HW1 - 20 CHAPTER1 FUNCTlONS AND LIMITS 7—10 Determine...

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Unformatted text preview: 20 CHAPTER1 FUNCTlONS AND LIMITS 7—10 Determine whether the curve is the graph of a function of x. If it is, state the domain and range of the function. 7. y 8.F M “ll a. / ,7, “1 L. a}? '0 i Tc %1 Tc 1. j. _' a. _/ LP 11 9. 10r TyA ’t 1T r 401—: + 11. The graph shown gives the weight of a certain person as a function of age. Describe in words how this person’s weight varies over time. What do you think happened when this person was 30 years old? 200 weight 150 d (poun s) .100 50 1 1 t «1- 1 l t + 0 10 20 30 40 50 60 70 age (years) 12. The graph shows the height of the water in a bathtub as a function of time. Give a verbal description of what you think happened. height (inches) 15 10 5 0 5 10 15 time (min) 13. ‘You put some ice cubes in a glass, fill the glass with cold water, and then let the glass sit on a table. Describe how the temperature of the water changes as time passes. Then sketch a rough graph of the temperature of the water as a function of the elapsed time. 14. Three runners compete in a lOO-meter race. The graph depicts the distance run as a function of time for each runner. Describe in words What the graph tells you about this race. Who won the race? Did each runner finish the race? y (In) 20 2(5) 15. The graph shows the power consumption for a clay in Septem- ber in San Francisco. (P is measured in megawatts; t is mea- sured in hours starting at midnight.) (a) What was the power consumption at 6 AM? At 6 PM? (b) When was the power consumption the lowest? When was it the highest? Do these times seem reasonable? P+— 1 f 300 a 1 L r f“ 7' ‘ A 600 j i .l 4 400 at J t .7. 200 l i .p . 1 7' 0 3 6 9 12 15 18 21 ? Pacific Gas & Electric 16. Sketch a rough graph of the number of hours of daylight as a function of the time of year. 1']. Sketch a rough graph of the outdoor temperature as a function of time during a typical spring day. 18. Sketch a rough graph of the market value of a new car as a function of time for a period of 20 years. Assume the car is well maintained. 19. Sketch the graph of the amount of a particular brand of coffee sold by a store as a function of the price of the coffee. 20. You place a frozen pie in an oven and bake it for an hour. Then you take it out and let it cool before eating it. Describe how the temperature of the pie changes as time passes. Then sketch a rough graph of the temperature of the pie as a function of time. 21. A homeowner mows the lawn every Wednesday afternoon. Sketch a rough graph of the height of the grass as a function of time over the course of a four—week period. 22. An airplane takes off from an airport and lands an hour later at another airport, 400 miles away. If t represents the time in min— utes since the plane has left the terminal building, let x(t) be the horizontal distance traveled and y(t) be the altitude of the plane. SECTION 1.1 FOUR WAYS TO REPRESENT A FUNCTION 21 (a) Sketch a possible graph of x(t). (b) Sketch a possible graph of y(t). (c) Sketch a possible graph of the ground speed. (d) Sketch a possible graph of the vertical velocity. 23. The number N (in millions) of US cellular phone subscribers is shown in the table. (Midyear estimates are given.) z‘ 1996 1998 2000 ‘i 2002 2004 2006 N '44 69 109 141 182 l 233 ‘ (a) Use the data to sketch a rough graph of N as a function of t. (b) Use your graph to estimate the number of cell-phone sub- scribers at midyear in 20071 and 2005. 