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Ron Larson, Calculus, 8th Edition. Boston, Houghton Mifflin, 2006.

3.1 Exercises

In Exercises 13–18, find any critical numbers of the function.
13. f(x) = x2(x – 3)
In Exercises 19–36, locate the absolute extrema of the function on the closed interval.
19. f(x) = 3(3 – x), [-1, 2]

59. Power. The formula for the power output P of a battery is P = VI – RI2, where V is the electromotive force in volts, R is the resistance, and I is the current. Find the current (measured in amperes) that corresponds to a maximum value of P in a battery for which V = 12 volts and R = 0.5 ohm. Assume that a 15-ampere fuse bounds the output in the interval 0 ≤ I ≤ 15. Could the power output be increased by replacing the 15-ampere fuse with a 20-ampere fuse? Explain.

60. Inventory Cost. A retailer has determined that the cost C of ordering and storing x units of a product is C = 2x + 300000/x, 1 ≤ x ≤ 300. The delivery truck can bring at most 300 units per order. Find the order size that will minimize cost. Could the cost be decreased if the truck were replaced with one that could bring at most 400 units? Explain.

61. Lawn Sprinkler. A lawn sprinkler is constructed in such a way that dθ/dt is constant, where θ ranges between 45º and 135º (see figure). The distance the water travels horizontally is x = v2 sin 2θ/32, 45º ≤ θ ≤ 135º where v is the speed of the water. Find dx/dt and explain why this lawn sprinkler does not water evenly. What part of the lawn receives the most water?

62. Honeycomb. The surface area of a cell in a honeycomb is S = 6hs + 3s2/2((√3 - cos θ)/sin θ) where  h and are positive constants and θ is the angle at which the upper faces meet the altitude of the cell (see figure). Find the angle θ (π/6 ≤ θ ≤ π/2) that minimizes the surface area S.

69. Highway Design. In order to build a highway, it is necessary to fill a section of a valley where the grades (slopes) of the sides are 9% and 6% (see figure). The top of the filled region will have the shape of a parabolic arc that is tangent to the two slopes at the points A and B. The horizontal distance between the points A and B is 1000 feet. (a) Find a quadratic function y = ax2 + bx + c, -500 ≤ x ≤ 500, that describes the top of the filled region. (b) Construct a table giving the depths x = -500, -400, -300, -200, -100, 0, 100, 200, 300, 400, and 500. (c)  What will be the lowest point on the completed highway? Will  it  be  directly  over  the  point  where  the  two  hillsides come together?


3.2 Exercises

29. Vertical Motion. The height of a ball t seconds after it is thrown upward from a height of 32 feet and with  an  initial velocity of 48 feet per second is f(t) = -16t2 + 48t + 32. (a) Verify that f(1) = f(2). (b) According to Rolle’s Theorem, what must be the velocity at some time in the interval (1, 2)? Find that time.

30. Reorder Costs. The ordering and transportation cost C for components used in a manufacturing process is approximated by C(x) = 10(1/x + x/(x + 3)), where C is measured in thousands of dollars and x is the order size in hundreds. (a) Verify that C(3) = C(6). (b) According to Rolle’s Theorem, the rate of change of the cost must be 0 for some order size in the interval. Find that order size.

37. Mean Value Theorem. Consider the graph of the function f(x) = x2 + 1. (a) Find the equation of the secant line joining the points (-1, 2) and (2, 5). (b) Use the Mean Value Theorem to determine a point c in the interval (-1, 2) such that the tangent line at c is parallel to the secant line. (c) Find the equation of the tangent line through c. (d) Then use a graphing utility to graph f, the secant line, and the tangent line.

51. Vertical Motion. The height of an object t seconds after it is dropped from a height of 500 meters is s(t) = -4.9t2 + 500. (a) Find the average velocity of the object during the first 3 seconds. (b) Use the Mean Value Theorem to verify that at some time during the first 3 seconds of fall the instantaneous velocity equals the average velocity. Find that time.

