions |||| When you memorize this table, it is helpful to notice that the minus signs go with the derivatives of the “cofunctions,” that is, cosine, cosecant, and cotangent. d ͑sin x͒ ෇ cos x dx d ͑cos x͒ ෇ Ϫsin x dx d ͑tan x͒ ෇ sec2x dx EXAMPLE 2 Differentiate f ͑x͒ ෇ have a horizontal tangent? d ͑csc x͒ ෇ Ϫcsc x cot x dx d ͑sec x͒ ෇ sec x tan x dx d ͑cot x͒ ෇ Ϫcsc 2x dx sec x . For what values of x does the graph of f 1 ϩ tan x SOLUTION The Quotient Rule gives ͑1 ϩ tan x͒ f Ј͑x͒ ෇ d d ͑sec x͒ Ϫ sec x ͑1 ϩ tan x͒ dx dx ͑1 ϩ tan x͒2 ෇ ͑1 ϩ tan x͒ sec x tan x Ϫ sec x ؒ sec2x ͑1 ϩ tan x͒2 ෇ sec x ͑tan x ϩ tan2x Ϫ sec2x͒ ͑1 ϩ tan x͒2 ෇ sec x ͑tan x Ϫ 1͒ ͑1 ϩ tan x͒2 5E-03(pp 166-175) 1/17/06 2:06 PM Page 173 SECTION 3.5 DERIVATIVES OF TRIGONOMETRIC FUNCTIONS ❙❙❙❙ 173 In simplifying the answer we have used the identity tan2x ϩ 1 ෇ sec2x. Since sec x is never 0, we see that f Ј͑x͒ ෇ 0 when tan x ෇ 1, and this occurs when x ෇ n␲ ϩ ␲͞4, where n is an integer (see Figure 4). 3 _3 5 Trigonometric functions are often used in modeling real-world phenomena. In particular, vibrations, waves, elastic motions, and other quantities that vary in a periodic manner can be described using trigonometric functions. In the following example we discuss an instance of simple harmonic motion. _3 FIGURE 4 The horizontal tangents in Example 2 EXAMPLE 3 An object at the end of a vertical spring is stretched 4 cm beyond its rest position and released at time t ෇ 0. (See Figure 5 and note that the downward direction is positive.) Its position at time t is s ෇ f ͑t͒ ෇ 4 cos t 0 Find the velocity at time t and use it to analyze the motion of the object. 4 SOLUTION The velocity is s √ s π The object oscillates from the lowest point ͑s ෇ 4 cm͒ to the highest point ͑s ෇ Ϫ4 cm͒. The period of the oscillation is 2␲, the period of cos t. The speed is v ෇ 4 sin t , which is greatest when sin t ෇ 1, that is, when cos t ෇ 0. So the object moves fastest as it passes through its equilibrium position ͑s ෇ 0͒. Its speed is 0 when sin t ෇ 0, that is, at the high and low points. See the graphs in Figure 6. Խ Խ 2 0 ds d d ෇ ͑4 cos t͒ ෇ 4 ͑cos t͒ ෇ Ϫ4 sin t dt dt dt v෇ FIGURE 5 2π t Խ Խ Խ Խ _2 Our main use for the limit in Equation 2 has been to prove the differentiation formula for the sine function. But this limit is also useful in finding certain other trigonometric limits, as the following two examples show. FIGURE 6 EXAMPLE 4 Find lim xl0 sin 7x . 4x SOLUTION In order to apply Equation 2, we first rewrite the function by multiplying and dividing by 7: Note that sin 7x 7 sin 7x ෇ 4x 4 7 sin x. ͩ ͪ sin 7x 7x Notice that as x l 0, we have 7x l 0, and so, by Equation 2 with ␪ ෇ 7x, lim xl0 Thus lim xl0 sin 7x sin͑7x͒ ෇ lim ෇1 7x l 0 7x 7x sin 7x 7 ෇ lim xl0 4 4x ෇ ͩ ͪ sin 7x 7x 7 sin 7x 7 7 lim ෇ ؒ1෇ 4 x l 0 7x 4 4 5E-03(pp 166-175) ❙❙❙❙ 174 1/17/06 2:07 PM Page 174 CHAPTER 3 DERIVATIVES EXAMPLE 5 Calculate lim x cot x. xl0 SOLUTION Here we divide numerator and denominator by x: x cos x x l 0 sin x lim cos x cos x xl0 ෇ lim ෇ x l 0 sin x sin x lim xl0 x x lim x cot x ෇ lim xl0 cos 0 1 ෇1 ෇ |||| 3.5 1–16 |||| Exercises Differentiate. 25. (a) Find an equation of the tangent line to the curve 1. f ͑x͒ ෇ x Ϫ 3 sin x 2. f ͑x͒ ෇ x sin x 3. y ෇ sin x ϩ 10 tan x 4. y ෇ 2 csc x ϩ 5 cos x 5. t͑t͒ ෇ t cos t 6. t͑t͒ ෇ 4 sec t ϩ tan t 7. h͑␪͒ ෇ ␪ csc ␪ Ϫ cot ␪ 8. y ෇ u͑a cos u ϩ b cot u͒ 3 9. y ෇ x cos x 10. y ෇ sec ␪ 11. f ͑␪ ͒ ෇ 1 ϩ sec ␪ 13. y ෇ ■ sin x x2 ■ ; ; 1 ϩ sin x x ϩ cos x ; 16. y ෇ x sin x cos x ■ ■ 17. Prove that ■ ■ ■ (b) Check to see that your answer to part (a) is reasonable by graphing both f and f Ј for 0 ഛ x ഛ 2␲. 29. For what values of x does the graph of f ͑x͒ ෇ x ϩ 2 sin x have a horizontal tangent? d ͑sec x͒ ෇ sec x tan x. dx 19. Prove that ■ d ͑csc x͒ ෇ Ϫcsc x cot x. dx 18. Prove that ■ (b) Check to see that your answer to part (a) is reasonable by graphing both f and f Ј for 0 Ͻ x Ͻ ␲. 28. (a) If f ͑x͒ ෇ sx sin x, find f Ј͑x͒. ; ■ y ෇ sec x Ϫ 2 cos x at the point ͑␲͞3, 1͒. (b) Illustrate part (a) by graphing the curve and the tangent line on the same screen. 27. (a) If f ͑x͒ ෇ 2x ϩ cot x, find f Ј͑x͒. 14. y ෇ csc ␪ ͑␪ ϩ cot ␪͒ ■ y ෇ x cos x at the point ͑␲, Ϫ␲͒. (b) Illustrate part (a) by graphing the curve and the tangent line on the same screen. 26. (a) Find an equation of the tangent line to the curve tan x Ϫ 1 12. y ෇ sec x 15. y ෇ sec ␪ tan ␪ ■ (by the continuity of cosine and Equation 2) d ͑cot x͒ ෇ Ϫcsc 2x. dx 30. Find the points on the curve y ෇ ͑cos x͒͑͞2 ϩ sin x͒ at which the tangent is horizontal. 31. A mass on a spring vibrates horizontally on a smooth level surface (see the figure). Its equation of motion is x͑t͒ ෇ 8 sin t, where t is in seconds and x in centimeters. (a) Find the velocity at time t. (b) Find the position and velocity of the mass at time t ෇ 2␲͞3. In what direction is it moving at that time? 20. Prove, using the definition of derivative, that if f ͑x͒ ෇ cos x, then f Ј͑x͒ ෇ Ϫsin x. 21–24 |||| Find an equation of the tangent line to the curve at the given point. 21. y ෇ tan x, ͑␲͞4, 1͒ 23. y ෇ x ϩ cos x, ■ ■ ■ ■ 22. y ෇ ͑1 ϩ x͒ cos x, ͑0, 1͒ ■ 24. y ෇ ■ ■ ■ ͑0, 1͒ 1 , sin x ϩ cos x ■ ■ equilibrium position ͑0, 1͒ ■ ■ 0 x x 5E-03(pp 166-175) 1/17/06 2:08 PM Page 175 SECTION 3.6 THE CHAIN RULE ; 32. An elastic band is hung on a hook and a mass is hung on the lower end of the band. When the mass is pulled downward and then released, it vibrates vertically. The equation of motion is s ෇ 2 cos t ϩ 3 sin t, t ജ 0, where s is measured in centimeters and t in seconds. (We take the positive direction to be downward.) (a) Find the velocity at time t. (b) Graph the velocity and position functions. (c) When does the mass pass through the equilibrium position for the first time? (d) How far from its equilibrium position does the mass travel? (e) When is the speed the greatest? 33. A ladder 10 ft long rests against a vertical wall. Let ␪ be the angle between the top of the ladder and the wall and let x be the distance from the bottom of the ladder to the wall. If the bottom of the ladder slides away from the wall, how fast does x change with respect to ␪ when ␪ ෇ ␲͞3? 43. lim ␪l0 ■ ■ sin ␪ ␪ ϩ tan ␪ ■ 44. lim xl1 ■ ■ ■ ■ ; sin͑x Ϫ 1͒ x2 ϩ x Ϫ 2 ■ ■ ■ ■ ■ (or familiar) identity. sin x (a) tan x ෇ cos x 1 (b) sec x ෇ cos x (c) sin x ϩ cos x ෇ 1 ϩ cot x csc x 46. A semicircle with diameter PQ sits on an isosceles triangle PQR to form a region shaped like an ice-cream cone, as shown in the figure. If A͑␪ ͒ is the area of the semicircle and B͑␪ ͒ is the area of the triangle, find lim by a force acting along a rope attached to the object. If the rope makes an angle ␪ with the plane, then the magnitude of the force is ␮W ␮ sin ␪ ϩ cos ␪ 175 45. Differentiate each trigonometric identity to obtain a new 34. An object with weight W is dragged along a horizontal plane F෇ ❙❙❙❙ ␪ l 0ϩ A͑␪ ͒ B͑␪ ͒ A(¨ ) P Q B(¨ ) where ␮ is a constant called the coefficient of friction. (a) Find the rate of change of F with respect to ␪. (b) When is this rate of change equal to 0? (c) If W ෇ 50 lb and ␮ ෇ 0.6, draw the graph of F as a function of ␪ and use it to locate the value of ␪ for which dF͞d␪ ෇ 0. Is the value consistent with your answer to part (b)? ¨ R 47. The figure shows a circular arc of length s and a chord of 35–44 |||| length d, both subtended by a central angle ␪. Find Find the limit. 35. lim sin 3x x 36. lim sin 4x sin 6x 37. lim tan 6t sin 2t 38. lim cos ␪ Ϫ 1 sin ␪ 39. lim sin͑cos ␪͒ sec ␪ 40. lim sin2 3t t2 41. lim cot 2x csc x 42. lim xl0 tl0 ␪l0 xl0 |||| 3.6 xl0 ␪l0 tl0 x l ␲͞4 lim ␪ l 0ϩ d s d s ¨ sin x Ϫ cos x cos 2x The Chain Rule Suppose you are asked to differentiate the function F͑x͒ ෇ sx 2 ϩ 1 The differentiation formulas you learned in the previous sections of this chapter do not enable you to calculate FЈ͑x͒. 5E-03(pp 176-185) 176 ❙❙❙❙ 1/17/06 1:58 PM Page 176 CHAPTER 3 DERIVATIVES |||| See Section 1.3 for a review of composite functions. Resources / Module 4 / Trigonometric Models / The Chain Rule Observe that F is a composite function. In fact, if we let y ෇ f ͑u͒ ෇ su and let u ෇ t͑x͒ ෇ x 2 ϩ 1, then we can write y ෇ F͑x͒ ෇ f ͑t͑x͒͒, that is, F ෇ f ‫ ؠ‬t. We know how to differentiate both f and t, so it would be useful to have a rule that tells us how to find the derivative of F ෇ f ‫ ؠ‬t in terms of the derivatives of f and t. It turns out that the derivative of the composite function f ‫ ؠ‬t is the product of the derivatives of f and t. This fact is one of the most important of the differentiation rules and is called the Chain Rule. It seems plausible if we interpret derivatives as rates of change. Regard du͞dx as the rate of change of u with respect to x, dy͞du as the rate of change of y with respect to u, and dy͞dx as the rate of change of y with respect to x. If u changes twice as fast as x and y changes three times as fast as u, then it seems reasonable that y changes six times as fast as x, and so we expect that dy dy du ෇ dx du dx The Chain Rule If f and t are both differentiable and F ෇ f ‫ ؠ‬t is the composite func- tion defined by F͑x͒ ෇ f ͑ t͑x͒͒, then F is differentiable and FЈ is given by the product FЈ͑x͒ ෇ f Ј͑t͑x͒͒tЈ͑x͒ In Leibniz notation, if y ෇ f ͑u͒ and u ෇ t͑x͒ are both differentiable functions, then dy du dy ෇ dx du dx Comments on the Proof of the Chain Rule Let ⌬u be the change in u corresponding to a change of ⌬x in x, that is, ⌬u ෇ t͑x ϩ ⌬x͒ Ϫ t͑x͒ Then the corresponding change in y is ⌬y ෇ f ͑u ϩ ⌬u͒ Ϫ f ͑u͒ It is tempting to write dy ⌬y ෇ lim ⌬x l 0 ⌬x dx ෇ lim ⌬y ⌬u ؒ ⌬u ⌬x ෇ lim ⌬y ⌬u ؒ lim ⌬x l 0 ⌬x ⌬u ෇ lim 1 ⌬y ⌬u ؒ lim ⌬x l 0 ⌬x ⌬u ⌬x l 0 ⌬x l 0 ⌬u l 0 ෇ (Note that ⌬u l 0 as ⌬x l 0 since t is continuous.) dy du du dx The only flaw in this reasoning is that in (1) it might happen that ⌬u ෇ 0 (even when 5E-03(pp 176-185) 1/17/06 1:58 PM Page 177 SECTION 3.6 THE CHAIN RULE ❙❙❙❙ 177 ⌬x 0) and, of course, we can’t divide by 0. Nonetheless, this reasoning does at least suggest that the Chain Rule is true. A full proof of the Chain Rule is given at the end of this section. The Chain Rule can be written either in the prime notation ͑ f ‫ ؠ‬t͒Ј͑x͒ ෇ f Ј͑t͑x͒͒tЈ͑x͒ 2 or, if y ෇ f ͑u͒ and u ෇ t͑x͒, in Leibniz notation: dy du dy ෇ dx du dx 3 Equation 3 is easy to remember because if dy͞du and du͞dx were quotients, then we could cancel du. Remember, however, that du has not been defined and du͞dx should not be thought of as an actual quotient. EXAMPLE 1 Find FЈ͑x͒ if F͑x͒ ෇ sx 2 ϩ 1. SOLUTION 1 (using Equation 2): At the beginning of this section we expressed F as F͑x͒ ෇ ͑ f ‫ ؠ‬t͒͑x͒ ෇ f ͑t͑x͒͒ where f ͑u͒ ෇ su and t͑x͒ ෇ x 2 ϩ 1. Since f Ј͑u͒ ෇ 1 uϪ1͞2 ෇ 2 1 2su and tЈ͑x͒ ෇ 2x FЈ͑x͒ ෇ f Ј͑t͑x͒͒tЈ͑x͒ we have ෇ 1 x ؒ 2x ෇ 2sx 2 ϩ 1 sx 2 ϩ 1 SOLUTION 2 (using Equation 3): If we let u ෇ x 2 ϩ 1 and y ෇ su, then FЈ͑x͒ ෇ ෇ dy du 1 ෇ ͑2x͒ du dx 2su 1 x ͑2x͒ ෇ 2sx 2 ϩ 1 sx 2 ϩ 1 When using Formula 3 we should bear in mind that dy͞dx refers to the derivative of y when y is considered as a function of x (called the derivative of y with respect to x), whereas dy͞du refers to the derivative of y when considered as a function of u (the derivative of y with respect to u). For instance, in Example 1, y can be considered as a function of x ( y ෇ sx 2 ϩ 1 ) and also as a function of u ( y ෇ su ). Note that dy x ෇ FЈ͑x͒ ෇ 2 ϩ 1 dx sx NOTE whereas dy 1 ෇ f Ј͑u͒ ෇ du 2su In using the Chain Rule we work from the outside to the inside. Formula 2 says that we differentiate the outer function f [at the inner function t͑x͒] and then we multiply by the derivative of the inner function. ■ d dx f ͑t͑x͒͒ outer function evaluated at inner function ෇ fЈ ͑t͑x͒͒ derivative of outer function evaluated at inner function ؒ tЈ͑x͒ derivative of inner function 5E-03(pp 176-185) 178 ❙❙❙❙ 1/17/06 1:58 PM Page 178 CHAPTER 3 DERIVATIVES EXAMPLE 2 Differentiate (a) y ෇ sin͑x 2 ͒ and (b) y ෇ sin2x. SOLUTION (a) If y ෇ sin͑x 2 ͒, then the outer function is the sine function and the inner function is the squaring function, so the Chain Rule gives dy d ෇ dx dx sin ͑x 2 ͒ outer function evaluated at inner function ෇ 2x cos͑x 2 ͒ ෇ cos ͑x 2 ͒ derivative of outer function ؒ evaluated at inner function 2x derivative of inner function (b) Note that sin2x ෇ ͑sin x͒2. Here the outer function is the squaring function and the inner function is the sine function. So dy d ෇ ͑sin x͒2 dx dx inner function |||| See Reference Page 2 or Appendix D. ෇ 2 ؒ derivative of outer function ͑sin x͒ ؒ evaluated at inner function cos x derivative of inner function The answer can be left as 2 sin x cos x or written as sin 2x (by a trigonometric identity known as the double-angle formula). In Example 2(a) we combined the Chain Rule with the rule for differentiating the sine function. In general, if y ෇ sin u, where u is a differentiable function of x, then, by the Chain Rule, dy dy du du ෇ ෇ cos u dx du dx dx d du ͑sin u͒ ෇ cos u dx dx Thus In a similar fashion, all of the formulas for differentiating trigonometric functions can be combined with the Chain Rule. Let’s make explicit the special case of the Chain Rule where the outer function f is a power function. If y ෇ ͓t͑x͔͒ n, then we can write y ෇ f ͑u͒ ෇ u n where u ෇ t͑x͒. By using the Chain Rule and then the Power Rule, we get dy dy du du ෇ ෇ nu nϪ1 ෇ n͓t͑x͔͒ nϪ1tЈ͑x͒ dx du dx dx 4 The Power Rule Combined with the Chain Rule If n is any real number and u ෇ t͑x͒ is differentiable, then d du ͑u n ͒ ෇ nu nϪ1 dx dx Alternatively, d ͓t͑x͔͒ n ෇ n͓t͑x͔͒ nϪ1 ؒ tЈ͑x͒ dx Notice that the derivative in Example 1 could be calculated by taking n ෇ 1 in Rule 4. 