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Unformatted text preview: 5E03(pp 126135) 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 5E03(pp 126135) 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 5E03(pp 126135) 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 xa 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 ƒf(a) P a x x ƒf(a)
xa
x=a
=slope of tangent at P
=slope of curve at P (b) f ª(a)=lim If we use the pointslope 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 5E03(pp 126135) 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 xscale and the yscale 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 5E03(pp 126135) 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 yvalues 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 yvalues change rapidly. When the derivative is small, the
curve is relatively ﬂat and the yvalues 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. 5E03(pp 126135) 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 largescale 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 5E03(pp 126135) 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 tЈ͑͞4͒ 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. ■ 5E03(pp 126135) 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. ■ 5E03(pp 126135) 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: SpringerVerlag, 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 5E03(pp 126135) 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
/ SlopeaScope
/ 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 xaxis 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. 5E03(pp 136145) 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 mid1988. 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 5E03(pp 136145) 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. 5E03(pp 136145) 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„„„„
x1 EXAMPLE 5 Find f Ј if f ͑x͒ SOLUTION f Ј͑x͒ lim hl0 0 x 1 (b) f ª(x)= 1
2œ„„„„
x1 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. 5E03(pp 136145) 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
xa dy
dx ͬ xa 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 5E03(pp 136145) 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 5E03(pp 136145) 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 5E03(pp 136145) 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 5E03(pp 136145) 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. 5E03(pp 136145) 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. 5E03(pp 136145) 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 hotwater 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 lefthand and righthand 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 xaxis. 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 5E03(pp 146155) 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 righthand 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 5E03(pp 146155) 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 5E03(pp 146155) 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. 5E03(pp 146155) 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. 5E03(pp 146155) 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 5E03(pp 146155) 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. 5E03(pp 146155) 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 5E03(pp 146155) 1/17/06 2:19 PM Page 153 SECTION 3.3 DIFFERENTIATION FORMULAS ❙❙❙❙ 153 5E03(pp 146155) ❙❙❙❙ 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 5E03(pp 146155) 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 a4 ■ 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 ͒ ͩ a1 ■ ; 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 x2 5E03(pp 156165) 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 RichmondPetersburg, 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 5E03(pp 156165) 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 RichmondPetersburg
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 yaxis 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 5E03(pp 156165) 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 t2 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 5E03(pp 156165) 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 saxis).
(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 x1 1
2sx Ϳ x1 0.50 kg͞m 5E03(pp 156165) 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 5E03(pp 156165) 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
P50 Ϫ 5.3
P2 Ϳ P50 5.3
Ϫ0.00212 m 3͞kPa
2500 The compressibility at that pressure is Ϫ 1 dV
V dP Ϳ P50 0.00212
0.02 ͑m 3͞kPa͒͞m 3
5.3
50 5E03(pp 156165) 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. 5E03(pp 156165) 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 JeanLouisMarie 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
4l
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 5E03(pp 156165) 164 ❙❙❙❙ 1/17/06 2:12 PM Page 164 CHAPTER 3 DERIVATIVES and the velocity gradient at that point is
dv
dr Ϳ r0.002 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 largescale 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 5E03(pp 156165) 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 socalled 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.” 5E03(pp 166175) 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 5E03(pp 166175) 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 thirddegree 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 fourthdegree 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. 5E03(pp 166175) 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, predatorprey 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? 5E03(pp 166175) 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
/ SlopeAScope 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 5E03(pp 166175) 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
5E03(pp 166175) 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, 5E03(pp 166175) 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 5E03(pp 166175) 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 realworld 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 5E03(pp 166175) ❙❙❙❙ 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. h͑͒ 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 ͑, Ϫ͒.
(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 2͞3. 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 5E03(pp 166175) 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 icecream 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. 5E03(pp 176185) 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 5E03(pp 176185) 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 5E03(pp 176185) 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 doubleangle 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 5E03(pp 176185) 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 5E03(pp 176185) 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 5E03(pp 176185) 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 5E03(pp 176185) 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 2͑3͒ 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 bulletnose 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 xcoordinates 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 5E03(pp 176185) 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 xaxis 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.) 5E03(pp 176185) 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 . 5E03(pp 176185) 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 5E03(pp 186195) 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 5E03(pp 186195) 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 xϩy6 (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 ﬁfthdegree 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
nthdegree 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 implicitplotting 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. 5E03(pp 186195) 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 velocityequipotential 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
ϩy0
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 ■ ■ ■ ■ ■ 5E03(pp 186195) 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 xcoordinates of these points.
(b) Find equations of the tangent lines at the points (0, 1)
and (0, 2).
(c) Find the exact xcoordinates 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 xcoordinates 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 yintercepts 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 ■ ■ ■ ■ ■ ■ ■ ■ ■ 5E03(pp 186195) 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 xaxis
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 yaxis 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 xaxis 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 5E03(pp 186195) 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? 5E03(pp 186195) 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 5E03(pp 186195) 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. 5E03(pp 186195) 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 5E03(pp 186195) 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. t͑͒ 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. 5E03(pp 196205) 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 tٞ͑͞4͒.
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 xaxis, 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. 5E03(pp 196205) 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 seconddegree 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 thirddegree 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 ͒ ■ ptimes differentiable and f ͑ p͒ f . Using mathematical induction, show that f is in fact ntimes 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. 5E03(pp 196205) 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 5E03(pp 196205) 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
4r 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 1͑͞25͒ cm͞s. 5E03(pp 196205) 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 5E03(pp 196205) 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 8͑͞9͒ 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 problemsolving 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 ͪ 5E03(pp 196205) 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 d͞dt 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 ycoordinate is increasing at a rate of
4 cm͞s. How fast is the xcoordinate 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. 5E03(pp 196205) 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 15fttall 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 xcoordinate 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 crosssection 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? 5E03(pp 196205) 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? 5E03(pp 196205) 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. 5E03(pp 206217) 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 5E03(pp 206217) 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+30.1 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 xcoordinate of P is about
Ϫ2.66 and the xcoordinate 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+30.5 _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 5E03(pp 206217) 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: AddisonWesley, 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. 5E03(pp 206217) 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 5E03(pp 206217) 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.  a1 6. f ͑x͒ 1͞s2 ϩ x, a ͞2 7. f ͑x͒ cos x,
8. f ͑x͒ sx,
3 ■ ■ ■ a0 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 ■ 5E03(pp 206217) 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 balloontipped
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 5E03(pp 206217) 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 ﬁrstdegree (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 seconddegree (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 higherdegree polynomials. We look for an nthdegree
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 5E03(pp 206217) 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 nthdegree Taylor polynomial of f centered at a.
6. Find the 8thdegree 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. ■ TRUEFALSE 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 5E03(pp 206217) 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 5E03(pp 206217) 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 doubleangle formula cos 2x cos2x Ϫ sin2x 4 ■ ■ ■ ■ ■ ■ ■ ■ obtain the doubleangle 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 Љ͑͞6͒. 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, Ϫ͞2 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 5E03(pp 206217) 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 400fttall 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. 5E03(pp 206217) 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. 5E03(pp 218221) 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 xaxis). 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 xcoordinate. (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 ycoordinate 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 xaxis 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. 5E03(pp 218221) 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 n͞2͒.
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 xaxis 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 5E03(pp 218221) 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 yintercepts 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
xaxis. 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 5E03(pp 218221) 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 xcoordinates 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.
 Spring '10
 Lay
 Linear Algebra, Algebra, The Land

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