52. Sales. A company introduces a new product for which the number of units sold S is S(t) = 200(5 – 9/(2 + t)) where t is the time in months. (a) Find the average value of S(t) during the first year. (b)  During what month does S′(t) equal the average value during the first year?

57. Speed. A plane begins its takeoff at 2:00 P.M. on a 2500-mile flight. The plane arrives at its destination at 7:30 P.M. Explain why there are at least two times during the flight when the speed of the plane is 400 miles per hour.

58. Temperature. When an object is removed from a furnace and placed in an environment with a constant temperature of 90 ºF, its core temperature is 1500 ºF. Five hours later the core temperature is 390 ºF. Explain why there must exist a time in the interval when the temperature is decreasing at a rate of 222 ºF per hour.

59. Velocity. Two bicyclists begin a race at 8:00 A.M. They both finish the race 2 hours and 15 minutes later. Prove that at some time during the race, the bicyclists are traveling at the same velocity.

60. Acceleration. At 9:13 A.M., a sports car is traveling 35 miles per hour. Two minutes later, the car is traveling 85 miles per hour. Prove that at some time during this two-minute interval, the car’s acceleration is exactly 1500 miles per hour squared.


3.3 Exercises

137. Find a second-degree polynomial f(x) = ax2 + bx + c such that its graph has a tangent line with slope 10 at the point (2, 7) and an x-intercept at (1, 0).

138. Consider the third-degree polynomial f(x) = ax3 + bx2 + cx + d, a ≠ 0. Determine conditions for a, b, c, and d if the graph of f has (a) no horizontal tangent lines, (b) exactly one horizontal tangent line, and (c) exactly two horizontal tangent lines. Give an example for each case.

139. Find the derivative of f(x) = x|x|. Does f″(0) exists?


3.4 Exercises

161. Horizontal Tangent line. Determine the point(s) in the interval (0, 2π) at which the graph of f(x) = 2cosx + sin2x has a horizontal tangent line.

167. Doppler Effect. The frequency F of a fire truck siren heard by a stationary observer is F = 132400/(331 ± v) where v represent the velocity of the accelerating fire truck in meters per second. Find the rate of change of F with respect to v when (a) the fire truck is approaching at a velocity of 30 meters per second (b) the fire truck is moving away a velocity of 30 meters per second.

168. Harmonic Motion. The displacement from equilibrium of an object in harmonic motion on the end of a spring is y = 1/3 cos 12t – ¼ sin 12t where y is measured in feet and t is the time in seconds. Determine the position and velocity of the object when t = π/8.

169. A 15-centimeter pendulum moves according to the equation θ = 0.2 cos 8t, where θ is the angular displacement from the vertical in radians and t is the time in seconds. Determine the maximum angular displacement and the rate of change of θ when t = 3 seconds.

170. Wave motion. A buoy oscillates in simple harmonic motion y = A cos ωt as waves move past it. The buoy moves a total of 3.5 feet (vertically) from its low point to its high point. It returns to its high point every 10 seconds. (a) Write an equation describing the motion of the buoy if it is at its high point at t = 0. (b) Determine the velocity of the buoy as a function of t.

171. Circulatory System. The speed S of blood that is r centimeters from the center of an artery is S = C(R2 – r2) where C is a constant, R is the radius of the artery, and S is measured in centimeters per second. Suppose a drug is administered and the artery begins to dilate at a rate of dR/dt. At a constant distance r, find the rate at which S changes with respect to t for C = 1.76 × 105, R = 1.2 × 10-2, and dR/dt = 10-5.

172. Modeling Data. The normal daily maximum temperatures (in degrees Fahrenheit) for Chicago, Illinois are shown in the table. (a) Use a graphing utility to plot the data and find a model for the data of the form T(t) = a + b sin(ct – d) where T is the temperature and t is the time in months, with t = 1 corresponding to January. (b) Use a graphing utility to graph the model. How well does the model fit the data? (c)  Find T′ and use a graphing utility to graph the derivative. (d) Based on the graph of the derivative, during what times does the temperature change most rapidly? Most slowly? Do your answers agree with your observations of the temperature changes? Explain.