24. Temperature readings T (in °F) were recorded every two hours from midnight to 2:00 PM in Phoenix on September 10, 2008. The time t was measured in hours from midnight. t02l_468101214 T 82 75 74 75 84 90 L 93 94 (a) Use the readings to sketch a rough graph of T as a function of t. (b) Use your graph to estimate the temperature at 9:00 AM. 25. If f(x) = 3x2 — x + 2, find f(2), f(—2). f(a). f(—a). N + 1), 2f(a), fan). flaz). [f(a)]2, and f(a + h). 26. A spherical balloon with radius r inches has volume D V(r) = §7rr3. Find a function that represents the amount of air required to inflate the balloon from a radius of r inches to a radius of r + 1 inches. 27—30 Evaluate the difference quotient for the given function. Simplify your answer. f(3 + h) —f(3) Z7.f(x)=4+3x-x2, h f(a + h) —f(a) 28. f(x) = x3, h 29. re) = hf“: :1; (a) m M = j: + 3 f(x) ~ fa) + l ’ x — 1 31—37 Find the domain of the function. x+4 2x3—5 x2+x—6 34. g(t) =1/3 — t — d2 +t 31. f(x) = 32. f (x) = x2—9 3?. f(t) = €/2t — 1 l u + 1 35. h(x) fi— 36. = 1 1 + u+l 31. F(p) = 1/2 — y; Y 38. Find the domain and range and sketch the graph of the function h(x) = 1/4 - x2_ ' 39—50 Find the domain and sketch the graph of the function. 39. f(x) = 2 — 0.4x 4D. F(x) =x2 - 2): +1 41. f(t) = 2x + t2 42. H(t) = 42?: 43. g(x)=«/x—5 44. F(x)=|2x+ ll 45. G(x) = Ex—lx—l 4B. g(x) = Ix] - x x+2 ifx<0 l—x ifxBO 47- f (x) __ 3—%x ifx$2 48'f(x)_{2x—51fx>2 x+2 ifx<—l 49. = fix) 1:2 ifx>—l x+9 ifx<—3 50.f(x)= —2x if [xf$3 —6 if-x>3 51~56 Find an expression for the function whose graph is the given curve. 51. The line segment joining the points (1, —3) and (5, 7) ' 52. The line segment joining the points (—5, 10) and (7, ~10) 53. The bottom half of the parabola x + (y — l)2 = O 54. The top half of the circle 2:2 + (y — 2)2 = 4 55. r 3,— T A! I 56. ' 4_ yr I’— . l RV» 57—61 Find a formula for the described function and state its domain. 57. A rectangle has perimeter 20 m. Express the area of the rect~ angle as a function of the length of one of its sides. 22 58. 59. 60. 61. 62. 63. E4. 65. 66. CHAPTER 1 FUNCTIONS AND LIMITS A rectangle has area 16 m2. Express the perimeter of the rect— angle as a function of the length of one of its sides. Express the area of an equilateral triangle as a function of the length of a side. Express the surface area of a cube as a function of its volume. An open rectangular box with volume 2 In3 has a square base. Express the surface area of the box as a function of the length .of a side of the base. A Norman window has the shape of a rectangle surmounted by a semicircle. If the perimeter of the window is 30 ft, express the area A of the window as a function of the width x of the window. .. 16% A box with an open top is to be constructed from a rectangular piece of cardboard with dimensions 12 in. by 20 in. by cutting out equal squares of side x at each corner and then folding up the sides as in the figure. Express the volume V of the box as a function of x. A cell phone plan has a basic charge of $35 a month. The plan includes 400 free minutes and charges 10 cents for each addi— tional minute of usage. Write the monthly cost C as a function of the number x of minutes used and graph C as a function of x for 0 S x S 600. In a certain state the maximum speed permitted on freeways is 65 mi/ h and the minimum speed is '40 mi/ h. The fine for vio- lating'these limits is $15 for every mile per hour above the imaximum speed or below the minimum speed. Express the amount of the fine F as a function of the driving speed )6 and graph F(x) for O S x S 100. An electricity company charges its customers a base rate of $10 a month, plus 6 cents per kilowatt—hour (kWh) for the first 1200 kWh and 7’cents per kWh for all usage over 1200 kWh. Express the monthly cost E as a function of the amount .7: of electricity used. Then graph the function E for 0 S x S 2000. 