2 5E-03(pp 176-185) 1/17/06 1:59 PM Page 179 SECTION 3.6 THE CHAIN RULE ❙❙❙❙ EXAMPLE 3 Differentiate y ෇ ͑x 3 Ϫ 1͒100. SOLUTION Taking u ෇ t͑x͒ ෇ x 3 Ϫ 1 and n ෇ 100 in (4), we have dy d d ෇ ͑x 3 Ϫ 1͒100 ෇ 100͑x 3 Ϫ 1͒99 ͑x 3 Ϫ 1͒ dx dx dx ෇ 100͑x 3 Ϫ 1͒99 ؒ 3x 2 ෇ 300x 2͑x 3 Ϫ 1͒99 EXAMPLE 4 Find f Ј͑x͒ if f ͑x͒ ෇ SOLUTION First rewrite f : 1 . sx ϩ x ϩ 1 3 2 f ͑x͒ ෇ ͑x 2 ϩ x ϩ 1͒Ϫ1͞3. Thus f Ј͑x͒ ෇ Ϫ1 ͑x 2 ϩ x ϩ 1͒Ϫ4͞3 3 d ͑x 2 ϩ x ϩ 1͒ dx ෇ Ϫ1 ͑x 2 ϩ x ϩ 1͒Ϫ4͞3͑2x ϩ 1͒ 3 EXAMPLE 5 Find the derivative of the function t͑t͒ ෇ ͩ ͪ tϪ2 2t ϩ 1 9 SOLUTION Combining the Power Rule, Chain Rule, and Quotient Rule, we get ͩ ͪ ͩ ͪ ͩ ͪ tϪ2 2t ϩ 1 8 tЈ͑t͒ ෇ 9 d dt tϪ2 2t ϩ 1 tϪ2 2t ϩ 1 8 ෇9 ͑2t ϩ 1͒ ؒ 1 Ϫ 2͑t Ϫ 2͒ 45͑t Ϫ 2͒8 ෇ 2 ͑2t ϩ 1͒ ͑2t ϩ 1͒10 EXAMPLE 6 Differentiate y ෇ ͑2x ϩ 1͒5͑x 3 Ϫ x ϩ 1͒4. SOLUTION In this example we must use the Product Rule before using the Chain Rule: |||| The graphs of the functions y and yЈ in Example 6 are shown in Figure 1. Notice that yЈ is large when y increases rapidly and yЈ ෇ 0 when y has a horizontal tangent. So our answer appears to be reasonable. d d dy ෇ ͑2x ϩ 1͒5 ͑x 3 Ϫ x ϩ 1͒4 ϩ ͑x 3 Ϫ x ϩ 1͒4 ͑2x ϩ 1͒5 dx dx dx ෇ ͑2x ϩ 1͒5 ؒ 4͑x 3 Ϫ x ϩ 1͒3 d ͑x 3 Ϫ x ϩ 1͒ dx ϩ ͑x 3 Ϫ x ϩ 1͒4 ؒ 5͑2x ϩ 1͒4 10 d ͑2x ϩ 1͒ dx yª ෇ 4͑2x ϩ 1͒5͑x 3 Ϫ x ϩ 1͒3͑3x 2 Ϫ 1͒ ϩ 5͑x 3 Ϫ x ϩ 1͒4͑2x ϩ 1͒4 ؒ 2 _2 1 y _10 FIGURE 1 Noticing that each term has the common factor 2͑2x ϩ 1͒4͑x 3 Ϫ x ϩ 1͒3, we could factor it out and write the answer as dy ෇ 2͑2x ϩ 1͒4͑x 3 Ϫ x ϩ 1͒3͑17x 3 ϩ 6x 2 Ϫ 9x ϩ 3͒ dx 179 5E-03(pp 176-185) 180 ❙❙❙❙ 1/17/06 1:59 PM Page 180 CHAPTER 3 DERIVATIVES The reason for the name “Chain Rule” becomes clear when we make a longer chain by adding another link. Suppose that y ෇ f ͑u͒, u ෇ t͑x͒, and x ෇ h͑t͒, where f , t, and h are differentiable functions. Then, to compute the derivative of y with respect to t, we use the Chain Rule twice: dy dy dx dy du dx ෇ ෇ dt dx dt du dx dt EXAMPLE 7 If f ͑x͒ ෇ sin͑cos͑tan x͒͒, then f Ј͑x͒ ෇ cos͑cos͑tan x͒͒ d cos͑tan x͒ dx ෇ cos͑cos͑tan x͓͒͒Ϫsin͑tan x͔͒ d ͑tan x͒ dx ෇ Ϫcos͑cos͑tan x͒͒ sin͑tan x͒ sec2x Notice that the Chain Rule has been used twice. EXAMPLE 8 Differentiate y ෇ ssec x 3. SOLUTION Here the outer function is the square root function, the middle function is the secant function, and the inner function is the cubing function. So we have dy 1 d ෇ ͑sec x 3 ͒ 3 dx dx 2ssec x ෇ 1 d sec x 3 tan x 3 ͑x 3 ͒ 3 2ssec x dx ෇ 3x 2 sec x 3 tan x 3 2ssec x 3 How to Prove the Chain Rule Recall that if y ෇ f ͑x͒ and x changes from a to a ϩ ⌬x, we defined the increment of y as ⌬y ෇ f ͑a ϩ ⌬x͒ Ϫ f ͑a͒ According to the definition of a derivative, we have lim ⌬x l 0 ⌬y ෇ f Ј͑a͒ ⌬x So if we denote by ␧ the difference between the difference quotient and the derivative, we obtain lim ␧ ෇ lim ⌬x l 0 But ␧෇ ͩ ⌬x l 0 ⌬y Ϫ f Ј͑a͒ ⌬x ͪ ⌬y Ϫ f Ј͑a͒ ෇ f Ј͑a͒ Ϫ f Ј͑a͒ ෇ 0 ⌬x ? ⌬y ෇ f Ј͑a͒ ⌬x ϩ ␧ ⌬x 5E-03(pp 176-185) 1/17/06 2:00 PM Page 181 SECTION 3.6 THE CHAIN RULE ❙❙❙❙ 181 If we define ␧ to be 0 when ⌬x ෇ 0, then ␧ becomes a continuous function of ⌬x. Thus, for a differentiable function f, we can write ⌬y ෇ f Ј͑a͒ ⌬x ϩ ␧ ⌬x 5 where ␧ l 0 as ⌬x l 0 and ␧ is a continuous function of ⌬x. This property of differentiable functions is what enables us to prove the Chain Rule. Proof of the Chain Rule Suppose u ෇ t͑x͒ is differentiable at a and y ෇ f ͑u͒ is differentiable at b ෇ t͑a͒. If ⌬x is an increment in x and ⌬u and ⌬y are the corresponding increments in u and y, then we can use Equation 5 to write ⌬u ෇ tЈ͑a͒ ⌬x ϩ ␧1 ⌬x ෇ ͓tЈ͑a͒ ϩ ␧1 ͔ ⌬x 6 where ␧1 l 0 as ⌬x l 0. Similarly ⌬y ෇ f Ј͑b͒ ⌬u ϩ ␧2 ⌬u ෇ ͓ f Ј͑b͒ ϩ ␧2 ͔ ⌬u 7 where ␧2 l 0 as ⌬u l 0. If we now substitute the expression for ⌬u from Equation 6 into Equation 7, we get ⌬y ෇ ͓ f Ј͑b͒ ϩ ␧2 ͔͓tЈ͑a͒ ϩ ␧1 ͔ ⌬x ⌬y ෇ ͓ f Ј͑b͒ ϩ ␧2 ͔͓tЈ͑a͒ ϩ ␧1 ͔ ⌬x so As ⌬x l 0, Equation 6 shows that ⌬u l 0. So both ␧1 l 0 and ␧2 l 0 as ⌬x l 0. Therefore dy ⌬y ෇ lim ෇ lim ͓ f Ј͑b͒ ϩ ␧2 ͔͓tЈ͑a͒ ϩ ␧1 ͔ ⌬x l 0 ⌬x ⌬x l 0 dx ෇ f Ј͑b͒tЈ͑a͒ ෇ f Ј͑t͑a͒͒tЈ͑a͒ This proves the Chain Rule. |||| 3.6 Exercises Write the composite function in the form f ͑ t͑x͒͒. [Identify the inner function u ෇ t͑x͒ and the outer function y ෇ f ͑u͒.] Then find the derivative dy͞dx. 1–6 13. y ෇ cos͑a 3 ϩ x 3 ͒ 14. y ෇ a 3 ϩ cos3x 15. y ෇ cot͑x͞2͒ |||| 16. y ෇ 4 sec 5x 17. t͑x͒ ෇ ͑1 ϩ 4x͒5͑3 ϩ x Ϫ x 2 ͒8 1. y ෇ sin 4x 2. y ෇ s4 ϩ 3x 3. y ෇ ͑1 Ϫ x 2 ͒10 4. y ෇ tan͑sin x͒ 18. h͑t͒ ෇ ͑t 4 Ϫ 1͒3͑t 3 ϩ 1͒4 5. y ෇ ssin x 6. y ෇ sin sx 19. y ෇ ͑2x Ϫ 5͒4͑8x 2 Ϫ 5͒Ϫ3 ■ ■ 7–42 |||| ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ 21. y ෇ x 3 cos nx Find the derivative of the function. 7. F͑x͒ ෇ ͑x ϩ 4x͒ 3 9. F͑x͒ ෇ s1 ϩ 2x ϩ x 4 1 11. t͑t͒ ෇ 4 ͑t ϩ 1͒3 8. F͑x͒ ෇ ͑x Ϫ x ϩ 1͒ 7 2 3 3 20. y ෇ ͑x 2 ϩ 1͒ sx 2 ϩ 2 22. y ෇ x sin sx 23. y ෇ sin͑x cos x͒ 24. f ͑x͒ ෇ x s7 Ϫ 3x 26. G͑ y͒ ෇ ͑ y Ϫ 1͒4 ͑ y 2 ϩ 2y͒5 3 10. f ͑x͒ ෇ ͑1 ϩ x ͒ 4 2͞3 3 12. f ͑t͒ ෇ s1 ϩ tan t 25. F͑z͒ ෇ ͱ zϪ1 zϩ1 5E-03(pp 176-185) 182 ❙❙❙❙ 1/17/06 2:01 PM Page 182 CHAPTER 3 DERIVATIVES 28. y ෇ 29. y ෇ tan͑cos x͒ 30. y ෇ sin2x cos x 31. y ෇ sins1 ϩ x 2 32. y ෇ tan 2͑3␪͒ 33. y ෇ ͑1 ϩ cos2x͒6 34. y ෇ x sin 35. y ෇ sec 2x ϩ tan2x 36. y ෇ cot͑x 2 ͒ ϩ cot 2 x 37. y ෇ cot 2͑sin ␪͒ 38. y ෇ sin͑sin͑sin x͒͒ 39. y ෇ sx ϩ sx 40. y ෇ 41. y ෇ sin(tan ssin x ) 53. Suppose that F͑x͒ ෇ f ͑ t͑x͒͒ and t͑3͒ ෇ 6, tЈ͑3͒ ෇ 4, cos ␲ x sin ␲ x ϩ cos ␲ x 42. y ෇ scos͑sin x͒ 27. y ෇ ■ ■ 43–46 r sr ϩ 1 2 ■ ■ f Ј͑3͒ ෇ 2, and f Ј͑6͒ ෇ 7. Find FЈ͑3͒. 54. Suppose that w ෇ u ‫ ؠ‬v and u͑0͒ ෇ 1, v͑0͒ ෇ 2, uЈ͑0͒ ෇ 3, uЈ͑2͒ ෇ 4, vЈ͑0͒ ෇ 5, and vЈ͑2͒ ෇ 6. Find wЈ͑0͒. 55. A table of values for f , t, f Ј, and tЈ is given. 1 x x ■ ■ ■ ■ ■ ■ ■ Find an equation of the tangent line to the curve at the given point. |||| 43. y ෇ ͑1 ϩ 2x͒10, 45. y ෇ sin͑sin x͒, ■ ■ ■ ■ ͑0, 1͒ 44. y ෇ sin x ϩ sin2 x, ͑␲, 0͒ ■ 46. y ෇ s5 ϩ x 2, ■ ■ ■ ■ ■ f Ј͑x͒ tЈ͑x͒ 3 1 7 2 8 2 4 5 7 6 7 9 (a) If h͑x͒ ෇ f ͑t͑x͒͒, find hЈ͑1͒. (b) If H͑x͒ ෇ t͑ f ͑x͒͒, find HЈ͑1͒. 2 ■ t͑x͒ 1 2 3 sx ϩ sx ϩ sx f ͑x͒ ͑0, 0͒ (a) If F͑x͒ ෇ f ͑ f ͑x͒͒, find FЈ͑2͒. (b) If G͑x͒ ෇ t͑t͑x͒͒, find GЈ͑3͒. 57. If f and t are the functions whose graphs are shown, let u͑x͒ ෇ f ͑ t͑x͒͒, v͑x͒ ෇ t͑ f ͑x͒͒, and w͑x͒ ෇ t͑ t͑x͒͒. Find each derivative, if it exists. If it does not exist, explain why. (a) uЈ͑1͒ (b) vЈ͑1͒ (c) wЈ͑1͒ ͑2, 3͒ ■ 56. Let f and t be the functions in Exercise 55. ■ y 47. (a) Find an equation of the tangent line to the curve ; f y ෇ tan͑␲ x 2͞4͒ at the point ͑1, 1͒. (b) Illustrate part (a) by graphing the curve and the tangent line on the same screen. Խ Խ 48. (a) The curve y ෇ x ͞s2 Ϫ x 2 is called a bullet-nose curve. ; Find an equation of the tangent line to this curve at the point ͑1, 1͒. (b) Illustrate part (a) by graphing the curve and the tangent line on the same screen. 49. (a) If f ͑x͒ ෇ s1 Ϫ x 2͞x, find f Ј͑x͒. ; g 1 0 x 1 58. If f is the function whose graph is shown, let h͑x͒ ෇ f ͑ f ͑x͒͒ and t͑x͒ ෇ f ͑x 2 ͒. Use the graph of f to estimate the value of each derivative. (a) hЈ͑2͒ (b) tЈ͑2͒ (b) Check to see that your answer to part (a) is reasonable by comparing the graphs of f and f Ј. y y=ƒ ; 50. The function f ͑x͒ ෇ sin͑x ϩ sin 2x͒, 0 ഛ x ഛ ␲, arises in applications to frequency modulation (FM) synthesis. (a) Use a graph of f produced by a graphing device to make a rough sketch of the graph of f Ј. (b) Calculate f Ј͑x͒ and use this expression, with a graphing device, to graph f Ј. Compare with your sketch in part (a). 51. Find all points on the graph of the function f ͑x͒ ෇ 2 sin x ϩ sin x 1 0 x 1 59. Use the table to estimate the value of hЈ͑0.5͒, where h͑x͒ ෇ f ͑ t͑x͒͒. 2 at which the tangent line is horizontal. 52. Find the x-coordinates of all points on the curve y ෇ sin 2x Ϫ 2 sin x at which the tangent line is horizontal. x 0 0.1 0.2 0.3 0.4 0.5 0.6 f ͑x͒ 12.6 14.8 18.4 23.0 25.9 27.5 29.1 t͑x͒ 0.58 0.40 0.37 0.26 0.17 0.10 0.05 5E-03(pp 176-185) 1/17/06 2:01 PM Page 183 SECTION 3.6 THE CHAIN RULE 60. If t͑x͒ ෇ f ͑ f ͑x͒͒, use the table to estimate the value of tЈ͑1͒. x 0.0 0.5 1.0 1.5 2.0 1.7 1.8 2.0 2.4 3.1 4.4 61. Suppose f is differentiable on ‫ .ޒ‬Let F͑x͒ ෇ f ͑cos x͒ and G͑x͒ ෇ cos͑ f ͑x͒͒. Find expressions for (a) FЈ͑x͒ and (b) GЈ͑x͒. 62. Suppose f is differentiable on ‫ ޒ‬and ␣ is a real number. Let F͑x͒ ෇ f ͑x ␣ ͒ and G͑x͒ ෇ ͓ f ͑x͔͒ ␣. Find expressions for (a) FЈ͑x͒ and (b) GЈ͑x͒. 69. Computer algebra systems have commands that differentiate CAS 70. (a) Use a CAS to differentiate the function 63. Suppose L is a function such that LЈ͑x͒ ෇ 1͞x for x Ͼ 0. Find an expression for the derivative of each function. (a) f ͑x͒ ෇ L͑x 4 ͒ (b) t͑x͒ ෇ L͑4x͒ (c) F͑x͒ ෇ ͓L͑x͔͒ 4 (d) G͑x͒ ෇ L͑1͞x͒ 64. Let r͑x͒ ෇ f ͑ t͑h͑x͒͒͒, where h͑1͒ ෇ 2, t͑2͒ ෇ 3, hЈ͑1͒ ෇ 4, tЈ͑2͒ ෇ 5, and f Ј͑3͒ ෇ 6. Find rЈ͑1͒. 65. The displacement of a particle on a vibrating string is given by the equation f ͑x͒ ෇ 1 where s is measured in centimeters and t in seconds. Find the velocity of the particle after t seconds. 66. If the equation of motion of a particle is given by s ෇ A cos͑␻ t ϩ ␦͒, the particle is said to undergo simple harmonic motion. (a) Find the velocity of the particle at time t. (b) When is the velocity 0? x4 Ϫ x ϩ 1 x4 ϩ x ϩ 1 and to simplify the result. (b) Where does the graph of f have horizontal tangents? (c) Graph f and f Ј on the same screen. Are the graphs consistent with your answer to part (b)? (a) The derivative of an even function is an odd function. (b) The derivative of an odd function is an even function. 72. Use the Chain Rule and the Product Rule to give an alternative proof of the Quotient Rule. [Hint: Write f ͑x͒͞t͑x͒ ෇ f ͑x͓͒ t͑x͔͒ Ϫ1.] 73. (a) If n is a positive integer, prove that d ͑sinn x cos nx͒ ෇ n sinnϪ1x cos͑n ϩ 1͒x dx 67. A Cepheid variable star is a star whose brightness alternately increases and decreases. The most easily visible such star is Delta Cephei, for which the interval between times of maximum brightness is 5.4 days. The average brightness of this star is 4.0 and its brightness changes by Ϯ0.35. In view of these data, the brightness of Delta Cephei at time t, where t is measured in days, has been modeled by the function B͑t͒ ෇ 4.0 ϩ 0.35 sin͑2␲ t͞5.4͒ (a) Find the rate of change of the brightness after t days. (b) Find, correct to two decimal places, the rate of increase after one day. 68. In Example 4 in Section 1.3 we arrived at a model for the length of daylight (in hours) in Philadelphia on the t th day of the year: ͫ ͱ 71. Use the Chain Rule to prove the following. s͑t͒ ෇ 10 ϩ 4 sin͑10␲ t͒ L͑t͒ ෇ 12 ϩ 2.8 sin 183 functions, but the form of the answer may not be convenient and so further commands may be necessary to simplify the answer. (a) Use a CAS to find the derivative in Example 5 and compare with the answer in that example. Then use the simplify command and compare again. (b) Use a CAS to find the derivative in Example 6. What happens if you use the simplify command? What happens if you use the factor command? Which form of the answer would be best for locating horizontal tangents? 2.5 f ͑x͒ CAS ❙❙❙❙ ͬ 2␲ ͑t Ϫ 80͒ 365 Use this model to compare how the number of hours of daylight is increasing in Philadelphia on March 21 and May 21. (b) Find a formula for the derivative of y ෇ cosn x cos nx that is similar to the one in part (a). 74. Suppose y ෇ f ͑x͒ is a curve that always lies above the x-axis and never has a horizontal tangent, where f is differentiable everywhere. For what value of y is the rate of change of y 5 with respect to x eighty times the rate of change of y with respect to x ? 75. Use the Chain Rule to show that if ␪ is measured in degrees, then d ␲ ͑sin ␪͒ ෇ cos ␪ d␪ 180 (This gives one reason for the convention that radian measure is always used when dealing with trigonometric functions in calculus: The differentiation formulas would not be as simple if we used degree measure.) 5E-03(pp 176-185) 184 ❙❙❙❙ 1/17/06 2:02 PM Page 184 CHAPTER 3 DERVIATIVES Խ Խ Խ Խ 76. (a) Write x ෇ sx 2 and use the Chain Rule to show that (c) If t͑x͒ ෇ sin x , find tЈ͑x͒ and sketch the graphs of t and tЈ. Where is t not differentiable? d x ෇ dx 77. Suppose P and Q are polynomials and n is a positive integer. Խ Խ Խ x x Խ Խ Use mathematical induction to prove that the nth derivative of the rational function f ͑x͒ ෇ P͑x͒͞Q͑x͒ can be written as a rational function with denominator ͓Q͑x͔͒ nϩ1. In other words, there is a polynomial An such that f ͑n͒͑x͒ ෇ An͑x͓͒͞Q͑x͔͒ nϩ1. Խ (b) If f ͑x͒ ෇ sin x , find f Ј͑x͒ and sketch the graphs of f and f Ј. Where is f not differentiable? |||| 3.7 Implicit Differentiation The functions that we have met so far can be described by expressing one variable explicitly in terms of another variable—for example, y ෇ sx 3 ϩ 1 or y ෇ x sin x or, in general, y ෇ f ͑x͒. Some functions, however, are defined implicitly by a relation between x and y such as 1 x 2 ϩ y 2 ෇ 25 2 x 3 ϩ y 3 ෇ 6xy or In some cases it is possible to solve such an equation for y as an explicit function (or several functions) of x. For instance, if we solve Equation 1 for y, we get y ෇ Ϯs25 Ϫ x 2, so two of the functions determined by the implicit Equation l are f ͑x͒ ෇ s25 Ϫ x 2 and t͑x͒ ෇ Ϫs25 Ϫ x 2. The graphs of f and t are the upper and lower semicircles of the circle x 2 ϩ y 2 ෇ 25. (See Figure 1.) y 0 FIGURE 1 (a) ≈+¥=25 y x 0 25-≈ (b) ƒ=œ„„„„„„ y x 0 x 25-≈ (c) ©=_ œ„„„„„„ It’s not easy to solve Equation 2 for y explicitly as a function of x by hand. (A computer algebra system has no trouble, but the expressions it obtains are very complicated.) Nonetheless, (2) is the equation of a curve called the folium of Descartes shown in Figure 2 and it implicitly defines y as several functions of x. The graphs of three such functions are shown in Figure 3. When we say that f is a function defined implicitly by Equation 2, we mean that the equation x 3 ϩ ͓ f ͑x͔͒ 3 ෇ 6x f ͑x͒ is true for all values of x in the domain of f . 5E-03(pp 176-185) 1/17/06 2:02 PM Page 185 SECTION 3.7 IMPLICIT DIFFERENTIATION y y y ❙❙❙❙ 185 y ˛+Á=6xy 0 x FIGURE 2 The folium of Descartes 0 0 x x 0 x FIGURE 3 Graphs of three functions defined by the folium of Descartes Fortunately, we don’t need to solve an equation for y in terms of x in order to find the derivative of y. Instead we can use the method of implicit differentiation. This consists of differentiating both sides of the equation with respect to x and then solving the resulting equation for yЈ. In the examples and exercises of this section it is always assumed that the given equation determines y implicitly as a differentiable function of x so that the method of implicit differentiation can be applied. EXAMPLE 1 dy . dx (b) Find an equation of the tangent to the circle x 2 ϩ y 2 ෇ 25 at the point ͑3, 4͒. (a) If x 2 ϩ y 2 ෇ 25, find SOLUTION 1 (a) Differentiate both sides of the equation x 2 ϩ y 2 ෇ 25: d d ͑x 2 ϩ y 2 ͒ ෇ ͑25͒ dx dx d d ͑x 2 ͒ ϩ ͑y 2 ͒ ෇ 0 dx dx Remembering that y is a function of x and using the Chain Rule, we have d dy dy d ͑y 2 ͒ ෇ ͑y 2 ͒ ෇ 2y dx dy dx dx Thus 2x ϩ 2y dy ෇0 dx Now we solve this equation for dy͞dx : x dy ෇Ϫ dx y (b) At the point ͑3, 4͒ we have x ෇ 3 and y ෇ 4, so dy 3 ෇Ϫ dx 4 An equation of the tangent to the circle at ͑3, 4͒ is therefore y Ϫ 4 ෇ Ϫ3 ͑x Ϫ 3͒ 4 or 3x ϩ 4y ෇ 25 5E-03(pp 186-195) 186 ❙❙❙❙ 1/17/06 1:54 PM Page 186 CHAPTER 3 DERIVATIVES SOLUTION 2 (b) Solving the equation x 2 ϩ y 2 ෇ 25, we get y ෇ Ϯs25 Ϫ x 2. The point ͑3, 4͒ lies on the upper semicircle y ෇ s25 Ϫ x 2 and so we consider the function f ͑x͒ ෇ s25 Ϫ x 2. Differentiating f using the Chain Rule, we have f Ј͑x͒ ෇ 1 ͑25 Ϫ x 2 ͒Ϫ1͞2 2 d ͑25 Ϫ x 2 ͒ dx ෇ 1 ͑25 Ϫ x 2 ͒Ϫ1͞2͑Ϫ2x͒ ෇ Ϫ 2 f Ј͑3͒ ෇ Ϫ So x s25 Ϫ x 2 3 3 ෇Ϫ 2 4 s25 Ϫ 3 and, as in Solution 1, an equation of the tangent is 3x ϩ 4y ෇ 25. NOTE 1 Example 1 illustrates that even when it is possible to solve an equation explicitly for y in terms of x, it may be easier to use implicit differentiation. ■ The expression dy͞dx ෇ Ϫx͞y gives the derivative in terms of both x and y. It is correct no matter which function y is determined by the given equation. For instance, for y ෇ f ͑x͒ ෇ s25 Ϫ x 2 we have NOTE 2 ■ dy x x ෇Ϫ ෇Ϫ dx y s25 Ϫ x 2 whereas for y ෇ t͑x͒ ෇ Ϫs25 Ϫ x 2 we have dy x x x ෇Ϫ ෇Ϫ ෇ dx y Ϫs25 Ϫ x 2 s25 Ϫ x 2 EXAMPLE 2 (a) Find yЈ if x 3 ϩ y 3 ෇ 6xy. (b) Find the tangent to the folium of Descartes x 3 ϩ y 3 ෇ 6xy at the point ͑3, 3͒. (c) At what points on the curve is the tangent line horizontal? SOLUTION (a) Differentiating both sides of x 3 ϩ y 3 ෇ 6xy with respect to x, regarding y as a function of x, and using the Chain Rule on the y 3 term and the Product Rule on the 6xy term, we get 3x 2 ϩ 3y 2 yЈ ෇ 6y ϩ 6xyЈ or We now solve for yЈ : x 2 ϩ y 2 yЈ ෇ 2y ϩ 2xyЈ y 2 yЈ Ϫ 2xyЈ ෇ 2y Ϫ x 2 ͑y 2 Ϫ 2x͒yЈ ෇ 2y Ϫ x 2 yЈ ෇ 2y Ϫ x 2 y 2 Ϫ 2x (b) When x ෇ y ෇ 3, yЈ ෇ 2 ؒ 3 Ϫ 32 ෇ Ϫ1 32 Ϫ 2 ؒ 3 5E-03(pp 186-195) 1/17/06 1:54 PM Page 187 SECTION 3.7 IMPLICIT DIFFERENTIATION y 187 and a glance at Figure 4 confirms that this is a reasonable value for the slope at ͑3, 3͒. So an equation of the tangent to the folium at ͑3, 3͒ is (3, 3) y Ϫ 3 ෇ Ϫ1͑x Ϫ 3͒ 0 ❙❙❙ x or xϩy෇6 (c) The tangent line is horizontal if yЈ ෇ 0. Using the expression for yЈ from part (a), we see that yЈ ෇ 0 when 2y Ϫ x 2 ෇ 0. Substituting y ෇ 1 x 2 in the equation of the curve, 2 we get x 3 ϩ ( 1 x 2)3 ෇ 6x ( 1 x 2) 2 2 FIGURE 4 which simplifies to x 6 ෇ 16x 3. So either x ෇ 0 or x 3 ෇ 16. If x ෇ 16 1͞3 ෇ 2 4͞3, then y ෇ 1 ͑2 8͞3 ͒ ෇ 2 5͞3. Thus, the tangent is horizontal at (0, 0) and at ͑2 4͞3, 2 5͞3 ͒, which 2 is approximately (2.5198, 3.1748). Looking at Figure 5, we see that our answer is reasonable. 