173. Volume. Air is being pumped into a spherical balloon so that the radius is increasing at the rate of dr/dt = 3 inches per second. What is the rate of change of the volume of the balloon, in cubic inches per second, when r = 8 inches?

174. Think About It. The table shows some values of the derivative of an unknown function f. Complete the table by finding (if possible) the derivative of each transformation of f.

175.  Modeling Data. The table shows the temperatures T(°F) at which water boils at selected pressures p (pounds per square inch). A model that approximates the data is T = 87.97 + 34.96  ln p + 7.91√p. (a) Use a graphing utility to plot the data and graph the model. (b) Find the rates of change of T with respect to p when p = 10 and p = 70.

176. Depreciation. After t years, the value of a car purchased for $25,000 is V(t) = 25000(3/4)t. (a)  Use a graphing utility to graph the function and determine the value of the car 2 years after it was purchased. (b)  Find the rates of change of V with respect to t when t = 1 and r = 4.

177. Inflation. If the annual rate of inflation averages 5% over the next 10 years, the approximate cost C of goods or services during any year in that decade is C(t) = P(1.05)t where t is the time in years and P is the present cost. (a) If the price of an oil change for your car is presently $29.95, estimate the price 10 years from now. (b) Find the rates of change of C with respect to t when t = 1 and r = 8. (c) Verify that the rate of change of C is proportional to C. What is the constant of proportionality?

178. Finding a Pattern. Consider the function f(x) = sin βx, where β is a constant. (a) Find the first-, second-, third-, and fourth-order derivatives of the function. (b) Verify that the function and its second derivative satisfy the equation f″(x) + β2 f(x) = 0. (c) Use the results of part (a) to write general rules for the even- and odd-order derivatives f(2k)(x) and f(2k-1)(x).

179. Conjecture. Let f be a differentiable function of period p. (a) Is the function f′ periodic? Verify your answer. (b) Consider the function g(x) = f(2x). Is the function g′(x) periodic? Verify your answer.

3.5 Exercises


53. (a) Use implicit differentiation to find an equation of the tangent line to the ellipse x2/2 + y2/2 = 1 at (1, 2). (b) Show that the equation of the tangent line to the ellipse x2/a2 + y2/b2 = 1.

54. (a) Use implicit differentiation to find an equation of the tangent line to the hyperbola x2/6 - y2/8 = 1 at (3, -2). (b) Show that the equation of the tangent line to the hyperbola x2/a2 - y2/b2 = 1.

67. Show that the normal line at any point on the circle x2 + y2 = r2 passes through the origin.

68. Two circles of radius 4 are tangent to the graph of y2 = 4x at the point (1, 2). Find equations of these two circles.

101. Slope. Find all points on the circle x2 + y2 = 100 where the slope is ¾.

102. Horizontal tangent line. Determine the point(s) at which the graph of y4 = y2 – x2 has a horizontal tangent line.

103. Tangent lines. Find equations of both tangent lines to the ellipse x2/4 + y2/9 = 1that passes through the point (4, 0).

104. Normals to a Parabola. The graph shows the normal lines from the point (2, 0) to the graph of the parabola x = y2. How many normal lines are there from the point (x0, 0) to the graph of the parabola if (a) x0 = ¼, (b) x0 = ½, and (c) x0 = 1? For what value of  x0 are two of the normal lines perpendicular to each other?

105. Normal Lines. (a) Find an equation of the normal line to the ellipse x2/32 + y2/8 = 1 at the point (4, 2). (b) Use a graphing utility to graph the ellipse and the normal line. (c) At what other point does the normal line intersect the ellipse?