77. = 1 + 3x2 — x4 67. In a certain country, income tax is assessed as follows. There is no tax on income up to $10,000. Any income over $10,000 is taxed at a rate of 10%, up to an income of $20,000. Any income over $20,000 is taxed at 15%. _ (a) Sketch the graph of the tax rate R as a function of the income I. . (b) How much tax is assessed on an income of $14,000? On $26,000? (0) Sketch the graph of the total assessed tax T as a function of the income I. 58. The functions in Example 10 and Exercise 67 are called step functions because their graphs look like stairs. Give two other examples of step functions that arise in everyday life. 69—70 Graphs of f and g are shown. Decide whether each function is even, odd, or neither. Explain your reasoning. 59. 70. y 71. (a) If the point (5, 3) is on the graph of an even function, what other point must also be on the graph? (b) If the point (5, 3) is on the graph of an odd function, what other point must also be on the graph? 72. A function f has domain [—5, 5] and a portion of its graph is shown. (a) Complete the graph of f if it is known that f is even. (b) Complete the graph of f if it is known that f is odd. 73~7B Determine whether f is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually. Z 73. f(x) = x, + 1 74- f(x) = x4: 1 75.f(x)=x:1 7s.f(x)=x|x1 78. f(x)=1+ 3x3 - x5 SECTION 1.2 MATHEMATICAL MODELS: A CATALOG OF ESSENTIAL FUNCTIONS Exercises 33 1-2 Classify each function as a power function, root function, polynomial (state its degree), rational function, algebraic function, trigonometric function, exponential function, or logarithmic function. 1. (a)f(x)=10g2x (b) g(x) = v; 3 (c) h(x) = 1 _ x2 ((1) u(t) = 1 — 1.11 + 2.54!2 8‘ (e) 0(1) = 5’ (f) 16(0) = sin6 cos26 2-(a)y=77" (b)y=x" (c) y = x2(2 ~ x3) (d) y = tant — cos t s I V563 - l = f = _ (e) y 1 + s ( ) y 1 + {fac— 3—4 Match each equation with its graph. Explain your choices. 9- (Don’t use a computer or graphing calculator.) 3- (a)y=x2 (b) y=x5 (c) y =x8 4.(a)y=3x (b>y=3* (c) y = x3 (d) y T, 3/; 12. 5. (a) Find an equation for the family of linear functions with slope 2 and sketch several members of the family. (b) Find an equation for the family of linear functions such that f (2) = 1 and sketch several members of the family. (0) Which function belongs to both families? E3 Graphing calculator or computer required 7. 10. 11. 13. 14. B. What do all members of the family of linear functions f (x) = 1 + m(x + 3) have in common? Sketch several mem— bers of the family. What do all members of the family of linear functions f (x) = c — x have in common? Sketch several members of the family. Find expressions for the quadratic functions whose graphs are shown. Find an expression for a cubic function f if f (1) = 6 and f(—1)=f(0) =f(2) =0- Recent studies indicate that the average surface tempera— ture of the earth has been rising steadily. Some scientists have modeled the temperature by the linear function T = 0.02t + 8.50, where T is temperature in °C and t represents years since 1900. (a) What do the slope and T~intercept represent? (b) Use the equation to predict the average global surface temperature in 2100. If the recommended adult dosage for a drug is D (in mg), then to determine the appropriate dosage c for a child of age a, pharmacists use the equation 0 = 0.0417D(a + 1). Suppose the dosage for an adult is 200 mg. (a) Find the slope of the graph of c. What does it represent? (b) What is the dosage for a newborn? The manager of a weekend flea market knows from past expe— rience that if he Charges x dollars for a rental space at the mar- ket, then the number y of spaces he can rent is given by the equation y = 200 - 4x. (a) Sketch a graph of this linear function. (Remember that the rental charge per space and the number of spaces rented can’t be negative qtlantities.) , V (b) What do the slope, the y—intercept, and the x—intercept of the graph represent? The relationship between the Fahrenheit (F) and Celsius (C) temperature scales is given by the linear function F = §C + 32. (a) Sketch a graph of this function. (b) What is the slope of the graph and what does it represent? What is the F -intercept and what does it represent? Jason leaves Detroit at 2:00 PM and drives at a constant speed west along I—96. He passes Ann Arbor, 40 mi from Detroit, at 2:50 PM. (a) Express the distance traveled in terms of the time elapsed. 1. Homework Hints available at stewartcalculuscom “ ‘eu—NW ‘4“ 42 CHAPTER1 FUNCTIONS AND LIMITS Mere-hes 1- suppose the'gTaPh 0f f is gtVen- Write equations for the graphs 6—7 The graph of y = «3x — x2 is given. Use transformations to that are Obtained from the graph 0f f as fOHOWS- create a function whose graph is as shown. (a) Shift 3 units upward. (b) Shift 3 units downward. (c) Shift 3 units to the right. (d) Shift 3 units to the left. 3’ A (e) Reflect about the x-axis. (f) Reflect about the y-axis. _ r'——“2 (g) Stretch vertically by a factor of 3. 1‘5 " y _ 3x _ x (h) Shrink vertically by a factor of 3. f E 2. Explain how each graph is obtained from the graph of y = f (x). —0 3 x , (a)y=f(x)+3 (b)y=f(x+8) ‘ (c) y = 2W) . (d) y = mm 5. (e) y = —f<x) — 1 (f) y = 8f(%x) | 3. The graph of y = f (x) is given. Match each equation with its ' graph and give reasons for your choices. (a)y=f(x—4) (b) y=f(x)+3 (c) y = %f(x) (d) y = —f(x + 4) (6) y = 2f(x + 6) ® 8. (a) How is the graph of y = 2 sinx related to the graph of y = sin x? Use your answer and Figure 6 to sketch the graph of y = 2 sin x. (b) How is the graph of y = l + J); related to the graph of y = #7 Use your answer and Figure 4(a) to sketch the graphofy =1 + 9~24 Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Sec— tion 172, and then applying the appropriate transformations. 1 9.y= 111)):(115—1)3 4. The graph of f is given. Draw the graphs of the following 3‘ + 2 functions. 11. =—3x 12. =x2+6x+4 <a>y=f(x)—2 (b) y=f(x-2) y f y (c) y = —2f(x) (d) y = f(§x) + 1 13. y = 1/x — 2 — 1 14. y = 4sin 3x 3’ A ‘1 . 2 2 15. y = smex) 16. y = —x— — 2 __ , 17.y=%(1~cosx) 18.y=l—21/x+3 0 1 x ‘ 1S!.y=l—2x—x2 20;y=lx|—-2 5. The graph of f is given. Use it to graph the following 21' y __. l x _ 2' 22' y = i tan(x _ 1) functions. ‘ 4 4 z = i (3)3} f(2x) (my “236) 23_y=I\/;—1| 24.y=]cos'rrx{ (c) y =f(-x) (d) y = -f(—x) - _y.T 25. The city of New Orleans is located at latitude 30°N. Use Fig- ' 7 1 ure 9 to find a function that models the number of hours of 0 /————-> daylight at New Orleans as a function of the time of year. To $.41 \/ x check the accuracy of your model, use the fact that on March 31 L ' I the sun rises at 5:51 AM and sets at 6:18 PM in New Orleans. 