4 NOTE 3 There is a formula for the three roots of a cubic equation that is like the quadratic formula but much more complicated. If we use this formula (or a computer algebra system) to solve the equation x 3 ϩ y 3 ෇ 6xy for y in terms of x, we get three functions determined by the equation: 4 0 ■ 3 3 y ෇ f ͑x͒ ෇ sϪ1 x 3 ϩ s1 x 6 Ϫ 8x 3 ϩ sϪ1 x 3 Ϫ s1 x 6 Ϫ 8x 3 2 4 2 4 FIGURE 5 and [ ( 3 3 y ෇ 1 Ϫf ͑x͒ Ϯ sϪ3 sϪ1 x 3 ϩ s1 x 6 Ϫ 8x 3 Ϫ sϪ1 x 3 Ϫ s1 x 6 Ϫ 8x 3 2 2 4 2 4 |||| The Norwegian mathematician Niels Abel proved in 1824 that no general formula can be given for the roots of a fifth-degree equation in terms of radicals. Later the French mathematician Evariste Galois proved that it is impossible to find a general formula for the roots of an nth-degree equation (in terms of algebraic operations on the coefficients) if n is any integer larger than 4. )] (These are the three functions whose graphs are shown in Figure 3.) You can see that the method of implicit differentiation saves an enormous amount of work in cases such as this. Moreover, implicit differentiation works just as easily for equations such as y 5 ϩ 3x 2 y 2 ϩ 5x 4 ෇ 12 for which it is impossible to find a similar expression for y in terms of x. EXAMPLE 3 Find yЈ if sin͑x ϩ y͒ ෇ y 2 cos x. SOLUTION Differentiating implicitly with respect to x and remembering that y is a function of x, we get cos͑x ϩ y͒ ؒ ͑1 ϩ yЈ͒ ෇ 2yyЈ cos x ϩ y 2͑Ϫsin x͒ (Note that we have used the Chain Rule on the left side and the Product Rule and Chain Rule on the right side.) If we collect the terms that involve yЈ, we get 2 cos͑x ϩ y͒ ϩ y 2 sin x ෇ ͑2y cos x͒yЈ Ϫ cos͑x ϩ y͒ ؒ yЈ _2 2 _2 FIGURE 6 So yЈ ෇ y 2 sin x ϩ cos͑x ϩ y͒ 2y cos x Ϫ cos͑x ϩ y͒ Figure 6, drawn with the implicit-plotting command of a computer algebra system, shows part of the curve sin͑x ϩ y͒ ෇ y 2 cos x. As a check on our calculation, notice that yЈ ෇ Ϫ1 when x ෇ y ෇ 0 and it appears from the graph that the slope is approximately Ϫ1 at the origin. 5E-03(pp 186-195) 188 ❙❙❙❙ 1/17/06 1:54 PM Page 188 CHAPTER 3 DERIVATIVES Orthogonal Trajectories Two curves are called orthogonal if at each point of intersection their tangent lines are perpendicular. In the next example we use implicit differentiation to show that two families of curves are orthogonal trajectories of each other; that is, every curve in one family is orthogonal to every curve in the other family. Orthogonal families arise in several areas of physics. For example, the lines of force in an electrostatic field are orthogonal to the lines of constant potential. In thermodynamics, the isotherms (curves of equal temperature) are orthogonal to the flow lines of heat. In aerodynamics, the streamlines (curves of direction of airflow) are orthogonal trajectories of the velocity-equipotential curves. y EXAMPLE 4 The equation ≈-¥ =k xy ෇ c 3 c 0 represents a family of hyperbolas. (Different values of the constant c give different hyperbolas. See Figure 7.) The equation xy=c x 0 x2 Ϫ y2 ෇ k 4 k 0 represents another family of hyperbolas with asymptotes y ෇ Ϯx. Show that every curve in the family (3) is orthogonal to every curve in the family (4); that is, the families are orthogonal trajectories of each other. SOLUTION Implicit differentiation of Equation 3 gives FIGURE 7 x 5 dy ϩy෇0 dx dy y ෇Ϫ dx x so Implicit differentiation of Equation 4 gives 2x Ϫ 2y 6 dy ෇0 dx so dy x ෇ dx y From (5) and (6) we see that at any point of intersection of curves from each family, the slopes of the tangents are negative reciprocals of each other. Therefore, the curves intersect at right angles; that is, they are orthogonal. |||| 3.7 1–4 Exercises 5–20 |||| (a) Find yЈ by implicit differentiation. (b) Solve the equation explicitly for y and differentiate to get yЈ in terms of x. (c) Check that your solutions to parts (a) and (b) are consistent by substituting the expression for y into your solution for part (a). 1. xy ϩ 2x ϩ 3x 2 ෇ 4 3. ■ ■ Find dy͞dx by implicit differentiation. 5. x ϩ y 2 ෇ 1 6. x 2 Ϫ y 2 ෇ 1 7. x 3 ϩ x 2 y ϩ 4y 2 ෇ 6 8. x 2 Ϫ 2xy ϩ y 3 ෇ c 9. x 2 y ϩ xy 2 ෇ 3x 10. y 5 ϩ x 2 y 3 ෇ 1 ϩ x 4 y 2. 4x 2 ϩ 9y 2 ෇ 36 ■ ■ ■ ■ 11. x 2 y 2 ϩ x sin y ෇ 4 12. 1 ϩ x ෇ sin͑xy 2 ͒ 4. sx ϩ sy ෇ 4 1 1 ϩ ෇1 x y ■ |||| 2 13. 4 cos x sin y ෇ 1 14. y sin͑x 2 ͒ ෇ x sin͑ y 2 ͒ 15. tan͑x͞y͒ ෇ x ϩ y 16. sx ϩ y ෇ 1 ϩ x 2 y 2 ■ ■ ■ ■ ■ 5E-03(pp 186-195) 1/17/06 1:55 PM Page 189 SECTION 3.7 IMPLICIT DIFFERENTIATION y 1 ϩ x2 17. sxy ෇ 1 ϩ x 2 y 18. tan͑x Ϫ y͒ ෇ 19. xy ෇ cot͑xy͒ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ capabilities of computer algebra systems. (a) Graph the curve with equation y͑ y 2 Ϫ 1͒͑ y Ϫ 2͒ ෇ x͑x Ϫ 1͒͑x Ϫ 2͒ ■ 21. If 1 ϩ f ͑x͒ ϩ x ͓ f ͑x͔͒ ෇ 0 and f ͑1͒ ෇ 2, find f Ј͑1͒. 2 3 At how many points does this curve have horizontal tangents? Estimate the x-coordinates of these points. (b) Find equations of the tangent lines at the points (0, 1) and (0, 2). (c) Find the exact x-coordinates of the points in part (a). (d) Create even more fanciful curves by modifying the equation in part (a). 22. If t͑x͒ ϩ x sin t͑x͒ ෇ x 2 and t͑1͒ ෇ 0, find tЈ͑1͒. 23–24 |||| Regard y as the independent variable and x as the dependent variable and use implicit differentiation to find dx͞dy. 23. y 4 ϩ x 2 y 2 ϩ yx 4 ෇ y ϩ 1 ■ ■ 25–30 ■ ■ ■ 24. ͑x 2 ϩ y 2 ͒2 ෇ ax 2 y ■ ■ ■ ■ ■ ■ ■ CAS Use implicit differentiation to find an equation of the tangent line to the curve at the given point. |||| 2 26. x ϩ 2xy Ϫ y ϩ x ෇ 2, 2 34. (a) The curve with equation 2y 3 ϩ y 2 Ϫ y 5 ෇ x 4 Ϫ 2x 3 ϩ x 2 ͑1, 1͒ (ellipse) 25. x ϩ xy ϩ y ෇ 3, 2 has been likened to a bouncing wagon. Use a computer algebra system to graph this curve and discover why. (b) At how many points does this curve have horizontal tangent lines? Find the x-coordinates of these points. ͑1, 2͒ (hyperbola) 2 27. x 2 ϩ y 2 ෇ ͑2x 2 ϩ 2y 2 Ϫ x͒2 28. x 2͞3 ϩ y 2͞3 ෇ 4 (0, 1 ) 2 (Ϫ3 s3, 1) (cardioid) 189 33. Fanciful shapes can be created by using the implicit plotting 20. sin x ϩ cos y ෇ sin x cos y CAS ❙❙❙❙ (astroid) 35. Find the points on the lemniscate in Exercise 29 where the tangent is horizontal. y y 36. Show by implicit differentiation that the tangent to the ellipse y2 x2 ෇1 2 ϩ a b2 x 0 29. 2͑x 2 ϩ y 2 ͒2 ෇ 25͑x 2 Ϫ y 2 ͒ x 8 at the point ͑x 0 , y 0 ͒ is y0 y x0 x ϩ 2 ෇1 a2 b 30. y 2͑ y 2 Ϫ 4͒ ෇ x 2͑x 2 Ϫ 5͒ (0, Ϫ2) (devil’s curve) (3, 1) (lemniscate) 37. Find an equation of the tangent line to the hyperbola x2 y2 ෇1 2 Ϫ a b2 y y at the point ͑x 0 , y 0 ͒. x 0 38. Show that the sum of the x- and y-intercepts of any tangent x line to the curve sx ϩ sy ෇ sc is equal to c. 39. Show, using implicit differentiation, that any tangent line at ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ 31. (a) The curve with equation y ෇ 5x Ϫ x is called a 2 ; 4 2 kampyle of Eudoxus. Find an equation of the tangent line to this curve at the point ͑1, 2͒. (b) Illustrate part (a) by graphing the curve and the tangent line on a common screen. (If your graphing device will graph implicitly defined curves, then use that capability. If not, you can still graph this curve by graphing its upper and lower halves separately.) 32. (a) The curve with equation y 2 ෇ x 3 ϩ 3x 2 is called the ; a point P to a circle with center O is perpendicular to the radius OP. ■ Tschirnhausen cubic. Find an equation of the tangent line to this curve at the point ͑1, Ϫ2͒. (b) At what points does this curve have a horizontal tangent? (c) Illustrate parts (a) and (b) by graphing the curve and the tangent lines on a common screen. 40. The Power Rule can be proved using implicit differentiation for the case where n is a rational number, n ෇ p͞q, and y ෇ f ͑x͒ ෇ x n is assumed beforehand to be a differentiable function. If y ෇ x p͞q, then y q ෇ x p. Use implicit differentiation to show that yЈ ෇ 41–42 |||| Show that the given curves are orthogonal. 41. 2x ϩ y 2 ෇ 3, 2 42. x 2 Ϫ y 2 ෇ 5, ■ ■ p ͑ p͞q͒Ϫ1 x q ■ x ෇ y2 4x 2 ϩ 9y 2 ෇ 72 ■ ■ ■ ■ ■ ■ ■ ■ ■ 5E-03(pp 186-195) 190 ❙❙❙❙ 1/17/06 1:55 PM Page 190 CHAPTER 3 DERIVATIVES 46. x 2 ϩ y 2 ෇ ax, 43. Contour lines on a map of a hilly region are curves that join points with the same elevation. A ball rolling down a hill follows a curve of steepest descent, which is orthogonal to the contour lines. Given the contour map of a hill in the figure, sketch the paths of balls that start at positions A and B. 47. y ෇ cx 2, x 2 ϩ 2y 2 ෇ k 48. y ෇ ax 3, A 800 600 x 2 ϩ y 2 ෇ by x 2 ϩ 3y 2 ෇ b ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ 49. The equation x 2 Ϫ xy ϩ y 2 ෇ 3 represents a “rotated ellipse,” 400 that is, an ellipse whose axes are not parallel to the coordinate axes. Find the points at which this ellipse crosses the x-axis and show that the tangent lines at these points are parallel. 300 200 B 50. (a) Where does the normal line to the ellipse 400 ; 44. TV meteorologists often present maps showing pressure fronts. Such maps display isobars—curves along which the air pressure is constant. Consider the family of isobars shown in the figure. Sketch several members of the family of orthogonal trajectories of the isobars. Given the fact that wind blows from regions of high air pressure to regions of low air pressure, what does the orthogonal family represent? x 2 Ϫ xy ϩ y 2 ෇ 3 at the point ͑Ϫ1, 1͒ intersect the ellipse a second time? (See page 156 for the definition of a normal line.) (b) Illustrate part (a) by graphing the ellipse and the normal line. 51. Find all points on the curve x 2 y 2 ϩ xy ෇ 2 where the slope of the tangent line is Ϫ1. 52. Find equations of both the tangent lines to the ellipse x 2 ϩ 4y 2 ෇ 36 that pass through the point ͑12, 3͒. 53. The figure shows a lamp located three units to the right of the y-axis and a shadow created by the elliptical region x 2 ϩ 4y 2 ഛ 5. If the point ͑Ϫ5, 0͒ is on the edge of the shadow, how far above the x-axis is the lamp located? y ? 45–48 Show that the given families of curves are orthogonal trajectories of each other. Sketch both families of curves on the same axes. |||| 45. x 2 ϩ y 2 ෇ r 2, |||| 3.8 0 _5 3 x ≈+4¥=5 ax ϩ by ෇ 0 Higher Derivatives If f is a differentiable function, then its derivative f Ј is also a function, so f Ј may have a derivative of its own, denoted by ͑ f Ј͒Ј ෇ f Љ. This new function f Љ is called the second derivative of f because it is the derivative of the derivative of f . Using Leibniz notation, we write the second derivative of y ෇ f ͑x͒ as d dx Another notation is f Љ͑x͒ ෇ D 2 f ͑x͒. ͩ ͪ dy dx ෇ d2y dx 2 5E-03(pp 186-195) 1/17/06 1:55 PM Page 191 SECTION 3.8 HIGHER DERIVATIVES In Module 3.8A you can see how changing the coefficients of a polynomial f affects the appearance of the graphs of f , f Ј, and f Љ. ❙❙❙❙ 191 EXAMPLE 1 If f ͑x͒ ෇ x cos x, find and interpret f Љ͑x͒. SOLUTION Using the Product Rule, we have f Ј͑x͒ ෇ x d d ͑cos x͒ ϩ cos x ͑x͒ dx dx ෇ Ϫx sin x ϩ cos x To find f Љ͑x͒ we differentiate f Ј͑x͒: f Љ͑x͒ ෇ 3 d ͑Ϫx sin x ϩ cos x͒ dx f· ෇ Ϫx fª f _3 d d d ͑sin x͒ ϩ sin x ͑Ϫx͒ ϩ ͑cos x͒ dx dx dx ෇ Ϫx cos x Ϫ sin x Ϫ sin x 3 ෇ Ϫx cos x Ϫ 2 sin x _3 FIGURE 1 The graphs of ƒ=x cos x and its first and second derivatives The graphs of f, f Ј, and f Љ are shown in Figure 1. We can interpret f Љ͑x͒ as the slope of the curve y ෇ f Ј͑x͒ at the point ͑x, f Ј͑x͒͒. In other words, it is the rate of change of the slope of the original curve y ෇ f ͑x͒. Notice from Figure 1 that f Љ͑x͒ ෇ 0 whenever y ෇ f Ј͑x͒ has a horizontal tangent. Also, f Љ͑x͒ is positive when y ෇ f Ј͑x͒ has positive slope and negative when y ෇ f Ј͑x͒ has negative slope. So the graphs serve as a check on our calculations. In general, we can interpret a second derivative as a rate of change of a rate of change. The most familiar example of this is acceleration, which we define as follows. If s ෇ s͑t͒ is the position function of an object that moves in a straight line, we know that its first derivative represents the velocity v͑t͒ of the object as a function of time: v͑t͒ ෇ sЈ͑t͒ ෇ ds dt The instantaneous rate of change of velocity with respect to time is called the acceleration a͑t͒ of the object. Thus, the acceleration function is the derivative of the velocity function and is therefore the second derivative of the position function: a͑t͒ ෇ vЈ͑t͒ ෇ sЉ͑t͒ or, in Leibniz notation, a෇ dv d 2s ෇ 2 dt dt EXAMPLE 2 The position of a particle is given by the equation s ෇ f ͑t͒ ෇ t 3 Ϫ 6t 2 ϩ 9t where t is measured in seconds and s in meters. (a) Find the acceleration at time t. What is the acceleration after 4 s? (b) Graph the position, velocity, and acceleration functions for 0 ഛ t ഛ 5. (c) When is the particle speeding up? When is it slowing down? 5E-03(pp 186-195) 192 ❙❙❙❙ 1/17/06 1:56 PM Page 192 CHAPTER 3 DERIVATIVES SOLUTION (a) The velocity function is the derivative of the position function: s ෇ f ͑t͒ ෇ t 3 Ϫ 6t 2 ϩ 9t v͑t͒ ෇ ds ෇ 3t 2 Ϫ 12t ϩ 9 dt The acceleration is the derivative of the velocity function: a͑t͒ ෇ a͑4͒ ෇ 6͑4͒ Ϫ 12 ෇ 12 m͞s2 |||| The units for acceleration are meters per second per second, written as m/s2. 25 √ a s 0 5 _12 d 2s dv ෇ ෇ 6t Ϫ 12 dt 2 dt (b) Figure 2 shows the graphs of s, v, and a. (c) The particle speeds up when the velocity is positive and increasing (v and a are both positive) and also when the velocity is negative and decreasing (v and a are both negative). In other words, the particle speeds up when the velocity and acceleration have the same sign. (The particle is pushed in the same direction it is moving.) From Figure 2 we see that this happens when 1 Ͻ t Ͻ 2 and when t Ͼ 3. The particle slows down when v and a have opposite signs, that is, when 0 ഛ t Ͻ 1 and when 2 Ͻ t Ͻ 3. Figure 3 summarizes the motion of the particle. FIGURE 2 a √ In Module 3.8B you can see an animation of Figure 3 with an expression for s that you can choose yourself. 