Saturnino L. Sals, Garret J. Etgen, Einar Hille, Calculus: One and Several Variables, 10 edition, Wiley, 2006

Chapter 4. Review Exercises

11. Find the intervals on which f increases and the intervals on which f decreases; find the critical points and the local extreme values.

38. Given that the surface area of a sphere plus the surface area of a cube is constant, show that the sum of the volumes is minimized by letting the diameter of the sphere equal the length of a side of the cube. What dimensions maximize the sum of the volumes?

39. A closed rectangular box with a square base is to be built subject to the following conditions: the volume is to be 27 cubic feet, the area of the base may not exceed 18 square feet, the height of the box may not exceed 4 feet. Determine the dimensions of the box (a) for minimal surface area; (b) for maximal surface area.

40. The line through P(1, 2) intersects the positive x-axis at A(a, 0) and the positive y-axis at  B(0, b). Determine the values of a and b that minimize the area of the triangle OAB.

41. A right circular cylinder is generated by revolving a rectangle of given perimeter P about one of its sides. What dimensions of the rectangle will generate the cylinder of maximum volume?

42. A printed page is to have a total area of 80 square inches. The margins at the top and on the sides are to be 1 inch each; the bottom margin is to be 1.5 inches. Determine the dimensions of the page that maximize the area available for print.

43. An  object  moves  along  a  coordinate  line,  its  position  at time t given by the function x(t) = t + 2 cos t. Find those times  from  t = 0  to  t = 2π when  the  object  is  slowing down.

44. An object moves along a coordinate line, its position at time t given by the function x(t) = (4t − 1)(t − 1)2 , t ≥ 0. (a) When is the object moving to the right? When to the left? When does it change direction? (b) What is the maximum speed of the object when moving left?

45. An object moves along a coordinate line, its position at time t given by the function x(t) =√(t + 1), t ≥ 0. (a) Show that the acceleration is negative and proportional to the cube of the velocity. (b) Use differentials to obtain numerical estimates for the position, velocity, and acceleration at time t = 17. Base your estimate on t = 15.

46. A rocket is fired from the ground straight up with an initial velocity of 128 feet per second. (a) When does the rocket reach maximum height? What is maximum height? (b) When does the rocket hit the ground and at what speed?

47. Ballast dropped from a balloon that was rising at the rate of 8 feet per second reached the ground in 10 seconds. How high was the balloon when the ballast was released?

48. A ball thrown straight up from the ground reaches a height of 24 feet in 1 second. How high will the ball go?

49. A boy walks on a straight, horizontal path away from a light that hangs 12 feet above the path. How fast does his shadow lengthen if he is 5 feet tall and walks at the rate of 168 feet per minute?

50. The radius of a cone increases at the rate of 0.3 inches per minute, but the volume remains constant. At what rate does the height of the cone change when the radius is 4 inches and the height is 15 inches?

51. A railroad track crosses a highway at an angle of 60◦ . A locomotive is 500 feet from the intersection and moving away from it at the rate of 60 miles per hour. A car is 500 feet from the intersection and moving toward it at the rate of 30 miles per hour. What is the rate of change of the distance between them?

52. A square is inscribed in a circle. Given that the radius of the circle is increasing at the rate of 5 centimeters per minute, at what rate is the area of the square changing when the radius is 10 centimeters?

53. A horizontal water trough 12 feet long has a vertical cross section in the form of an isosceles triangle (vertex down). The base and height of the triangle are each 2 feet. Given that water is being drained out of the trough at the rate of 3 cubic feet per minute, how fast is the water level falling when the water is 1.5 feet deep?

54. Use a differential to estimate f (3.8) given that f (4) = 2 and

57. Use a differential to estimate the value of the expression tan 43o

58. A spherical tank with a diameter of 20 feet will be given a coat of paint 0.05 inches thick. Estimate by a differential the amount of paint needed. (Assume that there are 231 cubic inches in a gallon.)