1. Homework Hints available at stewartcalculuscom 26. A variable star is one whose brightness alternately increases and decreases. For the most visible variable star, Delta Cephei, the time between periods of maximum brightness is 5.4 days, the aVerage brightness (or magnitude) of the star is 4.0, and its brightness varies by :035 magnitude. Find a function that models the brightness of Delta Cephei as a function of time. 27. (a) How is the graph of y = f x related to the graph ‘of f? (b) Sketch the graph of y = sin [x (c) Sketch the graph of y = «Ix 23. Use the given graph of f to sketch the graph of y = l/ f (x). Which features of f are the most important in sketching y = 1/ f (x)? Explain how they are used. 29—30 Find (a) f + g, (b) f — g, (c) fg, and (d) f/g and state their domains. 29."if(x) = x3 + 2x2, g(x) = 3x2 — 1 30. f(x) = V3 —- x, g(x) = «Dc2 —1 31—36 Find the functions (a) f° g, Ch) 9 of, (c) fof, and (d) g 0 g and their domains. 31. f(x) = x2 — 1, g(x) = 2x +1 32. f(x) = x - 2, g(x) = x2 + 3x +n4 33., f(x) = l — 3x, g(x) = cosx 34. an = fi. 90:) = J31 — x l x+1 35‘ = +__ = f(x) x 3, go) H2 x 1+x’ 36. f (x) = g(x) = sin 2x 37—40 Find f0 g 0 h. 37. f(x) = 3x — 2, g(x) = sin x, 4106) = x2 38. f(x) = Ix — 41, g(x) = 2", h(x) = x5 39-f(x)=x/x— 3, g(x)=x2, h(x) =x3 +2 x . htx)=e/£ x-l fin-fa) = tan x, go.) = SECTION 1.3 NEW FUNCTIONS FROM OLD FUNCTIONS 43 41—46 Express the function in the form f O g. 41. F(x) = (2x + xz)4 42. F(x) = coszx 43. F(x) = 1 jg; 44. G(x) = .311 :x tant 1+ tant 45. v(t) = sec(t2) tan(t2) 46. 140?) = 47—49 Express the function in the form f 0 g 0 h. 47. R(x) = ./\/“ — 1 43. Hot) = .8/2 + m 49. H(x) = sec4(\/'x_) 50. Use the table to evaluate each expression. (a) f(g(1)) (b) 9(f(1)) (d) {MOD (6) (9 °f)(3) (C) f(f(1)) (f) (f° 9X6) x l 2 i 3 4 5 6 f(x) 3 1 4 2 2 5 g(x) 6 3 2 1 i 2 J 3 1 51. Use the given graphs of f and g to evaluate each expression, or explain why it is undefined. (a) f(9(2)) (b) 9(f(0)) (d) (9°f)(6) (e) (9°9)(r2) f (C) (f°g)(0) (f) (f °f)(4) l 52. Use the given graphs of f and g to estimate the value of f(g(x)) for x = —5, -4, —3, . . . , 5...Use these estimates to sketch a rough graph of f o g. T RY 44 CHAPTER 1 FUNCTIONS AND LIMITS 53. A stone is dropped into a lake, creating a circular ripple that travels outward at a speed of 60 cm/s. (a) Express the radius r of this circle as a function of the time t (in seconds). (b) If A is the area of this circle as a function of the radius, find A 0 r and interpret it. 54. A spherical balloon is being inflated and the radius of the bal- loon is increasing at a rate of 2 cm/s. ,. (a) Express the radius r of the balloon as a function of the time t (in seconds). . (b) If V is the volume of the balloon as a function of the radius, find V 0 r and interpret it. “I 55. A ship is moving at a speed of 30 km/ h parallel to a straight shoreline The ship is 6 km from shore and it passes a light« house at noon. ‘ (a) Express the distance s between the lighthouse and the ship as a function of d, the distance the ship has traveled since noon; that is, find f so that s =f(d). (b) Express d as a function of t, the time elapsed since noon; that is, find 9 so that d = 90‘). i (c) Find f 0 9. What does this function represent? 56. An airplane is flying at a speed of 350 mi/h at an altitude of one mile and passes directly over a radar station at time t = 0. (a) Express the horizontal distance d (in miles) that the plane has flown as a function of t. (b) Express the distance s between the plane and the radar station as a function of d. (c) Use composition to express s as a function of t. 57. The Heaviside function H is defined by 0 ift<0 Ht: 0 {1 ift>0 It is used in the study of electric cifcuits to represent the sudden surge of electric current, or voltage, when a switch is instantaneously turned on. (a) Sketch the graph of the Heaviside function. (b) Sketch the graph of the voltage V(t) in a circuit if the switch is turned on at time t = 0 and 120 volts are applied instantaneously to the circuit. Write a formula for V(t) in terms of H (t). The Tangent and Velocity Problems 58. 