5 s 0 1 t _5 forward FIGURE 3 slows down backward speeds up forward slows down speeds up The third derivative f ٞ is the derivative of the second derivative: f ٞ ෇ ͑ f Љ͒Ј. So f ٞ͑x͒ can be interpreted as the slope of the curve y ෇ f Љ͑x͒ or as the rate of change of f Љ͑x͒. If y ෇ f ͑x͒, then alternative notations for the third derivative are yٞ ෇ f ٞ͑x͒ ෇ d dx ͩ ͪ d2y dx 2 ෇ d 3y ෇ D 3f ͑x͒ dx 3 The process can be continued. The fourth derivative f ٣ is usually denoted by f ͑4͒. In general, the nth derivative of f is denoted by f ͑n͒ and is obtained from f by differentiating n times. If y ෇ f ͑x͒, we write dny y ͑n͒ ෇ f ͑n͒͑x͒ ෇ ෇ D n f ͑x͒ dx n 5E-03(pp 186-195) 1/17/06 1:56 PM Page 193 SECTION 3.8 HIGHER DERIVATIVES ❙❙❙❙ 193 We can interpret the third derivative physically in the case where the function is the position function s ෇ s͑t͒ of an object that moves along a straight line. Because sٞ ෇ ͑sЉ͒Ј ෇ aЈ, the third derivative of the position function is the derivative of the acceleration function and is called the jerk: j෇ da d 3s ෇ 3 dt dt Thus, the jerk j is the rate of change of acceleration. It is aptly named because a large jerk means a sudden change in acceleration, which causes an abrupt movement in a vehicle. y ෇ x 3 Ϫ 6x 2 Ϫ 5x ϩ 3 EXAMPLE 3 If yЈ ෇ 3x 2 Ϫ 12x Ϫ 5 then yЉ ෇ 6x Ϫ 12 yٞ ෇ 6 y ͑4͒ ෇ 0 and in fact y ͑n͒ ෇ 0 for all n ജ 4. EXAMPLE 4 If f ͑x͒ ෇ 1 , find f ͑n͒͑x͒. x SOLUTION f ͑x͒ ෇ 1 ෇ xϪ1 x f Ј͑x͒ ෇ ϪxϪ2 ෇ Ϫ1 x2 f Љ͑x͒ ෇ ͑Ϫ2͒͑Ϫ1͒x Ϫ3 ෇ 2 x3 f ٞ͑x͒ ෇ Ϫ3 ؒ 2 ؒ 1 ؒ x Ϫ4 f ͑4͒͑x͒ ෇ 4 ؒ 3 ؒ 2 ؒ 1 ؒ xϪ5 |||| The factor ͑Ϫ1͒ n occurs in the formula for f ͑n͒͑x͒ because we introduce another negative sign every time we differentiate. Since the successive values of ͑Ϫ1͒n are Ϫ1, 1, Ϫ1, 1, Ϫ1, 1, . . . , the presence of ͑Ϫ1͒ n indicates that the sign changes with each successive derivative. f ͑5͒͑x͒ ෇ Ϫ5 ؒ 4 ؒ 3 ؒ 2 ؒ 1 ؒ xϪ6 ෇ Ϫ5! xϪ6 . . . f ͑n͒͑x͒ ෇ ͑Ϫ1͒n n͑n Ϫ 1͒͑n Ϫ 2͒ и и и 2 ؒ 1 ؒ xϪ͑nϩ1͒ or f ͑n͒͑x͒ ෇ ͑Ϫ1͒n n! x nϩ1 Here we have used the factorial symbol n! for the product of the first n positive integers. n! ෇ 1 ؒ 2 ؒ 3 ؒ и и и ؒ ͑n Ϫ 1͒ ؒ n The following example shows how to find the second derivative of a function that is defined implicitly. 5E-03(pp 186-195) 194 ❙❙❙❙ 1/17/06 1:56 PM Page 194 CHAPTER 3 DERIVATIVES EXAMPLE 5 Find yЉ if x 4 ϩ y 4 ෇ 16. SOLUTION Differentiating the equation implicitly with respect to x, we get 4x 3 ϩ 4y 3 yЈ ෇ 0 |||| Figure 4 shows the graph of the curve x 4 ϩ y 4 ෇ 16 of Example 5. Notice that it’s a stretched and flattened version of the circle x 2 ϩ y 2 ෇ 4. For this reason it’s sometimes called a fat circle. It starts out very steep on the left but quickly becomes very flat. This can be seen from the expression yЈ ෇ Ϫ y ͩͪ x3 x ෇Ϫ y3 y 3 Solving for yЈ gives yЈ ෇ Ϫ 1 To find yЉ we differentiate this expression for yЈ using the Quotient Rule and remembering that y is a function of x : yЉ ෇ x$+y$=16 2 x3 y3 d dx ෇Ϫ ͩ ͪ Ϫ x3 y3 ෇Ϫ y 3 ͑d͞dx͒͑x 3 ͒ Ϫ x 3 ͑d͞dx͒͑y 3 ͒ ͑y 3 ͒2 y 3 ؒ 3x 2 Ϫ x 3͑3y 2 yЈ͒ y6 If we now substitute Equation 1 into this expression, we get 0 2 x ͩ ͪ 3x 2 y 3 Ϫ 3x 3 y 2 Ϫ yЉ ෇ Ϫ ෇Ϫ x3 y3 y 3͑x 2 y 4 ϩ x 6 ͒ 3x 2͑y 4 ϩ x 4 ͒ ෇Ϫ 7 y y7 But the values of x and y must satisfy the original equation x 4 ϩ y 4 ෇ 16. So the answer simplifies to 3x 2͑16͒ x2 yЉ ෇ Ϫ ෇ Ϫ48 7 7 y y FIGURE 4 EXAMPLE 6 Find D 27 cos x. SOLUTION The first few derivatives of cos x are as follows: D cos x ෇ Ϫsin x |||| Look for a pattern. D 2 cos x ෇ Ϫcos x D 3 cos x ෇ sin x D 4 cos x ෇ cos x D 5 cos x ෇ Ϫsin x We see that the successive derivatives occur in a cycle of length 4 and, in particular, D n cos x ෇ cos x whenever n is a multiple of 4. Therefore D 24 cos x ෇ cos x and, differentiating three more times, we have D 27 cos x ෇ sin x 5E-03(pp 186-195) 1/17/06 1:56 PM Page 195 ❙❙❙❙ SECTION 3.8 HIGHER DERIVATIVES 195 We have seen that one application of second and third derivatives occurs in analyzing the motion of objects using acceleration and jerk. We will investigate another application of second derivatives in Exercise 60 and in Section 4.3, where we show how knowledge of f Љ gives us information about the shape of the graph of f. In Chapter 12 we will see how second and higher derivatives enable us to represent functions as sums of infinite series. |||| 3.8 Exercises 1. The figure shows the graphs of f , f Ј, and f Љ. Identify each 4. The figure shows the graphs of four functions. One is the position function of a car, one is the velocity of the car, one is its acceleration, and one is its jerk. Identify each curve, and explain your choices. curve, and explain your choices. y a y b d a b x c c 0 t 2. The figure shows graphs of f, f Ј, f Љ, and f ٞ. Identify each curve, and explain your choices. 5–20 |||| Find the first and second derivatives of the function. 5. f ͑x͒ ෇ x 5 ϩ 6x 2 Ϫ 7x a b c d 6. f ͑t͒ ෇ t 8 Ϫ 7t 6 ϩ 2t 4 7. y ෇ cos 2␪ y 8. y ෇ ␪ sin ␪ 9. F͑t͒ ෇ ͑1 Ϫ 7t͒6 x 11. h͑u͒ ෇ 10. t͑x͒ ෇ 1 Ϫ 4u 1 ϩ 3u 2x ϩ 1 xϪ1 12. H͑s͒ ෇ a ss ϩ 13. h͑x͒ ෇ sx 2 ϩ 1 y a 20. h͑x͒ ෇ ■ b c ■ ■ ■ ■ ■ ■ ■ 4x sx ϩ 1 xϩ3 x 2 ϩ 2x ■ ■ ■ ■ 21. (a) If f ͑x͒ ෇ 2 cos x ϩ sin x, find f Ј͑x͒ and f Љ͑x͒. 2 ; 0 18. t͑s͒ ෇ s 2 cos s 19. t͑␪͒ ෇ ␪ csc ␪ tion function of a car, one is the velocity of the car, and one is its acceleration. Identify each curve, and explain your choices. 16. y ෇ 17. H͑t͒ ෇ tan 3t 3. The figure shows the graphs of three functions. One is the posi- 14. y ෇ x n 15. y ෇ ͑x 3 ϩ 1͒2͞3 b ss t (b) Check to see that your answers to part (a) are reasonable by comparing the graphs of f , f Ј, and f Љ. 22. (a) If f ͑x͒ ෇ x͑͞x 2 ϩ 1͒, find f Ј͑x͒ and f Љ͑x͒. ; (b) Check to see that your answers to part (a) are reasonable by comparing the graphs of f , f Ј, and f Љ. 5E-03(pp 196-205) 196 ❙❙❙❙ 1/17/06 1:47 PM CHAPTER 3 DERIVATIVES 23–24 |||| Find yٞ. 23. y ෇ s2x ϩ 3 ■ Page 196 ■ ■ 24. y ෇ ■ ■ ■ ■ velocity and acceleration of the car. What is the acceleration at t ෇ 10 seconds? x 2x Ϫ 1 ■ s ■ ■ ■ ■ 25. If f ͑t͒ ෇ t cos t, find f ٞ͑0͒. 26. If t͑x͒ ෇ s5 Ϫ 2x, find t ٞ͑2͒. 100 27. If f ͑␪ ͒ ෇ cot ␪, find f ٞ͑␲͞6͒. 0 10 20 t 28. If t͑x͒ ෇ sec x, find tٞ͑␲͞4͒. 29–32 |||| (b) Use the acceleration curve from part (a) to estimate the jerk at t ෇ 10 seconds. What are the units for jerk? Find yЉ by implicit differentiation. 29. 9x ϩ y 2 ෇ 9 30. sx ϩ sy ෇ 1 31. x ϩ y ෇ 1 32. x ϩ y ෇ a 2 3 ■ 3 ■ 33–37 ■ |||| 4 ■ ■ ■ 37. f ͑x͒ ෇ 38–40 ■ |||| The equation of motion is given for a particle, where s is in meters and t is in seconds. Find (a) the velocity and acceleration as functions of t, (b) the acceleration after 1 second, and (c) the acceleration at the instants when the velocity is 0. ■ ■ ■ 43. s ෇ 2t 3 Ϫ 15t 2 ϩ 36t ϩ 2, 44. s ෇ 2t Ϫ 3t Ϫ 12t, 3 1 5x Ϫ 1 34. f ͑x͒ ෇ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ 47. s ෇ t 4 Ϫ 4t 3 ϩ 2 ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ 48. s ෇ 2t 3 Ϫ 9t 2 ■ ■ ■ ■ ■ ■ ■ ■ 49. A particle moves according to a law of motion 38. D74 sin x 39. D103 cos 2x ■ ■ |||| An equation of motion is given, where s is in meters and t in seconds. Find (a) the times at which the acceleration is 0 and (b) the displacement and velocity at these times. |||| ■ ■ 0ഛtഛ2 tജ0 2 47–48 Find the given derivative by finding the first few derivatives and observing the pattern that occurs. ■ ■ tജ0 tജ0 46. s ෇ 2t Ϫ 7t ϩ 4t ϩ 1, 3 1 3x 3 ■ 2 45. s ෇ sin͑␲ t͞6͒ ϩ cos͑␲ t͞6͒, 36. f ͑x͒ ෇ sx 35. f ͑x͒ ෇ ͑1 ϩ x͒Ϫ1 ■ ■ 43–46 4 Find a formula for f ͑n͒͑x͒. 33. f ͑x͒ ෇ x n ■ ■ 4 40. D 35 x sin x ■ ■ ■ ■ ■ ■ ■ ■ ■ ; 41. A car starts from rest and the graph of its position function is shown in the figure, where s is measured in feet and t in seconds. Use it to graph the velocity and estimate the acceleration at t ෇ 2 seconds from the velocity graph. Then sketch a graph of the acceleration function. 50. A particle moves along the x-axis, its position at time t given s ; 120 100 80 60 40 20 0 s ෇ f ͑t͒ ෇ t 3 Ϫ 12t 2 ϩ 36t, t ജ 0, where t is measured in seconds and s in meters. (a) Find the acceleration at time t and after 3 s. (b) Graph the position, velocity, and acceleration functions for 0 ഛ t ഛ 8. (c) When is the particle speeding up? When is it slowing down? by x͑t͒ ෇ t͑͞1 ϩ t 2 ͒, t ജ 0, where t is measured in seconds and x in meters. (a) Find the acceleration at time t. When is it 0? (b) Graph the position, velocity, and acceleration functions for 0 ഛ t ഛ 4. (c) When is the particle speeding up? When is it slowing down? 51. A mass attached to a vertical spring has position function given 1 t 42. (a) The graph of a position function of a car is shown, where s is measured in feet and t in seconds. Use it to graph the by y͑t͒ ෇ A sin ␻ t, where A is the amplitude of its oscillations and ␻ is a constant. (a) Find the velocity and acceleration as functions of time. (b) Show that the acceleration is proportional to the displacement y. (c) Show that the speed is a maximum when the acceleration is 0. 5E-03(pp 196-205) 1/17/06 1:47 PM Page 197 APPLIED PROJECT WHERE SHOULD A PILOT START DESCENT? 52. A particle moves along a straight line with displacement s͑t͒, velocity v͑t͒, and acceleration a͑t͒. Show that 1 1 1 ෇ Ϫ x ͑x ϩ 1͒ x xϩ1 to compute the derivatives much more easily. Then find an expression for f ͑n͒͑x͒. This method of splitting up a fraction in terms of simpler fractions, called partial fractions, will be pursued further in Section 8.4. Explain the difference between the meanings of the derivatives dv͞dt and dv͞ds. 53. Find a second-degree polynomial P such that P͑2͒ ෇ 5, PЈ͑2͒ ෇ 3, and P Љ͑2͒ ෇ 2. CAS 62. (a) Use a computer algebra system to compute f ٞ , where 54. Find a third-degree polynomial Q such that Q͑1͒ ෇ 1, QЈ͑1͒ ෇ 3, QЉ͑1͒ ෇ 6, and Qٞ͑1͒ ෇ 12. f ͑x͒ ෇ 55. The equation yЉ ϩ yЈ Ϫ 2y ෇ sin x is called a differential 63. Suppose p is a positive integer such that the function f is 56. Find constants A, B, and C such that the function y ෇ Ax 2 ϩ Bx ϩ C satisfies the differential equation yЉ ϩ yЈ Ϫ 2y ෇ x 2. 57–59 |||| The function t is a twice differentiable function. Find f Љ in terms of t, tЈ, and tЉ. 57. f ͑x͒ ෇ xt͑x 2 ͒ ■ p-times differentiable and f ͑ p͒ ෇ f . Using mathematical induction, show that f is in fact n-times differentiable for every positive integer n and that each of its higher derivatives f ͑n͒ equals one of the p functions f , f Ј, f Љ, . . . , f ͑ pϪ1͒. 64. (a) If F͑x͒ ෇ f ͑x͒t͑x͒, where f and t have derivatives of all orders, show that F Љ ෇ f Љt ϩ 2 f ЈtЈ ϩ ftЉ t͑x͒ x (b) Find similar formulas for Fٞ and F ͑4͒. (c) Guess a formula for F ͑n͒. 59. f ͑x͒ ෇ t (sx ) ■ 7x ϩ 17 2x 2 Ϫ 7x Ϫ 4 (b) Find a much simpler expression for f ٞ by first splitting f into partial fractions. [In Maple, use the command convert(f,parfrac,x); in Mathematica, use Apart[f].] equation because it involves an unknown function y and its derivatives yЈ and yЉ. Find constants A and B such that the function y ෇ A sin x ϩ B cos x satisfies this equation. (Differential equations will be studied in detail in Chapter 10.) ■ 197 (b) Use the identity dv a͑t͒ ෇ v͑t͒ ds 58. f ͑x͒ ෇ ❙❙❙❙ ■ ■ ■ ■ ■ ■ ■ ■ 5 3 ; 60. If f ͑x͒ ෇ 3x Ϫ 10x ϩ 5, graph both f and f Љ. On what intervals is f Љ͑x͒ Ͼ 0? On those intervals, how is the graph of f related to its tangent lines? What about the intervals where f Љ͑x͒ Ͻ 0? 61. (a) Compute the first few derivatives of the function f ͑x͒ ෇ 1͑͞x 2 ϩ x͒ until you see that the computations are becoming algebraically unmanageable. ■ 65. If y ෇ f ͑u͒ and u ෇ t͑x͒, where f and t are twice differen- tiable functions, show that d2y d2y 2 ෇ dx du 2 ͩ ͪ du dx 2 ϩ dy d 2u du dx 2 66. If y ෇ f ͑u͒ and u ෇ t͑x͒, where f and t possess third derivatives, find a formula for d 3 y͞dx 3 similar to the one given in Exercise 65. APPLIED PROJECT Where Should a Pilot Start Descent? y y=P (x) 0 ᐉ An approach path for an aircraft landing is shown in the figure and satisfies the following conditions: (i) The cruising altitude is h when descent starts at a horizontal distance ᐉ from touchdown at the origin. (ii) The pilot must maintain a constant horizontal speed v throughout descent. (iii) The absolute value of the vertical acceleration should not exceed a constant k (which is much less than the acceleration due to gravity). h x 1. Find a cubic polynomial P͑x͒ ෇ ax 3 ϩ bx 2 ϩ cx ϩ d that satisfies condition (i) by imposing suitable conditions on P͑x͒ and PЈ͑x͒ at the start of descent and at touchdown. 5E-03(pp 196-205) 198 ❙❙❙❙ 1/17/06 1:47 PM Page 198 CHAPTER 3 DERIVATIVES 2. Use conditions (ii) and (iii) to show that 6h v 2 ഛk ᐉ2 3. Suppose that an airline decides not to allow vertical acceleration of a plane to exceed k ෇ 860 mi͞h2. If the cruising altitude of a plane is 35,000 ft and the speed is 300 mi͞h, how far away from the airport should the pilot start descent? ; 4. Graph the approach path if the conditions stated in Problem 3 are satisfied. APPLIED PROJECT Building a Better Roller Coaster L¡ P Suppose you are asked to design the first ascent and drop for a new roller coaster. By studying photographs of your favorite coasters, you decide to make the slope of the ascent 0.8 and the slope of the drop Ϫ1.6. You decide to connect these two straight stretches y ෇ L 1͑x͒ and y ෇ L 2 ͑x͒ with part of a parabola y ෇ f ͑x͒ ෇ a x 2 ϩ bx ϩ c, where x and f ͑x͒ are measured in feet. For the track to be smooth there can’t be abrupt changes in direction, so you want the linear segments L 1 and L 2 to be tangent to the parabola at the transition points P and Q. (See the figure.) To simplify the equations, you decide to place the origin at P. f Q L™ 1. (a) Suppose the horizontal distance between P and Q is 100 ft. Write equations in a, b, and ; c that will ensure that the track is smooth at the transition points. (b) Solve the equations in part (a) for a, b, and c to find a formula for f ͑x͒. (c) Plot L 1, f , and L 2 to verify graphically that the transitions are smooth. (d) Find the difference in elevation between P and Q. 2. The solution in Problem 1 might look smooth, but it might not feel smooth because the piecewise defined function [consisting of L 1͑x͒ for x Ͻ 0, f ͑x͒ for 0 ഛ x ഛ 100, and L 2͑x͒ for x Ͼ 100] doesn’t have a continuous second derivative. So you decide to improve the design by using a quadratic function q͑x͒ ෇ ax 2 ϩ bx ϩ c only on the interval 10 ഛ x ഛ 90 and connecting it to the linear functions by means of two cubic functions: t͑x͒ ෇ k x 3 ϩ lx 2 ϩ m x ϩ n 0 ഛ x Ͻ 10 h͑x͒ ෇ px ϩ qx ϩ rx ϩ s 90 Ͻ x ഛ 100 3 CAS |||| 3.9 2 (a) Write a system of equations in 11 unknowns that ensure that the functions and their first two derivatives agree at the transition points. (b) Solve the equations in part (a) with a computer algebra system to find formulas for q͑x͒, t͑x͒, and h͑x͒. (c) Plot L 1, t, q, h, and L 2, and compare with the plot in Problem 1(c). Related Rates Explore an expanding balloon interactively. Resources / Module 5 / Related Rates / Start of Related Rates If we are pumping air into a balloon, both the volume and