Exercises 5.4

35. Calculate  the  derivative  with  respect  to  x (a) without integrating; that is, using the results of Section 5.3; (b) by integrating and then differentiating the result.

41. Verify that the function is nonnegative on the given interval, and then calculate the area below the graph on that interval.
f (x) = 4x − x2 ;    [0, 4].

53. An object starts at the origin and moves along the x-axis with velocity v(t) = 10t − t2,   0 ≤ t ≤ 10.
(a) What is the position of the object at any time t, 0 ≤ t ≤ 10?
(b) When is the object’s velocity a maximum, and what is its position at that time?

54. The velocity of a bob suspended on a spring is given: v(t) = 3 sin t + 4 cos t,   t ≥ 0. At time t = 0, the bob is one unit below the equilibrium position. (See the figure.)
(a) Determine the position of the bob at each time t ≥ 0.
(b) What is the bob’s maximum displacement from the equilibrium position?

Exercises 5.6

19. Find f from the information given.
f ¢(x) = 2x − 1,    f (3) = 4

35. An object moves along a coordinate line with velocity v(t) = 6t2 − 6 units per second. Its initial position (position at time t = 0) is 2 units to the left of the origin. (a) Find the position of the object 3 seconds later. (b) Find the total distance traveled by the object during those 3 seconds.

36. An object moves along a coordinate line with acceleration a(t) = (t + 2)3 units per second per second. (a) Find the velocity function given that the initial velocity is 3 units per second. (b) Find the position function given that the initial velocity is 3 units per second and the initial position is the origin.

37. An object moves along a coordinate line with acceleration a(t) = (t + 1)−1/2 units per second per second. (a) Find the velocity function given that the initial velocity is 1 unit per second. (b) Find the position function given that the initial velocity is 1 unit per second and the initial position is the origin.

38. An  object  moves  along  a  coordinate  line  with  velocity v(t) = t(1 − t) units per second. Its initial position is 2 units to the left of the origin. (a) Find the position of the object 10 seconds later. (b) Find the total distance traveled by the object during those 10 seconds.

39. A car traveling at 60 mph decelerates at 20 feet per second per second. (a) How long does it take for the car to come to a complete stop? (b) What distance is required to bring the car to a complete stop?

40. An object moves along the x-axis with constant acceleration. Express the position x(t) in terms of the initial position x0, the initial velocity v0 , the velocity v(t), and the elapsed time t.

41. An object moves along the x-axis with constant acceleration a. Verify that
[v(t)]2 = v02 + 2a[x(t) − x0].

42. A bobsled moving at 60 mph decelerates at a constant rate to 40 mph over a distance of 264 feet and continues to decelerate at that same rate until it comes to a full stop. (a) What is the acceleration of the sled in feet per second per second? (b) How long does it take to reduce the speed to 40 mph? (c) How long does it take to bring the sled to a complete stop from 60 mph? (d) Over what distance does the sled come to a complete stop from 60 mph?

43. In the AB-run, minicars start from a standstill at point A, race along a straight track, and come to a full stop at point B one-half mile away. Given that the cars can accelerate uniformly to a maximum speed of 60 mph in 20 seconds and can brake at a maximum rate of 22 feet per second per second, what is the best possible time for the completion of the AB-run?

44. Find the general law of motion of an object that moves in a straight line with acceleration a(t). Write x0 for
the initial position and v0 for the initial velocity.
a(t) = sin t

47. As a particle moves about the plane, its x-coordinate changes at the rate of t2 − 5 units per second and its y-coordinate changes at the rate of 3t units per second. If the particle is at the point (4, 2) when t = 2 seconds, where is the particle 4 seconds later?

48. As a particle moves about the plane, its x-coordinate changes at the rate of t − 2 units per second and its y-coordinate changes at the rate of √t units per second. If the particle is at the point (3, 1) when t = 4 seconds, where is the particle 5 second later?

49. A particle moves along the x-axis with velocity v(t) = At + B. Determine A and B given that the initial velocity of the particle is 2 units per second and the position of the particle after 2 seconds of motion is 1 unit to the left of the initial position.