3.6 Exercises

53. Find equations of all tangent lines to the graph f(x) = arcos x that have slope -2.

54. Find an equation of the tangent line to the graph of g(x) = arctan x when x = 1.

67. Angular Rate of Change. An airplane flies at an altitude of 5 miles toward a point directly over an observer. Consider θ and x as shown in the figure. (a) Write θ as a function of x. (b) The speed of the plane is 400 miles per hour. Find dθ/dt when x = 10 miles and x = 3 miles.

68. Writing. Repeat Exercise 67 if the altitude of the plane is 3 miles and describe how the altitude affects the rate of change of θ.

69. Angular Rate of Change. In a free-fall experiment, an object is dropped from a height of 256 feet. A camera on the ground 500 feet from the point of impact records the fall of the object. (a) Find the position function giving the height of the object at time t, assuming the object is released at time t = 0.  At what time will the object reach ground level? (b) Find the rates of change of the angle of elevation of the camera when t = 1 and t = 2.

70. Angular Rate of Change. A television camera at ground level is filming the lift-off of a space shuttle at a point 800 meters from the launch pad. Let θ be the angle of elevation of the shuttle and let s be the distance between the camera and the shuttle. Write θ as a function of s for the period of time when the shuttle is moving vertically. Differentiate the result to find dθ/dt in terms of s and ds/dt.

71. Angular Rate of Change. An observer is standing 300 feet from the point at which a balloon is released. The balloon rises at a rate of 5 feet per second. How fast is the angle of elevation of the observer’s line of sight increasing when the balloon is 100 feet high?

72. Angular Speed. A patrol car is parked 50 feet from a long warehouse. The revolving light on top of the car turns at a rate of 30 revolutions per minute. Write θ as a function of x. How fast is the light beam moving along the wall when the beam makes an angle of θ = 45° with the line perpendicular from the light to the wall?

74. Existence of an Inverse. Determine the values of k such that the function f(x) = kx + sin x has an inverse function.


3.7 Exercises


11. Find the rate of change of the distance between the origin and a moving point on the graph of y = x2 + 1 if dx/dt = 2 centimeters per second.

12. Find the rate of change of the distance between the origin and a moving point on the graph of y = sin x if dx/dt = 2 centimeters per second.

13. Area. The radius r of a circle is increasing at a rate of 4 centimeters per minute. Find the rates of change of the area when (a) r = 8 centimeters and (b) r = 32 centimeters.

14.  Area. Let A be the area of a circle of radius r that is changing with respect to time. If dr/dt is constant, is dA/dt constant? Explain.

15. Area. The included angle of the two sides of constant equal length s of an isosceles triangle is θ. (a) Show that the area of the triangle is given by A = ½ s2 sinθ. (b) If θ is increasing at the rate of ½ radian per minute, find the rates of change of the area when θ =π/6 and θ =π/3. (c) Explain why the rate of change of the area of the triangle is not constant even though dθ/dt is constant.

16. Volume. The radius r of a sphere is increasing at a rate of 3 inches per minute. (a) Find the rates of change of the volume when r = 9 inches and r = 36 inches. (b) Explain why the rate of change of the volume of the sphere is not constant even though dr/dt is constant.

17. Volume. A hemispherical water tank with radius 6 meters is filled to a depth of h meters. The volume of water in the tank is given by V = 1/3πh(108 – h2), 0<h<6. If water is being pumped into the tank at the rate of 3 cubic meters per minute, find the rate of change of the depth of the water when h=2 meters.

18. Volume. All edges of a cube are expanding at a rate of 6 centimeters per second. How fast is the volume changing when each edge is (a) 2 centimeters and (b) 10 centimeters?

19. Surface Area. The conditions are the same as in Exercise 18. Determine how fast the surface area is changing when each edge is (a) 2 centimeters and (b) 10 centimeters.