the radius of the balloon are increasing and their rates of increase are related to each other. But it is much easier to measure directly the rate of increase of the volume than the rate of increase of the radius. In a related rates problem the idea is to compute the rate of change of one quantity in terms of the rate of change of another quantity (which may be more easily measured). The 5E-03(pp 196-205) 1/17/06 1:47 PM Page 199 SECTION 3.9 RELATED RATES ❙❙❙❙ 199 procedure is to find an equation that relates the two quantities and then use the Chain Rule to differentiate both sides with respect to time. EXAMPLE 1 Air is being pumped into a spherical balloon so that its volume increases at a rate of 100 cm3͞s. How fast is the radius of the balloon increasing when the diameter is 50 cm? SOLUTION We start by identifying two things: |||| According to the Principles of Problem Solving discussed on page 58, the first step is to understand the problem. This includes reading the problem carefully, identifying the given and the unknown, and introducing suitable notation. the given information: the rate of increase of the volume of air is 100 cm3͞s and the unknown: the rate of increase of the radius when the diameter is 50 cm In order to express these quantities mathematically, we introduce some suggestive notation: Let V be the volume of the balloon and let r be its radius. The key thing to remember is that rates of change are derivatives. In this problem, the volume and the radius are both functions of the time t. The rate of increase of the volume with respect to time is the derivative dV͞dt, and the rate of increase of the radius is dr͞dt. We can therefore restate the given and the unknown as follows: Given: Unknown: |||| The second stage of problem solving is to think of a plan for connecting the given and the unknown. dV ෇ 100 cm3͞s dt dr dt when r ෇ 25 cm In order to connect dV͞dt and dr͞dt, we first relate V and r by the formula for the volume of a sphere: V ෇ 4 ␲r 3 3 In order to use the given information, we differentiate each side of this equation with respect to t. To differentiate the right side, we need to use the Chain Rule: dV dV dr dr ෇ ෇ 4␲ r 2 dt dr dt dt Now we solve for the unknown quantity: |||| Notice that, although dV͞dt is constant, dr͞dt is not constant. dr 1 dV ෇ dt 4␲r 2 dt If we put r ෇ 25 and dV͞dt ෇ 100 in this equation, we obtain dr 1 1 ෇ 2 100 ෇ dt 4␲ ͑25͒ 25␲ The radius of the balloon is increasing at the rate of 1͑͞25␲͒ cm͞s. 5E-03(pp 196-205) 200 ❙❙❙❙ 1/17/06 1:47 PM Page 200 CHAPTER 3 DERIVATIVES How high will a fireman get while climbing a sliding ladder? Resources / Module 5 / Related Rates / Start of the Sliding Fireman wall EXAMPLE 2 A ladder 10 ft long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 1 ft͞s, how fast is the top of the ladder sliding down the wall when the bottom of the ladder is 6 ft from the wall? SOLUTION We first draw a diagram and label it as in Figure 1. Let x feet be the distance from the bottom of the ladder to the wall and y feet the distance from the top of the ladder to the ground. Note that x and y are both functions of t (time). We are given that dx͞dt ෇ 1 ft͞s and we are asked to find dy͞dt when x ෇ 6 ft (see Figure 2). In this problem, the relationship between x and y is given by the Pythagorean Theorem: 10 y x 2 ϩ y 2 ෇ 100 Differentiating each side with respect to t using the Chain Rule, we have x ground 2x FIGURE 1 dy dx ϩ 2y ෇0 dt dt and solving this equation for the desired rate, we obtain dy dt dy x dx ෇Ϫ dt y dt =? When x ෇ 6, the Pythagorean Theorem gives y ෇ 8 and so, substituting these values and dx͞dt ෇ 1, we have y dy 6 3 ෇ Ϫ ͑1͒ ෇ Ϫ ft͞s dt 8 4 x dx dt =1 FIGURE 2 The fact that dy͞dt is negative means that the distance from the top of the ladder to the ground is decreasing at a rate of 3 ft͞s. In other words, the top of the ladder is sliding 4 down the wall at a rate of 3 ft͞s. 4 EXAMPLE 3 A water tank has the shape of an inverted circular cone with base radius 2 m and height 4 m. If water is being pumped into the tank at a rate of 2 m3͞min, find the rate at which the water level is rising when the water is 3 m deep. 2 r 4 SOLUTION We first sketch the cone and label it as in Figure 3. Let V , r, and h be the volume of the water, the radius of the surface, and the height at time t, where t is measured in minutes. We are given that dV͞dt ෇ 2 m3͞min and we are asked to find dh͞dt when h is 3 m. The quantities V and h are related by the equation h FIGURE 3 V ෇ 1 ␲ r 2h 3 but it is very useful to express V as a function of h alone. In order to eliminate r, we use the similar triangles in Figure 3 to write r 2 ෇ h 4 and the expression for V becomes V෇ r෇ ͩͪ 1 h ␲ 3 2 2 h෇ h 2 ␲ 3 h 12 5E-03(pp 196-205) 1/17/06 1:47 PM Page 201 SECTION 3.9 RELATED RATES ❙❙❙❙ 201 Now we can differentiate each side with respect to t : dV ␲ 2 dh ෇ h dt 4 dt dh 4 dV ෇ dt ␲ h 2 dt so Substituting h ෇ 3 m and dV͞dt ෇ 2 m3͞min, we have dh 4 8 ෇ ؒ2෇ dt ␲ ͑3͒2 9␲ The water level is rising at a rate of 8͑͞9␲͒ Ϸ 0.28 m͞min. |||| Look back: What have we learned from Examples 1–3 that will help us solve future problems? Strategy It is useful to recall some of the problem-solving principles from page 58 and adapt them to related rates in light of our experience in Examples 1–3: 1. Read the problem carefully. 2. Draw a diagram if possible. 3. Introduce notation. Assign symbols to all quantities that are functions of time. | WARNING: A common error is to substitute the given numerical information (for quantities that vary with time) too early. This should be done only after the differentiation. (Step 7 follows Step 6.) For instance, in Example 3 we dealt with general values of h until we finally substituted h ෇ 3 at the last stage. (If we had put h ෇ 3 earlier, we would have gotten dV͞dt ෇ 0, which is clearly wrong.) 4. Express the given information and the required rate in terms of derivatives. 5. Write an equation that relates the various quantities of the problem. If necessary, use the geometry of the situation to eliminate one of the variables by substitution (as in Example 3). 6. Use the Chain Rule to differentiate both sides of the equation with respect to t. 7. Substitute the given information into the resulting equation and solve for the unknown rate. The following examples are further illustrations of the strategy. EXAMPLE 4 Car A is traveling west at 50 mi͞h and car B is traveling north at 60 mi͞h. Both are headed for the intersection of the two roads. At what rate are the cars approaching each other when car A is 0.3 mi and car B is 0.4 mi from the intersection? C y B x z A SOLUTION We draw Figure 4, where C is the intersection of the roads. At a given time t, let x be the distance from car A to C, let y be the distance from car B to C, and let z be the distance between the cars, where x, y, and z are measured in miles. We are given that dx͞dt ෇ Ϫ50 mi͞h and dy͞dt ෇ Ϫ60 mi͞h. (The derivatives are negative because x and y are decreasing.) We are asked to find dz͞dt. The equation that relates x, y, and z is given by the Pythagorean Theorem: z2 ෇ x 2 ϩ y 2 FIGURE 4 Differentiating each side with respect to t, we have 2z dz dx dy ෇ 2x ϩ 2y dt dt dt dz 1 ෇ dt z ͩ x dx dy ϩy dt dt ͪ 5E-03(pp 196-205) 202 ❙❙❙❙ 1/17/06 1:48 PM Page 202 CHAPTER 3 DERIVATIVES When x ෇ 0.3 mi and y ෇ 0.4 mi, the Pythagorean Theorem gives z ෇ 0.5 mi, so dz 1 ෇ ͓0.3͑Ϫ50͒ ϩ 0.4͑Ϫ60͔͒ dt 0.5 ෇ Ϫ78 mi͞h The cars are approaching each other at a rate of 78 mi͞h. EXAMPLE 5 A man walks along a straight path at a speed of 4 ft͞s. A searchlight is located on the ground 20 ft from the path and is kept focused on the man. At what rate is the searchlight rotating when the man is 15 ft from the point on the path closest to the searchlight? x SOLUTION We draw Figure 5 and let x be the distance from the man to the point on the path closest to the searchlight. We let ␪ be the angle between the beam of the searchlight and the perpendicular to the path. We are given that dx͞dt ෇ 4 ft͞s and are asked to find d␪͞dt when x ෇ 15. The equation that relates x and ␪ can be written from Figure 5: x ෇ tan ␪ 20 20 ¨ x ෇ 20 tan ␪ Differentiating each side with respect to t, we get dx d␪ ෇ 20 sec2␪ dt dt FIGURE 5 so d␪ dx 1 1 ෇ 20 cos2␪ ෇ 20 cos2␪ ͑4͒ ෇ 1 cos2␪ 5 dt dt 4 When x ෇ 15, the length of the beam is 25, so cos ␪ ෇ 5 and d␪ 1 ෇ dt 5 ͩͪ 4 5 2 ෇ 16 ෇ 0.128 125 The searchlight is rotating at a rate of 0.128 rad͞s. |||| 3.9 Exercises 1. If V is the volume of a cube with edge length x and the cube 6. A particle moves along the curve y ෇ s1 ϩ x 3. As it reaches the point ͑2, 3͒, the y-coordinate is increasing at a rate of 4 cm͞s. How fast is the x-coordinate of the point changing at that instant? expands as time passes, find dV͞dt in terms of dx͞dt. 2. (a) If A is the area of a circle with radius r and the circle expands as time passes, find dA͞dt in terms of dr͞dt. (b) Suppose oil spills from a ruptured tanker and spreads in a circular pattern. If the radius of the oil spill increases at a constant rate of 1 m͞s, how fast is the area of the spill increasing when the radius is 30 m? 3. If y ෇ x 3 ϩ 2x and dx͞dt ෇ 5, find dy͞dt when x ෇ 2. 4. If x 2 ϩ y 2 ෇ 25 and dy͞dt ෇ 6, find dx͞dt when y ෇ 4. 5. If z 2 ෇ x 2 ϩ y 2, dx͞dt ෇ 2, and dy͞dt ෇ 3, find dz͞dt when x ෇ 5 and y ෇ 12. 7–10 (a) (b) (c) (d) (e) |||| What quantities are given in the problem? What is the unknown? Draw a picture of the situation for any time t. Write an equation that relates the quantities. Finish solving the problem. 7. A plane flying horizontally at an altitude of 1 mi and a speed of 500 mi͞h passes directly over a radar station. Find the rate at which the distance from the plane to the station is increasing when it is 2 mi away from the station. 5E-03(pp 196-205) 1/17/06 1:48 PM Page 203 SECTION 3.9 RELATED RATES 8. If a snowball melts so that its surface area decreases at a rate of 1 cm2͞min, find the rate at which the diameter decreases when the diameter is 10 cm. 6 ft tall walks away from the pole with a speed of 5 ft͞s along a straight path. How fast is the tip of his shadow moving when he is 40 ft from the pole? 10. At noon, ship A is 150 km west of ship B. Ship A is sailing east at 35 km͞h and ship B is sailing north at 25 km͞h. How fast is the distance between the ships changing at 4:00 P.M.? ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ 203 17. At noon, ship A is 100 km west of ship B. Ship A is sailing south at 35 km͞h and ship B is sailing north at 25 km͞h. How fast is the distance between the ships changing at 4:00 P.M.? 9. A street light is mounted at the top of a 15-ft-tall pole. A man ■ ❙❙❙❙ ■ 11. Two cars start moving from the same point. One travels south at 60 mi͞h and the other travels west at 25 mi͞h. At what rate is the distance between the cars increasing two hours later? 12. A spotlight on the ground shines on a wall 12 m away. If a man 2 m tall walks from the spotlight toward the building at a speed of 1.6 m͞s, how fast is the length of his shadow on the building decreasing when he is 4 m from the building? 13. A man starts walking north at 4 ft͞s from a point P. Five min- utes later a woman starts walking south at 5 ft͞s from a point 500 ft due east of P. At what rate are the people moving apart 15 min after the woman starts walking? 14. A baseball diamond is a square with side 90 ft. A batter hits the ball and runs toward first base with a speed of 24 ft͞s. (a) At what rate is his distance from second base decreasing when he is halfway to first base? (b) At what rate is his distance from third base increasing at the same moment? 18. A particle is moving along the curve y ෇ sx. As the particle passes through the point ͑4, 2͒, its x-coordinate increases at a rate of 3 cm͞s. How fast is the distance from the particle to the origin changing at this instant? 19. Water is leaking out of an inverted conical tank at a rate of 10,000 cm3͞min at the same time that water is being pumped into the tank at a constant rate. The tank has height 6 m and the diameter at the top is 4 m. If the water level is rising at a rate of 20 cm͞min when the height of the water is 2 m, find the rate at which water is being pumped into the tank. 20. A trough is 10 ft long and its ends have the shape of isosceles triangles that are 3 ft across at the top and have a height of 1 ft. If the trough is being filled with water at a rate of 12 ft 3͞min, how fast is the water level rising when the water is 6 inches deep? 21. A water trough is 10 m long and a cross-section has the shape of an isosceles trapezoid that is 30 cm wide at the bottom, 80 cm wide at the top, and has height 50 cm. If the trough is being filled with water at the rate of 0.2 m3͞min, how fast is the water level rising when the water is 30 cm deep? 22. A swimming pool is 20 ft wide, 40 ft long, 3 ft deep at the shallow end, and 9 ft deep at its deepest point. A crosssection is shown in the figure. If the pool is being filled at a rate of 0.8 ft 3͞min, how fast is the water level rising when the depth at the deepest point is 5 ft? 3 6 6 12 16 6 90 ft 23. Gravel is being dumped from a conveyor belt at a rate of 15. The altitude of a triangle is increasing at a rate of 1 cm͞min while the area of the triangle is increasing at a rate of 2 cm2͞min. At what rate is the base of the triangle changing when the altitude is 10 cm and the area is 100 cm2 ? 16. A boat is pulled into a dock by a rope attached to the bow of the boat and passing through a pulley on the dock that is 1 m higher than the bow of the boat. If the rope is pulled in at a rate of 1 m͞s, how fast is the boat approaching the dock when it is 8 m from the dock? 30 ft 3͞min, and its coarseness is such that it forms a pile in the shape of a cone whose base diameter and height are always equal. How fast is the height of the pile increasing when the pile is 10 ft high? 5E-03(pp 196-205) 204 ❙❙❙❙ 1/17/06 1:48 PM Page 204 CHAPTER 3 DERVIATIVES 24. A kite 100 ft above the ground moves horizontally at a speed of 8 ft͞s. At what rate is the angle between the string and the horizontal decreasing when 200 ft of string have been let out? 25. Two sides of a triangle are 4 m and 5 m in length and the angle between them is increasing at a rate of 0.06 rad͞s. Find the rate at which the area of the triangle is increasing when the angle between the sides of fixed length is ␲͞3. how fast is the angle between the top of the ladder and the wall changing when the angle is ␲͞4 rad? 32. Two carts, A and B, are connected by a rope 39 ft long that passes over a pulley P (see the figure). The point Q is on the floor 12 ft directly beneath P and between the carts. Cart A is being pulled away from Q at a speed of 2 ft͞s. How fast is cart B moving toward Q at the instant when cart A is 5 ft from Q ? 