59. 60. 61. 62. 63. B4. (c) Sketch the graph of the voltage V(t) in a circuit if the switch is turned on at time t = 5 seconds and 240 volts are applied instantaneously to the circuit. Write a formula for V(t) in terms of H(t). (Note that starting at t = 5 corre- sponds to a translation.) The Heaviside function defined in Exercise 57 can also be used to define the ramp function y = ctH(t), which represents a gradual increase in voltage or current in a circuit. (a) Sketch the graph of the ramp function y = tH(t). (b) Sketch the graph of the voltage V0) in a circuit if the switch is turned on at time t = 0 and the voltage is gradu— ally increased to 120 volts over a 60—second time interval. Write a formula for V(t) in terms of H(t) for t < 60. (0) Sketch the graph of the voltage V(t) in a Circuit if the switch is turned on at time t = 7 seconds and the voltage is gradually increased to 100 volts over a period of 25 seconds. Write a formula for V(t) in terms of H (t) for t S 32. Let f and g be linear functions with equations f (x) = mlx + In and g(x) = mgx + b2. Is f 0 9 also a linear function? If so, what is the slope of its graph? If you invest x dollars at 4% interest compounded annually, then the amount A(x) of the investment after one year is A(x)=1.04x.FindA 0A, A DA 0A, andA C>A C’A 0A. What do these compositions represent? Find a formula for the com— position of 11 copies of A. (a) If g(x) = 2x + l and h(x) = 4x2 + 4x + 7, find a function f such that f 0 g = h. (Think about what operations you would have to perform on the formula for g to end up with the formula for h.) (b) If f(x) = 3x + 5 and h(x) = 3x? + 3x + 2, find a function 9 such that f0 g = h. Iff(x) = x + 4 and h(x) = 4x — 1, find a function 9 such that g of: 11. Suppose g is an even function and let h = f 0 g. Is It always an even function? Suppose g is an odd function and let h = f 0 g. Is h always an odd function? What if f is odd? What if f is even? In this section we see how limits arise when we attempt to find the tangent to a curve or the velocity of an object. The Tangent Problem The word tangent is derived from the Latin word tangens, which means “touching.” Thus a tangent to a curve is a line that touches the curve. In other words, a tangent line should have the same direction as the curve at the point of contact. How can this idea be made precise? Exercises SECTION 1.4 THE TANGENT AND VELOCITY PROBLEMS 49 , A tank holds 1000 gallons of water, which drains from the bottom of the tank in half an hour. The values in the table show the volume V of water remaining in the tank (in gallons) after t minutes. t(min) I 5 V(ga1) I 694 10 I 15 20 25 30 444 I 250‘ 111 28 0 (a) If P is the point (15, 250) on the graph of V, find the slopes of the secant lines PQ when Q is the point on the graph with t = 5, 10, 20, 25, and 30. (b) Estimate the slope of the tangent line at P by averaging the slopes of two secant lines. (c) Use a graph of the function to estimate the slope of the tangent line at P. (This slope represents the rate at which the water is flowing from the tank after 15 minutes.) . A cardiac monitor is used to measure the heart rate of a patient after surgery. It compiles the number of heartbeats after t min- utes. When the data in the table are graphed, the slope of the tangent line