50. A  particle  moves  along  the  x-axis  with  velocity  v(t) = At2 + 1. Determine A given that x(1) = x(0). Compute the total distance traveled by the particle during the first second.

51. An object moves along a coordinate line with velocity v(t) = sin t units per second. The object passes through the origin at time t = π/6 seconds. When is the next time: (a) that the object passes through the origin? (b) that the object passes through the origin moving from left to right?

53. An automobile with varying velocity v(t) moves in a fixed direction for 5 minutes and covers a distance of 4 miles. What theorem would you invoke to argue that for at least one instant the speedometer must have read 48 miles per hour?

54. A speeding motorcyclist sees his way blocked by a haywagon some distance s ahead and slams on his brakes. Given that the brakes impart to the motorcycle a constant negative acceleration a and that the haywagon is moving with speed ν1 in the same direction as the motorcycle, show that the motorcyclist can avoid collision only if he is traveling at a speed less than ν1 + √(2|a|s).

Exercises 5.9

1. Determine the average value of the function on the indicated interval and find an interior point of this interval at which the function takes on its average value.
f (x) = mx + b,   x Î[0, c].

17. Let  P(x, y)  be  an  arbitrary  point  on  the  curve  y = x2. Express as a function of x the distance from P to the origin and calculate the average of this distance as x ranges from 0 to √3.

18. Let P(x, y) be an arbitrary point on the line y = mx. Express as a function of x the distance from P to the origin and calculate the average of this distance as x ranges from 0 to 1.

19. A stone falls from rest in a vacuum for t seconds. (Section 4.9). (a) Compare its terminal velocity to its average velocity; (b) compare its average velocity during the first seconds to its average velocity during the next  seconds.
20. Let f be continuous. Show that, if f is an odd function, then its average value on every interval of the form [−a, a] is zero.

21. Suppose that f is continuous on [a, b] and . Prove that there is at least one number c in (a, b) for which f (c) = 0.

22. Show that the average value of the functions  f (x) = sin πx and g(x) = cos πx is zero on every interval of length 2n, n a positive integer.

23. An object starts from rest at the point x 0 and moves along the x-axis with constant acceleration a.
(a) Derive formulas for the velocity and position of the object at each time t ≥ 0.
(b) Show that the average velocity over any time interval [t1 , t2 ] is the arithmetic average of the initial and final
velocities on that interval.

24. Find the point on the rod of Example 1 that breaks up that rod into two pieces of equal mass. (Observe that this point is not the center of mass.)

25. A rod 6 meters long is placed on the x-axis from x = 0 to x = 6. The mass density is 12/√(x+1) kilograms per meter.
(a) Find the mass of the rod and the center of mass.
(b) What is the average mass density of the rod?

26. For a rod that extends from x = a to x = b and has mass density λ = λ(x), the integral
gives what is called the mass moment of the rod about the point x = c. Show that the mass moment about the center of mass is zero. (The center of mass can be defined as the point about which the mass moment is zero.)

27. A rod of length L is placed on the x-axis from x = 0 to x = L. Find the mass of the rod and the center of mass if the mass density of the rod varies directly: (a) as the square root of the distance from x = 0; (b) as the square of the distance from x = L.

28. A  rod  of  varying  mass  density,  mass  M,  and  center  of mass x M , extends from x = a to x = b. A partition P = {x0 , x1 , . . . , xn } of [a, b] decomposes the rod into n pieces in the obvious way. Show that, if the n pieces have masses M1 , M2 , . . . , Mn and centers of mass , then

29. A rod that has mass M and extends from x = 0 to x = L consists of two pieces with masses M1, M2. Given that the center of mass of the entire rod is at  and the center of mass of the first piece is at , determine the center of mass of the second piece.

30. A rod that has mass M and extends from x = 0 to x = L consists of two pieces. Find the mass of each piece given that the center of mass of the entire rod is at , the center of mass of the first piece is at , and the center of mass of the second piece is at .

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