26. Two sides of a triangle have lengths 12 m and 15 m. The angle between them is increasing at a rate of 2Њ͞min. How fast is the length of the third side increasing when the angle between the sides of fixed length is 60Њ ? P 27. Boyle’s Law states that when a sample of gas is compressed at a constant temperature, the pressure P and volume V satisfy the equation PV ෇ C, where C is a constant. Suppose that at a certain instant the volume is 600 cm3, the pressure is 150 kPa, and the pressure is increasing at a rate of 20 kPa͞min. At what rate is the volume decreasing at this instant? 28. When air expands adiabatically (without gaining or losing heat), its pressure P and volume V are related by the equation PV 1.4 ෇ C, where C is a constant. Suppose that at a certain instant the volume is 400 cm3 and the pressure is 80 kPa and is decreasing at a rate of 10 kPa͞min. At what rate is the volume increasing at this instant? 29. If two resistors with resistances R1 and R2 are connected in parallel, as in the figure, then the total resistance R, measured in ohms (⍀), is given by 1 1 1 ෇ ϩ R R1 R2 If R1 and R2 are increasing at rates of 0.3 ⍀͞s and 0.2 ⍀͞s, respectively, how fast is R changing when R1 ෇ 80 ⍀ and R2 ෇ 100 ⍀? 12 f t A B Q 33. A television camera is positioned 4000 ft from the base of a rocket launching pad. The angle of elevation of the camera has to change at the correct rate in order to keep the rocket in sight. Also, the mechanism for focusing the camera has to take into account the increasing distance from the camera to the rising rocket. Let’s assume the rocket rises vertically and its speed is 600 ft͞s when it has risen 3000 ft. (a) How fast is the distance from the television camera to the rocket changing at that moment? (b) If the television camera is always kept aimed at the rocket, how fast is the camera’s angle of elevation changing at that same moment? 34. A lighthouse is located on a small island 3 km away from the nearest point P on a straight shoreline and its light makes four revolutions per minute. How fast is the beam of light moving along the shoreline when it is 1 km from P ? 35. A plane flying with a constant speed of 300 km͞h passes over a R¡ R™ ground radar station at an altitude of 1 km and climbs at an angle of 30Њ. At what rate is the distance from the plane to the radar station increasing a minute later? 36. Two people start from the same point. One walks east at 30. Brain weight B as a function of body weight W in fish has been modeled by the power function B ෇ 0.007W 2͞3, where B and W are measured in grams. A model for body weight as a function of body length L (measured in centimeters) is W ෇ 0.12L2.53. If, over 10 million years, the average length of a certain species of fish evolved from 15 cm to 20 cm at a constant rate, how fast was this species’ brain growing when the average length was 18 cm? 31. A ladder 10 ft long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a speed of 2 ft͞s, 3 mi͞h and the other walks northeast at 2 mi͞h. How fast is the distance between the people changing after 15 minutes? 37. A runner sprints around a circular track of radius 100 m at a constant speed of 7 m͞s. The runner’s friend is standing at a distance 200 m from the center of the track. How fast is the distance between the friends changing when the distance between them is 200 m? 38. The minute hand on a watch is 8 mm long and the hour hand is 4 mm long. How fast is the distance between the tips of the hands changing at one o’clock? 5E-03(pp 196-205) 1/17/06 1:48 PM Page 205 SECTION 3.10 LINEAR APPROXIMATIONS AND DIFFERENTIALS |||| 3.10 205 Linear Approximations and Differentials Resources / Module 3 / Linear Approximation / Start of Linear Approximation y y=ƒ {a, f(a)} ❙❙❙❙ We have seen that a curve lies very close to its tangent line near the point of tangency. In fact, by zooming in toward a point on the graph of a differentiable function, we noticed that the graph looks more and more like its tangent line. (See Figure 2 in Section 2.6 and Figure 3 in Section 3.1.) This observation is the basis for a method of finding approximate values of functions. The idea is that it might be easy to calculate a value f ͑a͒ of a function, but difficult (or even impossible) to compute nearby values of f. So we settle for the easily computed values of the linear function L whose graph is the tangent line of f at ͑a, f ͑a͒͒. (See Figure 1.) In other words, we use the tangent line at ͑a, f ͑a͒͒ as an approximation to the curve y ෇ f ͑x͒ when x is near a. An equation of this tangent line is y ෇ f ͑a͒ ϩ f Ј͑a͒͑x Ϫ a͒ y=L(x) and the approximation 1 0 FIGURE 1 x f ͑x͒ Ϸ f ͑a͒ ϩ f Ј͑a͒͑x Ϫ a͒ is called the linear approximation or tangent line approximation of f at a. The linear function whose graph is this tangent line, that is, 2 L͑x͒ ෇ f ͑a͒ ϩ f Ј͑a͒͑x Ϫ a͒ is called the linearization of f at a. The following example is typical of situations in which we use a linear approximation to predict the future behavior of a function given by empirical data. EXAMPLE 1 Suppose that after you stuff a turkey its temperature is 50ЊF and you then put it in a 325ЊF oven. After an hour the meat thermometer indicates that the temperature of the turkey is 93ЊF and after two hours it indicates 129ЊF. Predict the temperature of the turkey after three hours. SOLUTION If T͑t͒ represents the temperature of the turkey after t hours, we are given that T͑0͒ ෇ 50, T͑1͒ ෇ 93, and T͑2͒ ෇ 129. In order to make a linear approximation with a ෇ 2, we need an estimate for the derivative TЈ͑2͒. Because TЈ͑2͒ ෇ lim t l2 T͑t͒ Ϫ T͑2͒ tϪ2 we could estimate TЈ͑2͒ by the difference quotient with t ෇ 1: TЈ͑2͒ Ϸ T͑1͒ Ϫ T͑2͒ 93 Ϫ 129 ෇ ෇ 36 1Ϫ2 Ϫ1 This amounts to approximating the instantaneous rate of temperature change by the average rate of change between t ෇ 1 and t ෇ 2, which is 36ЊF͞h. With this estimate, the linear approximation (1) for the temperature after 3 h is T͑3͒ Ϸ T͑2͒ ϩ TЈ͑2͒͑3 Ϫ 2͒ Ϸ 129 ϩ 36 ؒ 1 ෇ 165 So the predicted temperature after three hours is 165ЊF. 5E-03(pp 206-217) 206 ❙❙❙❙ 1/17/06 1:45 PM Page 206 CHAPTER 3 DERIVATIVES T We obtain a more accurate estimate for TЈ͑2͒ by plotting the given data, as in Figure 2, and estimating the slope of the tangent line at t ෇ 2 to be 150 TЈ͑2͒ Ϸ 33 Then our linear approximation becomes 100 50 L T͑3͒ Ϸ T͑2͒ ϩ TЈ͑2͒ ؒ 1 Ϸ 129 ϩ 33 ෇ 162 T 0 and our improved estimate for the temperature is 162ЊF. Because the temperature curve lies below the tangent line, it appears that the actual temperature after three hours will be somewhat less than 162ЊF, perhaps closer to 160ЊF. 1 2 3 t EXAMPLE 2 Find the linearization of the function f ͑x͒ ෇ sx ϩ 3 at a ෇ 1 and use it to approximate the numbers s3.98 and s4.05. Are these approximations overestimates or underestimates? FIGURE 2 SOLUTION The derivative of f ͑x͒ ෇ ͑x ϩ 3͒1͞2 is f Ј͑x͒ ෇ 1 ͑x ϩ 3͒Ϫ1͞2 ෇ 2 1 2sx ϩ 3 and so we have f ͑1͒ ෇ 2 and f Ј͑1͒ ෇ 1 . Putting these values into Equation 2, we see that 4 the linearization is L͑x͒ ෇ f ͑1͒ ϩ f Ј͑1͒͑x Ϫ 1͒ ෇ 2 ϩ 1 ͑x Ϫ 1͒ ෇ 4 7 x ϩ 4 4 The corresponding linear approximation (1) is sx ϩ 3 Ϸ 7 x ϩ 4 4 (when x is near 1) In particular, we have 7 0.98 s3.98 Ϸ 4 ϩ 4 ෇ 1.995 and 7 1.05 s4.05 Ϸ 4 ϩ 4 ෇ 2.0125 The linear approximation is illustrated in Figure 3. We see that, indeed, the tangent line approximation is a good approximation to the given function when x is near l. We also see that our approximations are overestimates because the tangent line lies above the curve. y 7 x y= 4 + 4 (1, 2) FIGURE 3 _3 0 1 y= x+3 œ„„„„ x Of course, a calculator could give us approximations for s3.98 and s4.05, but the linear approximation gives an approximation over an entire interval. In the following table we compare the estimates from the linear approximation in Example 2 with the true values. Notice from this table, and also from Figure 3, that the 5E-03(pp 206-217) 1/17/06 1:45 PM Page 207 SECTION 3.10 LINEAR APPROXIMATIONS AND DIFFERENTIALS ❙❙❙❙ 207 tangent line approximation gives good estimates when x is close to 1 but the accuracy of the approximation deteriorates when x is farther away from 1. x s6 Actual value 0.9 0.98 1 1.05 1.1 2 3 s3.9 s3.98 s4 s4.05 s4.1 s5 From L͑x͒ 1.975 1.995 2 2.0125 2.025 2.25 2.5 1.97484176 . . . 1.99499373 . . . 2.00000000 . . . 2.01246117 . . . 2.02484567 . . . 2.23606797 . . . 2.44948974 . . . How good is the approximation that we obtained in Example 2? The next example shows that by using a graphing calculator or computer we can determine an interval throughout which a linear approximation provides a specified accuracy. EXAMPLE 3 For what values of x is the linear approximation sx ϩ 3 Ϸ 7 x ϩ 4 4 accurate to within 0.5? What about accuracy to within 0.1? SOLUTION Accuracy to within 0.5 means that the functions should differ by less than 0.5: Ϳ 4.3 Q y= œ„„„„ x+3+0.5 L(x) P 10 _1 FIGURE 4 Q y= œ„„„„ x+3+0.1 _2 y= œ„„„„ x+3-0.1 FIGURE 5 1 7 x ϩ 4 4 Ͻ 0.5 sx ϩ 3 Ϫ 0.5 Ͻ 7 x ϩ Ͻ sx ϩ 3 ϩ 0.5 4 4 This says that the linear approximation should lie between the curves obtained by shifting the curve y ෇ sx ϩ 3 upward and downward by an amount 0.5. Figure 4 shows the tangent line y ෇ ͑7 ϩ x͒͞4 intersecting the upper curve y ෇ sx ϩ 3 ϩ 0.5 at P and Q. Zooming in and using the cursor, we estimate that the x-coordinate of P is about Ϫ2.66 and the x-coordinate of Q is about 8.66. Thus, we see from the graph that the approximation 7 x sx ϩ 3 Ϸ ϩ 4 4 3 P ͩ ͪͿ Equivalently, we could write y= œ„„„„ x+3-0.5 _4 sx ϩ 3 Ϫ 5 is accurate to within 0.5 when Ϫ2.6 Ͻ x Ͻ 8.6. (We have rounded to be safe.) Similarly, from Figure 5 we see that the approximation is accurate to within 0.1 when Ϫ1.1 Ͻ x Ͻ 3.9. Applications to Physics Linear approximations are often used in physics. In analyzing the consequences of an equation, a physicist sometimes needs to simplify a function by replacing it with its linear approximation. For instance, in deriving a formula for the period of a pendulum, physics 5E-03(pp 206-217) 208 ❙❙❙❙ 1/17/06 1:45 PM Page 208 CHAPTER 3 DERIVATIVES textbooks obtain the expression a T ෇ Ϫt sin ␪ for tangential acceleration and then replace sin ␪ by ␪ with the remark that sin ␪ is very close to ␪ if ␪ is not too large. [See, for example, Physics: Calculus, 2d ed., by Eugene Hecht (Pacific Grove, CA: Brooks/Cole, 2000), p. 431.] You can verify that the linearization of the function f ͑x͒ ෇ sin x at a ෇ 0 is L͑x͒ ෇ x and so the linear approximation at 0 is sin x Ϸ x (see Exercise 46). So, in effect, the derivation of the formula for the period of a pendulum uses the tangent line approximation for the sine function. Another example occurs in the theory of optics, where light rays that arrive at shallow angles relative to the optical axis are called paraxial rays. In paraxial (or Gaussian) optics, both sin ␪ and cos ␪ are replaced by their linearizations. In other words, the linear approximations sin ␪ Ϸ ␪ and cos ␪ Ϸ 1 are used because ␪ is close to 0. The results of calculations made with these approximations became the basic theoretical tool used to design lenses. [See Optics, 4th ed., by Eugene Hecht (Reading, MA: Addison-Wesley, 2002), p. 154.] In Section 12.12 we will present several other applications of the idea of linear approximations to physics. Differentials The ideas behind linear approximations are sometimes formulated in the terminology and notation of differentials. If y ෇ f ͑x͒, where f is a differentiable function, then the differential dx is an independent variable; that is, dx can be given the value of any real number. The differential dy is then defined in terms of dx by the equation |||| If dx 0, we can divide both sides of Equation 3 by dx to obtain dy ෇ f Ј͑x͒ dx 3 We have seen similar equations before, but now the left side can genuinely be interpreted as a ratio of differentials. y Q R Îy P dx=Îx 0 x y=ƒ FIGURE 6 dy S x+Î x dy ෇ f Ј͑x͒ dx So dy is a dependent variable; it depends on the values of x and dx. If dx is given a specific value and x is taken to be some specific number in the domain of f , then the numerical value of dy is determined. The geometric meaning of differentials is shown in Figure 6. Let P͑x, f ͑x͒͒ and Q͑x ϩ ⌬x, f ͑x ϩ ⌬x͒͒ be points on the graph of f and let dx ෇ ⌬x. The corresponding change in y is ⌬y ෇ f ͑x ϩ ⌬x͒ Ϫ f ͑x͒ x The slope of the tangent line PR is the derivative f Ј͑x͒. Thus, the directed distance from S to R is f Ј͑x͒ dx ෇ dy. Therefore, dy represents the amount that the tangent line rises or falls (the change in the linearization), whereas ⌬y represents the amount that the curve y ෇ f ͑x͒ rises or falls when x changes by an amount dx. EXAMPLE 4 Compare the values of ⌬y and dy if y ෇ f ͑x͒ ෇ x 3 ϩ x 2 Ϫ 2x ϩ 1 and x changes (a) from 2 to 2.05 and (b) from 2 to 2.01. 5E-03(pp 206-217) 1/17/06 1:45 PM Page 209 SECTION 3.10 LINEAR APPROXIMATIONS AND DIFFERENTIALS |||| Figure 7 shows the function in Example 4 and a comparison of dy and ⌬y when a ෇ 2. The viewing rectangle is ͓1.8, 2.5͔ by ͓6, 18͔. (2, 9) FIGURE 7 209 SOLUTION (a) We have f ͑2͒ ෇ 2 3 ϩ 2 2 Ϫ 2͑2͒ ϩ 1 ෇ 9 y=˛+≈-2x+1 dy ❙❙❙❙ f ͑2.05͒ ෇ ͑2.05͒3 ϩ ͑2.05͒2 Ϫ 2͑2.05͒ ϩ 1 ෇ 9.717625 ⌬y ෇ f ͑2.05͒ Ϫ f ͑2͒ ෇ 0.717625 Îy dy ෇ f Ј͑x͒ dx ෇ ͑3x 2 ϩ 2x Ϫ 2͒ dx In general, When x ෇ 2 and dx ෇ ⌬x ෇ 0.05, this becomes dy ෇ ͓3͑2͒2 ϩ 2͑2͒ Ϫ 2͔0.05 ෇ 0.7 (b) f ͑2.01͒ ෇ ͑2.01͒3 ϩ ͑2.01͒2 Ϫ 2͑2.01͒ ϩ 1 ෇ 9.140701 ⌬y ෇ f ͑2.01͒ Ϫ f ͑2͒ ෇ 0.140701 When dx ෇ ⌬x ෇ 0.01, dy ෇ ͓3͑2͒2 ϩ 2͑2͒ Ϫ 2͔0.01 ෇ 0.14 Notice that the approximation ⌬y Ϸ dy becomes better as ⌬x becomes smaller in Example 4. Notice also that dy was easier to compute than ⌬y. For more complicated functions it may be impossible to compute ⌬y exactly. In such cases the approximation by differentials is especially useful. In the notation of differentials, the linear approximation (1) can be written as f ͑a ϩ dx͒ Ϸ f ͑a͒ ϩ dy For instance, for the function f ͑x͒ ෇ sx ϩ 3 in Example 2, we have dy ෇ f Ј͑x͒ dx ෇ dx 2sx ϩ 3 If a ෇ 1 and dx ෇ ⌬x ෇ 0.05, then dy ෇ and 0.05 ෇ 0.0125 2s1 ϩ 3 s4.05 ෇ f ͑1.05͒ Ϸ f ͑1͒ ϩ dy ෇ 2.0125 just as we found in Example 2. Our final example illustrates the use of differentials in estimating the errors that occur because of approximate measurements. EXAMPLE 5 The radius of a sphere was measured and found to be 21 cm with a possible error in measurement of at most 0.05 cm. What is the maximum error in using this value of the radius to compute the volume of the sphere? ␲ r 3. If the error in the measured value of r is denoted by dr ෇ ⌬r, then the corresponding error in the calculated value of V is ⌬V , which can be approximated by the differential SOLUTION If the radius of the sphere is r, then its volume is V ෇ dV ෇ 4␲ r 2 dr 4 3 5E-03(pp 206-217) 210 ❙❙❙❙ 1/17/06 1:45 PM Page 210 CHAPTER 3 DERIVATIVES When r ෇ 21 and dr ෇ 0.05, this becomes dV ෇ 4␲ ͑21͒2 0.05 Ϸ 277 The maximum error in the calculated volume is about 277 cm3. NOTE Although the possible error in Example 5 may appear to be rather large, a better picture of the error is given by the relative error, which is computed by dividing the error by