20. Volume. The formula for the volume of a cone is V = 1/3πr2h. Find the rates of change of the volume if dr/dt is 2 inches per minute and h=3r when (a) r = 6 inches and (b) r=24 inches.

21. Volume. At a sand and gravel plant, sand is falling off a conveyor and onto a conical pile at a rate of 10 cubic feet per minute. The diameter of the base of the cone is approximately three times the altitude. At what rate is the height of the pile changing when the pile is 15 feet high?

22. Depth. A conical tank (with vertex down) is 10 feet across the top and 12 feet deep. If water is flowing into the tank at a rate of 10 cubic feet per minute, find the rate of change of the depth of the water when the water is 8 feet deep.

23. Depth. A swimming pool is 12 meters long, 6 meters wide, 1 meter deep at the shallow end, and 3 meters deep at the deep end. Water is being pumped into the pool at ¼ cubic meter per minute, and there is 1 meter of water at the deep end. (a)  What percent of the pool is filled? (b)  At what rate is the water level rising?

24. Depth. A trough is 12 feet long and 3 feet across the top. Its ends are isosceles triangles with altitudes of 3 feet. (a)  If water is being pumped into the trough at 2 cubic feet per minute, how fast is the water level rising when the depth is 1 foot? (b) If the water is rising at a rate of 3/8 inch per minute when determine the rate at which water is being pumped into the trough.

25. Moving Ladder. A ladder 25 feet long is leaning against the wall of a house. The base of the ladder is pulled away from the wall at a rate of 2 feet per second. (a) How fast is the top of the ladder moving down the wall when its base is 7 feet, 15 feet, and 24 feet from the wall? (b) Consider the triangle formed by the side of the house, the ladder, and the ground. Find the rate at which the area of the triangle is changing when the base of the ladder is 7 feet from the wall. (c) Find the rate at which the angle between the ladder and the wall of the house is changing when the base of the ladder is 7 feet from the wall.

26. Construction. A construction worker pulls a five-meter plank up the side of a building under construction by means of a rope tied to one end of the plank (see figure). Assume the opposite end of the plank follows a path perpendicular to the wall of the building and the worker pulls the rope at a rate of 0.15 meter per second. How fast is the end of the plank sliding along the ground when it is 2.5 meters from the wall of the building?

27. Construction. A winch at the top of a 12-meter building pulls a pipe of the same length to a vertical position, as shown in the figure. The winch pulls in rope at a rate of -0.2 meter per second. Find the rate of vertical change and the rate of horizontal change at the end of the pipe when y = 6.

28. Boating. A boat is pulled into a dock by means of a winch 12 feet above the deck of the boat. (a)  The  winch  pulls  in  rope  at  a  rate  of  4  feet  per  second. Determine the speed of the boat when there is 13 feet  of rope out. What happens to the speed of the boat as it gets closer to the dock? (b)  Suppose the boat is moving at a constant rate of 4 feet per second. Determine the speed at which the  winch pulls in rope when there is a total of 13 feet of rope out. What happens to the speed at which the winch pulls in rope as the boat gets closer to the dock?

29. Air Traffic Control. An air traffic controller spots two planes at the same altitude converging on a point as they fly at right angles to each other. One plane is 225 miles from the point moving at 450 miles per hour. The other plane is 300 miles from the point moving at 600 miles per hour. (a) At what rate is the distance between the planes decreasing? (b)  How much time does the air traffic controller have to get one of the planes on a different flight path?

30. Air Traffic Control. An airplane is flying at an altitude of 5 miles and passes directly over a radar antenna. When the plane is 10 miles away (s = 10), the radar detects that the distance s is changing at a rate of 240 miles per hour. What is the speed of the plane?

31. Sports. A baseball diamond has the shape of a square with sides 90 feet long. A player running from second base to third base at a speed of 25 feet per second is 20 feet from third base. At what rate is the player’s distance s from home plate changing?

32. Sports. For the baseball diamond in Exercise 31, suppose the player is running from first base to second base at a speed of 25 feet per second. Find the rate at which the distance from home plate is changing when the player is 20 feet from second base.