represents the heart rate in beats per minute. t (min) 36 Heartbeats 2530 38 I 40 42 44 2661 I2806 2948 3080 The monitor estimates this value by calculating the slope of a secant line. Use the data to estimate the patient’s heart rate after 42 minutes using the secant line between the points with the given values of t. (a)t=36 and t=42 (c)t=40 and t=42 What are your conclusions? (b) t = 38 and (d) t = 42 and t=42 t=44 . The point P(2, —1) lies on the curvey = 1/(1 — x). (a) If Q is the point (x, 1/(1 — x)), use your calculator to find the slope of the secant line PQ (correct to six decimal places) for the following values of x: (i) 1.5 (ii) 19 (iii) 1.99 (iv) 1.999 ' (v) 2.5 (vi) 2.1 (vii) 2.01 (viii) 2.001 (b) Using the results of part (a), guess the value of the slope of the tangent line to the curve at P(2, — 1). (c) Using the slope from part (b), find an equation of the tangent line to the curve at P(2, ~1). - The point P(0.5, 0) lies on the curve y = cos rn'x. (a) If Q is the point (x, cos 77x), use your calculator to find the slope of the secant line PQ (correct to six decimal places) for the following values of x: (i) 0 (ii) 0.4 (iii) 0.49 (iv) 0.499 (V) 1 (vi) 0.6 (vii) 0.51 (viii) 0.501 (b) Using the results of part (a), guess the value of the slope of the tangent line to the curve at P(0.5, 0). . Graphing calculator or computer required n (c) Using the slope from part (b), find an equation of the tangent line to the curve at P(0.5, 0). (d) Sketch the curve, two of the secant lines, and the tangent line. . If a ball is thrown into the air with a velocity of 40 ft/s, its height in feet 1‘ seconds later is given by y = 401 — 16t2. (a) Find the average velocity for the time period beginning when t = 2 and lasting * (i) 0.5 second (ii) 0.1 second (iii) 0.05 second (iv) 0.01 second (b) Estimate the instantaneous velocity when t = 2. . If a rock is thrown upward on the planet Mars with a velocity of 10 m/s, its height in meters t seconds later is given by y = 10: — 1.86%. (a) Find the average velocity over the given time intervals: (i) [1, 2] (ii) [1,15] (iii) [1,1.1] (iv) [1, 1.01] (v) [1, 1.001] (b) Estimate the instantaneous velocity when t = 1. . The table shows the position of a cyclist. t(seconds) 0 1 2 3 4 5 I 3 (meters) 0 1.4 5.1 10.7 17.7- 25.8 I (a) Find the average velocity for each time period: (1) [113] (ii) [2, 3] (iii) [3. 5] (iv) [3, 4] (b) Use the graph of s as a function of t to estimate the instan- taneous velocity when t = 3. . The displacement (in centimeters) of a particle moving back and forth along a straight line is given by the equation of motion 5 = 2 sin wt + 3 cos 171‘, where t is measured in ’ seconds. (a) Find the average velocity during each time period: (i) [1,2] (ii) [1,1.1] (iii) [1,1.01] (iv) [1,1001] (b) Estimate the instantaneous velocity of the particle when t = 1. - - . The point P(l, 0) lies on the curve y = sin(107T/x). (a) If Q is the point (x, sin(101T/x)), find the slope of the secant line PQ (correct to four decimal places) for x = 2, 1.5, 1.4, 1.3, 1.2, 1.1, 0.5, 0.6, 0.7, 0.8, and 0.9. Do the slopes appear to be approaching a limit? (b) Use a graph of the curve to explain why the slopes of the secant lines in part (a) are not close to the slope of the tan- gent line at P. (c) By choosing appropriate secant lines, estimate the slope of the tangent line at P. 1. Homework Hints available at stewartcalculuscorn ...
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HW1 - 20 CHAPTER1 FUNCTlONS AND LIMITS 7—10 Determine...

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