the total volume: ■ ⌬V dV 4␲ r 2 dr dr Ϸ ෇ 4 3 ෇3 V V r 3 ␲r Thus, the relative error in the volume is about three times the relative error in the radius. In Example 5 the relative error in the radius is approximately dr͞r ෇ 0.05͞21 Ϸ 0.0024 and it produces a relative error of about 0.007 in the volume. The errors could also be expressed as percentage errors of 0.24% in the radius and 0.7% in the volume. |||| 3.10 Exercises 1. The turkey in Example 1 is removed from the oven when its 4. The table shows the population of Nepal (in millions) as of temperature reaches 185ЊF and is placed on a table in a room where the temperature is 75ЊF. After 10 minutes the temperature of the turkey is 172ЊF and after 20 minutes it is 160ЊF. Use a linear approximation to predict the temperature of the turkey after half an hour. Do you think your prediction is an overestimate or an underestimate? Why? June 30 of the given year. Use a linear approximation to estimate the population at midyear in 1984. Use another linear approximation to predict the population in 2006. t temperature of 15ЊC, the pressure is 101.3 kilopascals (kPa) at sea level, 87.1 kPa at h ෇ 1 km, and 74.9 kPa at h ෇ 2 km. Use a linear approximation to estimate the atmospheric pressure at an altitude of 3 km. 5–8 1995 2000 15.0 17.0 19.3 22.0 24.9 Find the linearization L͑x͒ of the function at a. |||| a෇1 6. f ͑x͒ ෇ 1͞s2 ϩ x, a ෇ ␲͞2 7. f ͑x͒ ෇ cos x, 8. f ͑x͒ ෇ sx, 3 ■ ■ ■ a෇0 a ෇ Ϫ8 ■ ■ ■ ■ ■ ■ ■ ■ ; 9. Find the linear approximation of the function f ͑x͒ ෇ s1 Ϫ x at a ෇ 0 and use it to approximate the numbers s0.9 and s0.99. Illustrate by graphing f and the tangent line. P 3 ; 10. Find the linear approximation of the function t͑x͒ ෇ s1 ϩ x 20 Percent aged 65 and over 1990 5. f ͑x͒ ෇ x 3, 3. The graph indicates how Australia’s population is aging by showing the past and projected percentage of the population aged 65 and over. Use a linear approximation to predict the percentage of the population that will be 65 and over in the years 2040 and 2050. Do you think your predictions are too high or too low? Why? 1985 N͑t͒ 2. Atmospheric pressure P decreases as altitude h increases. At a 1980 3 at a ෇ 0 and use it to approximate the numbers s0.95 and 3 s1.1. Illustrate by graphing t and the tangent line. 10 |||| Verify the given linear approximation at a ෇ 0. Then determine the values of x for which the linear approximation is accurate to within 0.1. ; 11–14 0 1900 2000 t 3 11. s1 Ϫ x Ϸ 1 Ϫ 3 x 1 12. tan x Ϸ x ■ 5E-03(pp 206-217) 1/17/06 1:45 PM Page 211 SECTION 3.10 LINEAR APPROXIMATIONS AND DIFFERENTIALS 13. 1͑͞1 ϩ 2x͒4 Ϸ 1 Ϫ 8x 14. 1͞s4 Ϫ x Ϸ 2 ϩ 1 ■ ■ 15–20 ■ |||| 1 16 ■ x ■ ■ ■ ■ ■ ■ ■ ■ 17. y ෇ x tan x 18. y ෇ s1 ϩ t 2 40. The radius of a circular disk is given as 24 cm with a maxi- uϩ1 uϪ1 ■ mum error in measurement of 0.2 cm. (a) Use differentials to estimate the maximum error in the calculated area of the disk. (b) What is the relative error? What is the percentage error? 2 20. y ෇ ͑1 ϩ 2r͒Ϫ4 41. The circumference of a sphere was measured to be 84 cm with ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ 21–26 |||| (a) Find the differential dy and (b) evaluate dy for the given values of x and dx. 21. y ෇ x 2 ϩ 2x, x ෇ 3, dx ෇ 1 2 22. y ෇ x 3 Ϫ 6x 2 ϩ 5x Ϫ 7, 23. y ෇ s4 ϩ 5x, x ෇ 0, 24. y ෇ 1͑͞x ϩ 1͒, x ෇ Ϫ2, dx ෇ 0.1 x ෇ 1, apply a coat of paint 0.05 cm thick to a hemispherical dome with diameter 50 m. dx ෇ Ϫ0.01 x ෇ ␲͞4, x ෇ ␲͞3, 43. (a) Use differentials to find a formula for the approximate vol- dx ෇ Ϫ0.1 26. y ෇ cos x, dx ෇ 0.05 ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ x ෇ 1, 29. y ෇ 6 Ϫ x , 30. y ෇ 16͞x, x ෇ 4, ■ ■ F ෇ kR 4 ⌬x ෇ 1 x ෇ Ϫ2, 2 ■ of blood per unit time that flows past a given point) is proportional to the fourth power of the radius R of the blood vessel: x ෇ 1, ⌬x ෇ 0.5 28. y ෇ sx, ■ ⌬x ෇ 0.4 ⌬x ෇ Ϫ1 ■ ■ ■ ■ ■ ■ ■ ■ 31–36 |||| Use differentials (or, equivalently, a linear approximation) to estimate the given number. 31. ͑2.001͒5 32. s99.8 33. ͑8.06͒ 34. 1͞1002 35. tan 44Њ 36. cos 31.5Њ 2͞3 ■ ■ ■ ■ ■ ■ ■ ■ (a) dc ෇ 0 ■ (b) d͑cu͒ ෇ c du ■ ■ ■ |||| Explain, in terms of linear approximations or differentials, why the approximation is reasonable. 37. sec 0.08 Ϸ 1 ■ ■ ■ ■ ■ ■ ■ ■ (c) d͑u ϩ v͒ ෇ du ϩ dv (d) d͑uv͒ ෇ u dv ϩ v du ͩͪ (e) d 38. ͑1.01͒6 Ϸ 1.06 ■ (This is known as Poiseuille’s Law; we will show why it is true in Section 9.4.) A partially clogged artery can be expanded by an operation called angioplasty, in which a balloon-tipped catheter is inflated inside the artery in order to widen it and restore the normal blood flow. Show that the relative change in F is about four times the relative change in R. How will a 5% increase in the radius affect the flow of blood? 45. Establish the following rules for working with differentials (where c denotes a constant and u and v are functions of x). 37–38 ■ ume of a thin cylindrical shell with height h, inner radius r, and thickness ⌬r. (b) What is the error involved in using the formula from part (a)? 44. When blood flows along a blood vessel, the flux F (the volume |||| Compute ⌬y and dy for the given values of x and dx ෇ ⌬x. Then sketch a diagram like Figure 6 showing the line segments with lengths dx, dy, and ⌬y. 27–30 27. y ෇ x 2, a possible error of 0.5 cm. (a) Use differentials to estimate the maximum error in the calculated surface area. What is the relative error? (b) Use differentials to estimate the maximum error in the calculated volume. What is the relative error? 42. Use differentials to estimate the amount of paint needed to dx ෇ 0.04 25. y ෇ tan x, ■ in measurement of 0.1 cm. Use differentials to estimate the maximum possible error, relative error, and percentage error in computing (a) the volume of the cube and (b) the surface area of the cube. Find the differential of the function. 16. y ෇ cos ␲ x ■ 211 39. The edge of a cube was found to be 30 cm with a possible error 15. y ෇ x 4 ϩ 5x 19. y ෇ ❙❙❙❙ ■ ■ u v ෇ v du Ϫ u dv v2 (f) d͑x n ͒ ෇ nx nϪ1 dx 5E-03(pp 206-217) 212 ❙❙❙❙ 1/17/06 1:45 PM Page 212 CHAPTER 3 DERVIATIVES 46. On page 431 of Physics: Calculus, 2d ed., by Eugene Hecht (Pacific Grove, CA: Brooks/Cole, 2000), in the course of deriving the formula T ෇ 2␲ sL͞t for the period of a pendulum of length L, the author obtains the equation a T ෇ Ϫt sin ␪ for the tangential acceleration of the bob of the pendulum. He then says, “for small angles, the value of ␪ in radians is very nearly the value of sin ␪ ; they differ by less than 2% out to about 20°.” (a) Verify the linear approximation at 0 for the sine function: sin x Ϸ x ; (b) Are your estimates in part (a) too large or too small? Explain. y y=fª(x) 1 0 (b) Use a graphing device to determine the values of x for which sin x and x differ by less than 2%. Then verify Hecht’s statement by converting from radians to degrees. 1 x 48. Suppose that we don’t have a formula for t͑x͒ but we know that t͑2͒ ෇ Ϫ4 and tЈ͑x͒ ෇ sx 2 ϩ 5 for all x. (a) Use a linear approximation to estimate t͑1.95͒ and t͑2.05͒. (b) Are your estimates in part (a) too large or too small? Explain. 47. Suppose that the only information we have about a function f is that f ͑1͒ ෇ 5 and the graph of its derivative is as shown. (a) Use a linear approximation to estimate f ͑0.9͒ and f ͑1.1͒. LABORATORY PROJECT ; Taylor Polynomials The tangent line approximation L͑x͒ is the best first-degree (linear) approximation to f ͑x͒ near x ෇ a because f ͑x͒ and L͑x͒ have the same rate of change (derivative) at a. For a better approximation than a linear one, let’s try a second-degree (quadratic) approximation P͑x͒. In other words, we approximate a curve by a parabola instead of by a straight line. To make sure that the approximation is a good one, we stipulate the following: (i) P͑a͒ ෇ f ͑a͒ (P and f should have the same value at a.) (ii) PЈ͑a͒ ෇ f Ј͑a͒ (P and f should have the same rate of change at a.) (iii) P Љ͑a͒ ෇ f Љ͑a͒ (The slopes of P and f should change at the same rate.) 1. Find the quadratic approximation P͑x͒ ෇ A ϩ Bx ϩ Cx 2 to the function f ͑x͒ ෇ cos x that satisfies conditions (i), (ii), and (iii) with a ෇ 0. Graph P, f , and the linear approximation L͑x͒ ෇ 1 on a common screen. Comment on how well the functions P and L approximate f . 2. Determine the values of x for which the quadratic approximation f ͑x͒ ෇ P͑x͒ in Problem 1 is accurate to within 0.1. [Hint: Graph y ෇ P͑x͒, y ෇ cos x Ϫ 0.1, and y ෇ cos x ϩ 0.1 on a common screen.] 3. To approximate a function f by a quadratic function P near a number a, it is best to write P in the form P͑x͒ ෇ A ϩ B͑x Ϫ a͒ ϩ C͑x Ϫ a͒2 Show that the quadratic function that satisfies conditions (i), (ii), and (iii) is P͑x͒ ෇ f ͑a͒ ϩ f Ј͑a͒͑x Ϫ a͒ ϩ 1 f Љ͑a͒͑x Ϫ a͒2 2 4. Find the quadratic approximation to f ͑x͒ ෇ sx ϩ 3 near a ෇ 1. Graph f , the quadratic approximation, and the linear approximation from Example 3 in Section 3.10 on a common screen. What do you conclude? 5. Instead of being satisfied with a linear or quadratic approximation to f ͑x͒ near x ෇ a, let’s try to find better approximations with higher-degree polynomials. We look for an nth-degree polynomial Tn͑x͒ ෇ c0 ϩ c1 ͑x Ϫ a͒ ϩ c2 ͑x Ϫ a͒2 ϩ c3 ͑x Ϫ a͒3 ϩ и и и ϩ cn ͑x Ϫ a͒n 5E-03(pp 206-217) 1/17/06 1:45 PM Page 213 CHAPTER 3 REVIEW ❙❙❙❙ 213 such that Tn and its first n derivatives have the same values at x ෇ a as f and its first n derivatives. By differentiating repeatedly and setting x ෇ a, show that these conditions are satisfied if c0 ෇ f ͑a͒, c1 ෇ f Ј͑a͒, c2 ෇ 1 f Љ͑a͒, and in general 2 ck ෇ f ͑k͒͑a͒ k! where k! ෇ 1 ؒ 2 ؒ 3 ؒ 4 ؒ и и и ؒ k. The resulting polynomial Tn ͑x͒ ෇ f ͑a͒ ϩ f Ј͑a͒͑x Ϫ a͒ ϩ f Љ͑a͒ f ͑n͒͑a͒ ͑x Ϫ a͒2 ϩ и и и ϩ ͑x Ϫ a͒n 2! n! is called the nth-degree Taylor polynomial of f centered at a. 6. Find the 8th-degree Taylor polynomial centered at a ෇ 0 for the function f ͑x͒ ෇ cos x. Graph f together with the Taylor polynomials T2 , T4 , T6 , T8 in the viewing rectangle [Ϫ5, 5] by [Ϫ1.4, 1.4] and comment on how well they approximate f . |||| 3 Review ■ CONCEPT CHECK 1. Define the derivative f Ј͑a͒. Discuss two ways of interpreting 4. State the derivative of each function. this number. (a) (c) (e) (g) 2. (a) What does it mean for f to be differentiable at a? (b) What is the relation between the differentiability and continuity of a function? 3. State each of the following differentiation rules both in ■ y ෇ xn y ෇ cos x y ෇ csc x y ෇ cot x 5. Explain how implicit differentiation works. symbols and in words. (a) The Power Rule (b) The Constant Multiple Rule (c) The Sum Rule (d) The Difference Rule (e) The Product Rule (f) The Quotient Rule (g) The Chain Rule 6. What are the second and third derivatives of a function f ? If f is the position function of an object, how can you interpret f Љ and f ٞ ? 7. (a) Write an expression for the linearization of f at a. (b) If y ෇ f ͑x͒, write an expression for the differential dy. (c) If dx ෇ ⌬x, draw a picture showing the geometric meanings of ⌬y and dy. ■ TRUE-FALSE QUIZ Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. ■ 4. If f and t are differentiable, then d ͓ f ͑ t͑x͔͒͒ ෇ f Ј͑ t͑x͒͒tЈ͑x͒ dx 1. If f is continuous at a, then f is differentiable at a. 2. If f and t are differentiable, then d ͓ f ͑x͒ ϩ t͑x͔͒ ෇ f Ј͑x͒ ϩ tЈ͑x͒ dx 3. If f and t are differentiable, then d ͓ f ͑x͒t͑x͔͒ ෇ f Ј͑x͒tЈ͑x͒ dx (b) y ෇ sin x (d) y ෇ tan x (f) y ෇ sec x 5. If f is differentiable, then f Ј͑x͒ d . sf ͑x͒ ෇ dx 2 sf ͑x͒ 6. If f is differentiable, then d f Ј͑x͒ f (sx ) ෇ . dx 2 sx 7. d x 2 ϩ x ෇ 2x ϩ 1 dx Խ Խ Խ Խ 5E-03(pp 206-217) 214 ❙❙❙❙ 1/17/06 1:45 PM Page 214 CHAPTER 3 DERIVATIVES 8. If f Ј͑r͒ exists, then lim f ͑x͒ ෇ f ͑r͒. 11. An equation of the tangent line to the parabola y ෇ x 2 at x lr 9. If t͑x͒ ෇ x 5, then lim 10. d2y ෇ dx 2 xl2 ͩ ͪ ͑Ϫ2, 4͒ is y Ϫ 4 ෇ 2x͑x ϩ 2͒. t͑x͒ Ϫ t͑2͒ ෇ 80. xϪ2 12. 2 dy dx ■ EXERCISES 1. For the function f whose graph is shown, arrange the following numbers in increasing order: 0 f Ј͑2͒ 1 d d ͑tan2x͒ ෇ ͑sec 2x͒ dx dx ■ 7. The figure shows the graphs of f , f Ј, and f Љ. Identify each curve, and explain your choices. f Ј͑3͒ f Ј͑5͒ y f Ј͑7͒ a y b x 0 c 1 0 x 1 8. The total fertility rate at time t, denoted by F͑t͒, is an esti- 2. Find a function f and a number a such that mate of the average number of children born to each woman (assuming that current birth rates remain constant). The graph of the total fertility rate in the United States shows the fluctuations from 1940 to 1990. (a) Estimate the values of FЈ͑1950͒, FЈ͑1965͒, and FЈ͑1987͒. (b) What are the meanings of these derivatives? (c) Can you suggest reasons for the values of these derivatives? ͑2 ϩ h͒ Ϫ 64 ෇ f Ј͑a͒ h 6 lim h l0 3. The total cost of repaying a student loan at an interest rate of r% per year is C ෇ f ͑r͒. (a) What is the meaning of the derivative f Ј͑r͒? What are its units? (b) What does the statement f Ј͑10͒ ෇ 1200 mean? (c) Is f Ј͑r͒ always positive or does it change sign? 4–6 y 3.0 Trace or copy the graph of the function. Then sketch a graph of its derivative directly beneath. |||| 4. 5. y baby boom 3.5 baby bust 2.5 baby boomlet y=F(t) y 2.0 1.5 0 x x 0 1940 6. 1950 1960 1970 1980 1990 9. Let B͑t͒ be the total value of U.S. banknotes in circulation at y time t. The table gives values of this function from 1980 to 1998, at year end, in billions of dollars. Interpret and estimate the value of BЈ͑1990͒. x t ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ 1980 1985 1990 1995 1998 B͑t͒ ■ t 124.8 182.0 268.2 401.5 492.2 5E-03(pp 206-217) 1/17/06 1:46 PM Page 215 ❙❙❙❙ CHAPTER 3 REVIEW 10–11 |||| Find f Ј͑x͒ from first principles, that is, directly from the definition of a derivative. 10. f ͑x͒ ෇ ■ ■ 4Ϫx 3ϩx ■ 45–48 45. y ෇ 4 sin2 x, 11. f ͑x͒ ෇ x 3 ϩ 5x ϩ 4 |||| ■ ■ ■ ■ ■ ■ ■ ■ 46. y ෇ ■ 13. y ෇ ͑ x 4 Ϫ 3x 2 ϩ 5͒3 16. y ෇ 17. y ෇ 2xsx 2 ϩ 1 18. y ෇ 3 ■ ; t 1 Ϫ t2 ; ͩ ͪ ; 20. y ෇ sin͑cos x͒ 22. y ෇ 23. xy 4 ϩ x 2 y ෇ x ϩ 3y sec 2␪ 1 ϩ tan 2␪ 32. y ෇ 34. y ෇ sin mx x 35. y ෇ tan2͑sin ␪ ͒ 36. x tan y ෇ y Ϫ 1 5 37. y ෇ sx tan x 38. y ෇ ■ ■ ■ (b) Check to see that your answers to part (a) are reasonable by comparing the graphs of f , f Ј, and f Љ. 54. (a) By differentiating the double-angle formula cos 2x ෇ cos2x Ϫ sin2x 4 ■ ■ ■ ■ ■ ■ ■ ■ obtain the double-angle formula for the sine function. (b) By differentiating the addition formula sin͑x ϩ a͒ ෇ sin x cos a ϩ cos x sin a obtain the addition formula for the cosine function. 55. Suppose that h͑x͒ ෇ f ͑x͒t͑x͒ and F͑x͒ ෇ f ͑ t͑x͒͒, where ͑x Ϫ 1͒͑x Ϫ 4͒ ͑x Ϫ 2͒͑x Ϫ 3͒ ■ ■ ■ f ͑2͒ ෇ 3, t͑2͒ ෇ 5, tЈ͑2͒ ෇ 4, f Ј͑2͒ ෇ Ϫ2, and f Ј͑5͒ ෇ 11. Find (a) hЈ͑2͒ and (b) FЈ͑2͒. ■ 56. If f and t are the functions whose graphs are shown, let P͑x͒ ෇ f ͑x͒t͑x͒, Q͑x͒ ෇ f ͑x͒͞t͑x͒, and C͑x͒ ෇ f ͑ t͑x͒͒. Find (a) PЈ͑2͒, (b) QЈ͑2͒, and (c) CЈ͑2͒. 39. If f ͑t͒ ෇ s4t ϩ 1, find f Љ͑2͒. 