33. Shadow Length. A man 6 feet tall walks at a rate of 5 feet per second away from a light that is 15 feet above the ground. When he is 10 feet from the base of the light, (a) at what rate is the tip of his shadow moving? (b) at what rate is the length of his shadow changing?

34. Shadow Length. Repeat Exercise 33 for a man 6 feet tall walking at a rate of 5 feet per second toward a light that is 20 feet above the ground.

35. Machine Design. The endpoints of a movable rod of length 1 meter have coordinates (x, 0) and (0, y). The position of the end on the x-axis is x(t) = ½ sin πt/6 where t is the time in seconds. (a) Find the time of one complete cycle of the rod. (b) What is the lowest point reached by the end of the rod on the y-axis? (c) Find the speed of the y-axis endpoint when the x-axis endpoint is (1/4, 0).

36. Machine Design. Repeat Exercise 35 for a position function of x(t) =3/5 sin πt. Use the point (3/10, 0) for part (c).

37. Evaporation. As a spherical raindrop falls, it reaches a layer of dry air and begins to evaporate at a rate that is proportional to its surface area (S = 4πr2). Show that the radius of the raindrop decreases at a constant rate.

38. Electricity. The combined electrical resistance R of R1 and connected in parallel, is given by
1/R = 1/R1 + 1/R2 where R, R1 and R2 are measured in ohms. R1 and R2 are increasing at rates of 1 and 1.5 ohms per second, respectively. At what rate is R changing when R1 ohms and R2 ohms?

39. Adiabatic Expansion. When a certain polyatomic gas undergoes adiabatic expansion, its pressure and volume satisfy the equation pV1.3 = k, where k is a constant. Find the relationship between the related rates dp/dt and dV/dt.

40. Roadway Design. Cars on a certain roadway travel on a circular arc of radius r. In order not to rely on friction alone to overcome the centrifugal force, the road is banked at an angle of magnitude θ from the horizontal. The banking angle must satisfy the equation rg tan θ = v2, where v is the velocity of the cars and g=32 feet per second per second is the acceleration due to gravity. Find the relationship between the related rates dv/dt and dθ/dt.

41. Angle of Elevation. A balloon rises at a rate of 4 meters per second from a point on the ground 50 meters from an observer. Find the rate of change of the angle of elevation of the balloon from the observer when the balloon is 50 meters above the ground.

42. Angle of Elevation. A fish is reeled in at a rate of 1 foot per second from a point 10 feet above the water. At what rate is the angle θ between the line and the water changing when there is a total of 25 feet of line from the end of the rod to the water?

43. Relative Humidity. When the dewpoint is 65° Fahrenheit, the relative humidity H is H=(4347/400000000)e369444/(50t+19793) where t is the temperature in degrees Fahrenheit. (a) Determine the relative humidity when t = 65° and t = 80° (b) At 10 A.M., the temperature is 75° and increasing at the rate of 2° per hour. Find the rate at which the relative humidity is changing.

44. Linear vs. Angular Speed. A patrol car is parked 50 feet from a long warehouse. The revolving light on top of the car turns at a rate of 30 revolutions per minute. How fast is the light beam moving along the wall when the beam makes angles of (a) θ=30°, (b) θ=60°and (c) θ=70° with the line perpendicular from the light to the wall?

45. Linear vs. Angular Speed. A wheel of radius 30 centimeters revolves at a rate of 10 revolutions per second. A dot is painted at a point P on the rim of the wheel. (a) Find dx/dt as a function of θ. (b) Use a graphing utility to graph the function in part (a). (c) When is the absolute value of the rate of change of greatest? When is it least? (d) Find dx/dt when θ=30° and θ=60°.

46. Flight Control. An airplane is flying in still air with an air speed of 275 miles per hour. If it is climbing at an angle of 18°, find the rate at which it is gaining altitude.