40. If t͑␪ ͒ ෇ ␪ sin ␪, find t Љ͑␲͞6͒. y 41. Find y Љ if x 6 ϩ y 6 ෇ 1. g 42. Find f ͑n͒͑x͒ if f ͑x͒ ෇ 1͑͞2 Ϫ x͒. f 43–44 |||| 43. lim xl0 ■ ■ Find the limit. sec x 1 Ϫ sin x ■ 44. lim tl0 t3 tan3 2t 1 0 ■ ■ ■ ■ ■ ■ ■ ■ (b) Find equations of the tangent lines to the curve y ෇ x s5 Ϫ x at the points ͑1, 2͒ and ͑4, 4͒. (c) Illustrate part (b) by graphing the curve and tangent lines on the same screen. (d) Check to see that your answer to part (a) is reasonable by comparing the graphs of f and f Ј. f Ј͑x͒ 1 1 1 ෇ ϩ ϩ f ͑x͒ xϪa xϪb xϪc ͑x ϩ ␭͒ x 4 ϩ ␭4 33. y ෇ sin(tan s1 ϩ x 3 ) ■ 53. If f ͑x͒ ෇ ͑x Ϫ a͒͑x Ϫ b͒͑x Ϫ c͒, show that 30. y ෇ ssin sx 31. y ෇ cot͑3x 2 ϩ 5͒ ■ line has slope 1. 3 28. y ෇ 1͞sx ϩ sx 29. sin͑xy͒ ෇ x 2 Ϫ y ■ 52. Find the points on the ellipse x 2 ϩ 2y 2 ෇ 1 where the tangent 26. x 2 cos y ϩ sin 2y ෇ xy 27. y ෇ ͑1 Ϫ x Ϫ1 ͒Ϫ1 ͑2, 1͒ ■ is the tangent line horizontal? 24. y ෇ sec͑1 ϩ x 2 ͒ 25. y ෇ ■ 51. At what points on the curve y ෇ sin x ϩ cos x, 0 ഛ x ഛ 2␲, 1 sin͑x Ϫ sin x͒ 21. y ෇ tan s1 Ϫ x ■ 50. (a) If f ͑x͒ ෇ 4x Ϫ tan x, Ϫ␲͞2 Ͻ x Ͻ ␲͞2, find f Ј and f Љ. s7 1 x2 ■ 49. (a) If f ͑x͒ ෇ x s5 Ϫ x, find f Ј͑x͒. 3x Ϫ 2 s2x ϩ 1 xϩ ͑0, 1͒ 48. x 2 ϩ 4xy ϩ y 2 ෇ 13, 14. y ෇ cos͑tan x͒ 1 sx 4 19. y ෇ x Ϫ1 , ͑0, Ϫ1͒ x2 ϩ 1 47. y ෇ s1 ϩ 4 sin x, Calculate yЈ. 15. y ෇ sx ϩ ͑␲͞6, 1͒ 2 find f Ј͑x͒. (b) Find the domains of f and f Ј. (c) Graph f and f Ј on a common screen. Compare the graphs to see whether your answer to part (a) is reasonable. 13–38 Find an equation of the tangent to the curve at the given point. 12. (a) If f ͑x͒ ෇ s3 Ϫ 5x, use the definition of a derivative to ; |||| 215 ■ ■ 1 x 5E-03(pp 206-217) 216 ❙❙❙❙ 57–64 1/17/06 1:46 PM Page 216 CHAPTER 3 DERIVATIVES |||| Find f Ј in terms of tЈ. 73. The mass of part of a wire is x (1 ϩ sx ) kilograms, where x is measured in meters from one end of the wire. Find the linear density of the wire when x ෇ 4 m. 57. f ͑x͒ ෇ x t͑x͒ 58. f ͑x͒ ෇ t͑x ͒ 59. f ͑x͒ ෇ ͓ t͑x͔͒ 2 60. f ͑x͒ ෇ x at͑x b ͒ 61. f ͑x͒ ෇ t͑ t͑x͒͒ 62. f ͑x͒ ෇ sin͑ t͑x͒͒ 63. f ͑x͒ ෇ t͑sin x͒ 64. f ͑x͒ ෇ t(tan sx ) 2 ■ ■ 65–67 ■ |||| 2 ■ ■ ■ ■ ■ ■ Find hЈ in terms of f Ј and tЈ. 65. h͑x͒ ෇ f ͑x͒t͑x͒ f ͑x͒ ϩ t͑x͒ 66. h͑x͒ ෇ ͱ ■ 74. The cost, in dollars, of producing x units of a certain commod- ity is C͑x͒ ෇ 920 ϩ 2x Ϫ 0.02x 2 ϩ 0.00007x 3 ■ ■ (a) Find the marginal cost function. (b) Find CЈ͑100͒ and explain its meaning. (c) Compare CЈ͑100͒ with the cost of producing the 101st item. f ͑x͒ t͑x͒ 75. The volume of a cube is increasing at a rate of 10 cm3͞min. How fast is the surface area increasing when the length of an edge is 30 cm? 67. h͑x͒ ෇ f ͑ t͑sin 4x͒͒ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ 76. A paper cup has the shape of a cone with height 10 cm and ; 68. (a) Graph the function f ͑x͒ ෇ x Ϫ 2 sin x in the viewing rect- radius 3 cm (at the top). If water is poured into the cup at a rate of 2 cm3͞s, how fast is the water level rising when the water is 5 cm deep? angle ͓0, 8͔ by ͓Ϫ2, 8͔. (b) On which interval is the average rate of change larger: ͓1, 2͔ or ͓2, 3͔ ? (c) At which value of x is the instantaneous rate of change larger: x ෇ 2 or x ෇ 5? (d) Check your visual estimates in part (c) by computing f Ј͑x͒ and comparing the numerical values of f Ј͑2͒ and f Ј͑5͒. 77. A balloon is rising at a constant speed of 5 ft͞s. A boy is cycling along a straight road at a speed of 15 ft͞s. When he passes under the balloon, it is 45 ft above him. How fast is the distance between the boy and the balloon increasing 3 s later? 69. The graph of f is shown. State, with reasons, the numbers at 78. A waterskier skis over the ramp shown in the figure at a speed which f is not differentiable. of 30 ft͞s. How fast is she rising as she leaves the ramp? y 4 ft _1 0 2 4 6 x 15 ft 79. The angle of elevation of the Sun is decreasing at a rate of 0.25 rad͞h. How fast is the shadow cast by a 400-ft-tall building increasing when the angle of elevation of the Sun is ␲͞6? 70. A particle moves along a horizontal line so that its coordinate at time t is x ෇ sb 2 ϩ c 2 t 2, t ജ 0, where b and c are positive constants. (a) Find the velocity and acceleration functions. (b) Show that the particle always moves in the positive direction. ; 80. (a) Find the linear approximation to f ͑x͒ ෇ s25 Ϫ x 2 near 3. (b) Illustrate part (a) by graphing f and the linear approximation. (c) For what values of x is the linear approximation accurate to within 0.1? 71. A particle moves on a vertical line so that its coordinate at time t is y ෇ t 3 Ϫ 12t ϩ 3, t ജ 0. (a) Find the velocity and acceleration functions. (b) When is the particle moving upward and when is it moving downward? (c) Find the distance that the particle travels in the time interval 0 ഛ t ഛ 3. 72. The volume of a right circular cone is V ෇ ␲ r 2h͞3, where r is the radius of the base and h is the height. (a) Find the rate of change of the volume with respect to the height if the radius is constant. (b) Find the rate of change of the volume with respect to the radius if the height is constant. 3 81. (a) Find the linearization of f ͑x͒ ෇ s1 ϩ 3x at a ෇ 0. State ; the corresponding linear approximation and use it to give 3 an approximate value for s1.03. (b) Determine the values of x for which the linear approximation given in part (a) is accurate to within 0.1. 82. Evaluate dy if y ෇ x 3 Ϫ 2x 2 ϩ 1, x ෇ 2, and dx ෇ 0.2. 83. A window has the shape of a square surmounted by a semi- circle. The base of the window is measured as having width 60 cm with a possible error in measurement of 0.1 cm. Use differentials to estimate the maximum error possible in computing the area of the window. 5E-03(pp 206-217) 1/17/06 1:46 PM Page 217 CHAPTER 3 REVIEW 84–86 |||| x Ϫ1 84. lim x l1 x Ϫ 1 and f Ј͑x͒ ෇ 1 ϩ ͓ f ͑x͔͒ 2. Show that tЈ͑x͒ ෇ 1͑͞1 ϩ x 2 ͒. s16 ϩ h Ϫ 2 85. lim hl0 h 4 89. Find f Ј͑x͒ if it is known that cos ␪ Ϫ 0.5 86. lim ␪ l ␲͞3 ␪ Ϫ ␲͞3 ■ ■ ■ ■ 217 88. Suppose f is a differentiable function such that f ͑ t͑x͒͒ ෇ x Express the limit as a derivative and evaluate. 17 ❙❙❙❙ d ͓ f ͑2x͔͒ ෇ x 2 dx ■ ■ ■ ■ s1 ϩ tan x Ϫ s1 ϩ sin x 87. Evaluate lim . xl0 x3 ■ ■ ■ ■ 90. Show that the length of the portion of any tangent line to the astroid x 2͞3 ϩ y 2͞3 ෇ a 2͞3 cut off by the coordinate axes is constant. 5E-03(pp 218-221) 1/17/06 1:40 PM PROBLEMS PLUS Page 218 Before you look at the example, cover up the solution and try it yourself first. EXAMPLE How many lines are tangent to both of the parabolas y ෇ Ϫ1 Ϫ x 2 and y ෇ 1 ϩ x 2 ? Find the coordinates of the points at which these tangents touch the parabolas. SOLUTION To gain insight into this problem, it is essential to draw a diagram. So we sketch the parabolas y ෇ 1 ϩ x 2 (which is the standard parabola y ෇ x 2 shifted 1 unit upward) and y ෇ Ϫ1 Ϫ x 2 (which is obtained by reflecting the first parabola about the x-axis). If we try to draw a line tangent to both parabolas, we soon discover that there are only two possibilities, as illustrated in Figure 1. Let P be a point at which one of these tangents touches the upper parabola and let a be its x-coordinate. (The choice of notation for the unknown is important. Of course we could have used b or c or x 0 or x1 instead of a. However, it’s not advisable to use x in place of a because that x could be confused with the variable x in the equation of the parabola.) Then, since P lies on the parabola y ෇ 1 ϩ x 2, its y-coordinate must be 1 ϩ a 2. Because of the symmetry shown in Figure 1, the coordinates of the point Q where the tangent touches the lower parabola must be ͑Ϫa, Ϫ͑1 ϩ a 2 ͒͒. To use the given information that the line is a tangent, we equate the slope of the line PQ to the slope of the tangent line at P. We have y P 1 x _1 Q mPQ ෇ FIGURE 1 1 ϩ a 2 Ϫ ͑Ϫ1 Ϫ a 2 ͒ 1 ϩ a2 ෇ a Ϫ ͑Ϫa͒ a If f ͑x͒ ෇ 1 ϩ x 2, then the slope of the tangent line at P is f Ј͑a͒ ෇ 2a. Thus, the condition that we need to use is that 1 ϩ a2 ෇ 2a a Solving this equation, we get 1 ϩ a 2 ෇ 2a 2, so a 2 ෇ 1 and a ෇ Ϯ1. Therefore, the points are (1, 2) and (Ϫ1, Ϫ2). By symmetry, the two remaining points are (Ϫ1, 2) and (1, Ϫ2). 1. Find points P and Q on the parabola y ෇ 1 Ϫ x 2 so that the triangle ABC formed by the x-axis P RO B L E M S and the tangent lines at P and Q is an equilateral triangle. y 3 2 ; 2. Find the point where the curves y ෇ x Ϫ 3x ϩ 4 and y ෇ 3͑x Ϫ x͒ are tangent to each A other, that is, have a common tangent line. Illustrate by sketching both curves and the common tangent. 3. Suppose f is a function that satisfies the equation P B f ͑x ϩ y͒ ෇ f ͑x͒ ϩ f ͑ y͒ ϩ x 2 y ϩ xy 2 Q 0 C x for all real numbers x and y. Suppose also that lim FIGURE FOR PROBLEM 1 x l0 (a) Find f ͑0͒. 218 (b) Find f Ј͑0͒. f ͑x͒ ෇1 x (c) Find f Ј͑x͒. 5E-03(pp 218-221) 1/17/06 1:40 PM Page 219 y 4. A car is traveling at night along a highway shaped like a parabola with its vertex at the origin (see the figure). The car starts at a point 100 m west and 100 m north of the origin and travels in an easterly direction. There is a statue located 100 m east and 50 m north of the origin. At what point on the highway will the car’s headlights illuminate the statue? 5. Prove that x FIGURE FOR PROBLEM 4 dn ͑sin4 x ϩ cos4 x͒ ෇ 4nϪ1 cos͑4x ϩ n␲͞2͒. dx n 6. Find the n th derivative of the function f ͑x͒ ෇ x n͑͞1 Ϫ x͒. 7. The figure shows a circle with radius 1 inscribed in the parabola y ෇ x 2. Find the center of the circle. y y=≈ 1 1 0 x 8. If f is differentiable at a, where a Ͼ 0, evaluate the following limit in terms of f Ј͑a͒: f ͑x͒ Ϫ f ͑a͒ sx Ϫ sa lim xla 9. The figure shows a rotating wheel with radius 40 cm and a connecting rod AP with length 1.2 m. The pin P slides back and forth along the x-axis as the wheel rotates counterclockwise at a rate of 360 revolutions per minute. (a) Find the angular velocity of the connecting rod, d␣͞dt, in radians per second, when ␪ ෇ ␲͞3. (b) Express the distance x ෇ OP in terms of ␪. (c) Find an expression for the velocity of the pin P in terms of ␪. Խ Խ y A ¨ å P (x, 0) x O 10. Tangent lines T1 and T2 are drawn at two points P1 and P2 on the parabola y ෇ x 2 and they intersect at a point P. Another tangent line T is drawn at a point between P1 and P2 ; it intersects T1 at Q1 and T2 at Q2. Show that Խ PQ Խ ϩ Խ PQ Խ ෇ 1 Խ PP Խ Խ PP Խ 1 2 1 2 219 5E-03(pp 218-221) 1/17/06 1:41 PM Page 220 y 11. Let T and N be the tangent and normal lines to the ellipse x 2͞9 ϩ y 2͞4 ෇ 1 at any point P on yT the ellipse in the first quadrant. Let x T and y T be the x- and y-intercepts of T and x N and yN be the intercepts of N . As P moves along the ellipse in the first quadrant (but not on the axes), what values can x T , y T , x N , and yN take on? First try to guess the answers just by looking at the figure. Then use calculus to solve the problem and see how good your intuition is. T 2 P xT xN 0 3 N yN 12. Evaluate lim x xl0 sin͑3 ϩ x͒2 Ϫ sin 9 . x 13. (a) Use the identity for tan͑x Ϫ y͒ (see Equation 14b in Appendix D) to show that if two lines L 1 and L 2 intersect at an angle ␣, then tan ␣ ෇ FIGURE FOR PROBLEM 11 m2 Ϫ m1 1 ϩ m1 m2 where m1 and m2 are the slopes of L 1 and L 2 , respectively. (b) The angle between the curves C1 and C2 at a point of intersection P is defined to be the angle between the tangent lines to C1 and C2 at P (if these tangent lines exist). Use part (a) to find, correct to the nearest degree, the angle between each pair of curves at each point of intersection. (i) y ෇ x 2 and y ෇ ͑x Ϫ 2͒2 (ii) x 2 Ϫ y 2 ෇ 3 and x 2 Ϫ 4x ϩ y 2 ϩ 3 ෇ 0 14. Let P͑x 1, y1͒ be a point on the parabola y 2 ෇ 4px with focus F͑ p, 0͒. Let ␣ be the angle between the parabola and the line segment FP, and let ␤ be the angle between the horizontal line y ෇ y1 and the parabola as in the figure. Prove that ␣ ෇ ␤. (Thus, by a principle of geometrical optics, light from a source placed at F will be reflected along a line parallel to the x-axis. This explains why paraboloids, the surfaces obtained by rotating parabolas about their axes, are used as the shape of some automobile headlights and mirrors for telescopes.) y å 0 ∫ P(⁄, ›) y=› x F( p, 0) ¥=4px 15. Suppose that we replace the parabolic mirror of Problem 14 by a spherical mirror. Although Q P ¨ ¨ A R O the mirror has no focus, we can show the existence of an approximate focus. In the figure, C is a semicircle with center O. A ray of light coming in toward the mirror parallel to the axis along the line PQ will be reflected to the point R on the axis so that ЄPQO ෇ ЄOQR (the angle of incidence is equal to the angle of reflection). What happens to the point R as P is taken closer and closer to the axis? 16. If f and t are differentiable functions with f ͑0͒ ෇ t͑0͒ ෇ 0 and tЈ͑0͒ C lim xl0 FIGURE FOR PROBLEM 15 17. Evaluate lim xl0 220 f ͑x͒ f Ј͑0͒ ෇ t͑x͒ tЈ͑0͒ sin͑a ϩ 2x͒ Ϫ 2 sin͑a ϩ x͒ ϩ sin a . x2 0, show that 5E-03(pp 218-221) 1/17/06 1:41 PM Page 221 18. Given an ellipse x 2͞a 2 ϩ y 2͞b 2 ෇ 1, where a b, find the equation of the set of all points from which there are two tangents to the curve whose slopes are (a) reciprocals and (b) negative reciprocals. 19. Find the two points on the curve y ෇ x 4 Ϫ 2x 2 Ϫ x that have a common tangent line. 20. Suppose that three points on the parabola y ෇ x 2 have the property that their normal lines intersect at a common point. Show that the sum of their x-coordinates is 0. 21. A lattice point in the plane is a point with integer coordinates. Suppose that circles with radius r are drawn using all lattice points as centers. Find the smallest value of r such that any line with slope 2 intersects some of these circles. 5 22. A cone of radius r centimeters and height h centimeters is lowered point first at a rate of 1 cm͞s into a tall cylinder of radius R centimeters that is partially filled with water. How fast is the water level rising at the instant the cone is completely submerged? 23. A container in the shape of an inverted cone has height 16 cm and radius 5 cm at the top. It is partially filled with a liquid that oozes through the sides at a rate proportional to the area of the container that is in contact with the liquid. (The surface area of a cone is ␲ rl, where r is the radius and l is the slant height.) If we pour the liquid into the container at a rate of 2 cm3 ͞min, then the height of the liquid decreases at a rate of 0.3 cm͞min when the height is 10 cm. If our goal is to keep the liquid at a constant height of 10 cm, at what rate should we pour the liquid into the container? CAS 24. (a) The cubic function f ͑x͒ ෇ x͑x Ϫ 2͒͑x Ϫ 6͒ has three distinct zeros: 0, 2, and 6. Graph f and its tangent lines at the average of each pair of zeros. What do you notice? (b) Suppose the cubic function f ͑x͒ ෇ ͑x Ϫ a͒͑x Ϫ b͒͑x Ϫ c͒ has three distinct zeros: a, b, and c. Prove, with the help of a computer algebra system, that a tangent line drawn at the average of the zeros a and b intersects the graph of f at the third zero. 221 ...
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This note was uploaded on 02/08/2010 for the course M 340L taught by Professor Lay during the Spring '10 term at École Normale Supérieure.

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