47. Security Camera. A security camera is centered 50 feet above a 100-foot hallway. It is easiest to design the camera with a constant angular rate of rotation, but this results in a variable rate at which the images of the surveillance area are recorded. So, it is desirable to design a system with a variable rate of rotation and a constant rate of movement of the scanning beam along the hallway. Find a model for the variable rate of rotation if |dx/dt| = 2 feet per second.

49. Angle of Elevation. An airplane flies at an altitude of 5 miles toward a point directly over an observer. The speed of the plane is 600 miles per hour. Find the rates at which the angle of elevation    is changing when the angle is (a) θ=30°, (b) θ=60° and (c) θ=75°.

50. Moving Shadow. A ball is dropped from a height of 20 meters, 12 meters away from the top of a 20-meter lamppost. The ball’s shadow, caused by the light at the top of the lamppost, is moving along the level ground. How fast is the shadow moving 1 second after the ball is released?

51. Find the acceleration of the top of the ladder described in Exercise 25 when the base of the ladder is 7 feet from the wall.

52. Find the acceleration of the boat in Exercise 28(a) when there is a total of 13 feet of rope out.


3 Review exercises


33.  Vibrating String. When a guitar string is plucked, it vibrates with a frequency of F = 200√T where    F is measured in vibrations per second and the tension T is measured in pounds. Find the rates of change of F when (a) T = 4 and (b) T = 9.

34. Vertical Motion. A ball is dropped from a height of 100 feet. One second later, another ball is dropped from a height of 75 feet. Which ball hits the ground first?

35. Vertical Motion. To estimate the height of a building, a weight is dropped from the top of the building into a pool at ground level. How high (in feet) is the building if the splash is seen 9.2 seconds after the weight is dropped?

36. Vertical Motion. A bomb is dropped from an airplane at an altitude of 14,400 feet. How long will it take for the bomb to reach the ground? (Because of the motion of the plane, the fall will not be vertical, but the time will be the same as that for a vertical fall.) The plane is moving at 600 miles per hour. How far will the bomb move horizontally after it is released from the plane?

37. Projectile Motion. A thrown ball follows a path described by y = x – 0.02x2. (a) Sketch a graph of the path. (b) Find the total horizontal distance the ball is thrown. (c) At what value does the ball reach its maximum height? (Use the symmetry of the path.) (d) Find an equation that gives the instantaneous rate of change of the height of the ball with respect to the horizontal change. Evaluate the equation at 10, 25, 30, and 50. (e) What is the instantaneous rate of change of the height when the ball reaches its maximum height?

38. Projectile Motion. The path of a projectile thrown at an angle of 45° with level ground is y = x – (32/v02)x2 where the initial velocity is v0 feet per second. (a) Find the coordinate of the point where the projectile strikes the ground. Use the symmetry of the path of the projectile to locate the coordinate of the point where the projectile reaches its maximum height. (b) What is the instantaneous rate of change of the height when the projectile is at its maximum height? (c) Show that doubling the initial velocity of the projectile multiplies both the maximum height and the range by a factor of 4. (d) Find the maximum height and range of a projectile thrown with an initial velocity of 70 feet per second. Use a graphing utility to graph the path of the projectile.

39. Horizontal Motion. The position function of a particle moving along the x-axis is x(t) = t2 – 3t + 2 for -∞ < t < ∞. (a) Find the velocity of the particle. (b) Find the open interval(s) in which the particle is moving to the left. (c) Find the position of the particle when the velocity is 0. (d) Find the speed of the particle when the position is 0.

40. Modeling Data. The speed of a car in miles per hour and the stopping distance in feet are recorded in the table. (a) Use the regression capabilities of a graphing utility to find a quadratic model for the data. (b) Use a graphing utility to plot the data and graph the model. (c) Use a graphing utility to graph (d) Use the model to approximate the stopping distance at a speed of 65 miles per hour. (e)

 

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