AP Calculus Chapter 4 Testbank (Mr. Surowski)

Transcription

1 AP Calculus Chapter 4 Testbank (Mr. Surowski) Part I. Multiple-Choice Questions 1. Let f(x) = x 3 + 3x 2 45x + 4. Then the local extrema of f are (A) a local minimum of 179 at x = 5 and a local maximum of 77 at x = 3. (B) a local minimum of 77 at x = 3 and a local maximum of 179 at x = 5. (C) a local minimum of 179 at x = 5 and a local maximum of 77 at x = 3. (D) a local minimum of 77 at x = 3 and a local maximum of 77 at x = 5. (E) a local minimum of 77 at x = 5 and a local maximum of 179 at x = Consider the function f(x) = x 3 + 3x 2 45x + 4. Then I. f is decreasing on ( 5, 3) and increasing on (, 5) (3, ). II. f has a local minimum at x = 3. III. f has a local maximum at x = 5. (A) I only (B) II only (C) I and II only (D) I and III only (E) I, II, and III 3. For what value of x does the function f(x) = x 3 9x 2 120x + 6 have a local minimum? (A) 10 (B) 4 (C) 3 (D) 4 (E) 10

2 4. Use differentials to approximate the change in the volume of a sphere when the radius is increased from 10 to cm. (A) (B) (C) (D) (E) The graph of y = x 3 5x 2 + 4x + 2 has a local minimum at (A) (0.46, 2.87) (B) (0.46, 0) (C) (2.94, 4.05) (D) (4.06, 2.87) (E) (1.66, 0.59) 6. Find a positive value, c, for x, that satisfies the conclusion of the Mean Value Theorem for f(x) = 3x 2 5x+1 on the interval [2, 5]. (A) 1 (B) 13 6 (C) 11 6 (D) 23 6 (E) The graph of y = x 3 2x 2 5x + 2 has a local maximum at (A) (2.120, 0) (B) (2.120, 8.061) (C) ( 0.786, 0) (D) ( 0.786, 4.209) (E) (0.666, 1.926) 8. A 20-foot ladder slides down a wall at 5 ft/sec. At what speed is the bottom sliding out (in ft/sec) when the top is 10 feet from the floor? (A) (B) (C) (D) (E) 4.472

3 9. What are the coordinates of the point of inflection on the graph of y = x 3 15x x + 100? (A) (9, 0) (B) (5, 48) (C) (1, 119) (D) (9, 89) (E) (5, 15) 10. The graph of y = x 4 + 8x 3 72x is concave down for (A) 6 < x < 2 (B) x > 2 (C) x < 6 (D) x < or x > (E) < x < The function f is given by f(x) = x 4 + 4x 3. On which of the following intervals is f decreasing? (A) ( 3, 0) (B) (0, ) (C) ( 3, ) (D) (, 3) (E) (, 0) 12. The value of c that satisfies the Mean Value Theorem on the interval [0, 5] for the function f(x) = x 3 6x is (A) 5 (B) 0 (C) 1 (D) 5 5 (E) The graph of the function y = x x x + 3 has a relative maximum at x = (A) (B).248 (C) (D) (E) The side of a square is increasing at a constant rate of 0.4 cm/sec. In terms of the perimeter, P, what is the rate of change of the area of the square, in cm 2 /sec? (A) 0.05P (B) 0.2P (C) 0.4P (D) 6.4P (E) 51.2P

4 15. The second derivative of a function is given by f (x) = x sin x 2. How many points of inflection does f have on the interval ( 10, 10)? (A) Zero (B) Two (C) Four (D) Six (E) Eight

5 Part II. Free-Response Questions 16. Find the extreme values of the function f(x) = x 3 + 3x 2 36x + 4 on the interval [0, 4]. 17. Find the extreme values of the function f(x) = interval [ 3, 3]. x x on the 18. Find the extreme values of the function f(x) = interval (, ). x x on the 19. Find the extreme values of the function g(x) = x 2 x on the interval [ 2, 2]. 20. Find all relative extrema of the function h(x) = x 2 e 1/x. 21. Let f(x) = x 4 x 2. (a) Determine all relative extrema of f. (b) Determine all points of inflection of the graph of y = f(x). 22. Determine all relative extrema of the function f(x) = x + cos x.

6 23. Let f(x) = 2xe x, x 0 and determine the extrema of f(x) on [0, ). 24. Let f(x) = 2xe x, x 0. (a) Determine where f is increasing and where f is decreasing. (b) Determine where f is concave up and where f is concave down. (c) Sketch the graph of y = f(x), y x 25. Find the points of inflection of the graph y = e x.

7 26. A particle moves along the x-axis so that its position at time t 0 is given by the function x(t) = t 3 12t t, t 0, where x is measured in meters and t is measured in seconds. (a) Determine those values of t where the particle is not moving. (b) Determine when the particle is moving to the right and when the particle is moving to the left. (c) Determine when the particle is accelerating and when the particle is decelerating. (d) At what value(s) of t does the particle have zero acceleration? 27. Find all relative extrema of the function f(x) = e x sin x, x > Find the critical x-values of the function f(x) = x 2 3 x Find the open intervals on which the function f(x) = 1 4 x3 3x is increasing or decreasing. 30. Use the second derivative test to find the relative extrema of f(x) = x 4 4x As part of your work, be sure to identify all points of inflection. Sketch a graph, indicating all points of inflection. y x

8 31. The graph below depicts a function f defined on the interval (, ). In terms of the notation given in the graph, determine (i) the critical x values of f; (ii) the open intervals on which f is increasing and decreasing; (iii) the open intervals on which f is concave up and concave down; and (iv) the values of x at which f has points of inflection. y y = f(x) x 0 x 1 x 2 x 3 x 4 x 5 x 6 x 32. Find the production level that produces the maximum profit for hamburgers in a fast-food restaurant whose profit function (P dollars) in terms of hamburgers (x) is described as P = 2.44x x2 5, 000, 0 x 50, , Suppose that the semicircle y = 1 x 2 is drawn. Now draw a rectangle whose base is on the x-axis and height is such that the rectangle is incribed within the semicircle. What is the maximum area of this rectangle?

9 34. Suppose that on one side of a 2 km-wide river is an electricitygenerating plant. On the opposite side, and 10 km down the river is a small town that will be consuming the electricity. If it costs $80/m to lay cable under the river and $40/m to lay cable over land, find the most strategic method for laying the cable. Power Plant Town 35. Find the minimum distance from the point (2, 0) to a point on the parabola whose graph is y = 2x Write the linearization of the function f(x) = x valid near x = 0. Use this linearization to approximate Write the linearization of the function g(x) = cos x valid near x = 0. Use this to approximate cos(.01). (Are you surprised by this linearization?) 38. Write the linearization of f(x) = ln x valid near x = 1. Use this approximate ln(1.2).

10 39. Assume that y = x 2 ln x. (a) Compute dy in terms of x and dx. (b) Compute dy given that x = 1 and that dx = Let V be the volume of a sphere of radius r and let dv be its differential value (in terms of r and dr). For fixed dr, what would you say about dv : (a) that it is an increasing function of r, (b) that it is a decreasing function of r, or (c) that it doesn t depend on r? 41. Let Q be a quantity that depends on a measurable quantity, x, by the rule Q = 1 + x 2. Suppose that we measure x to be 5.2 ± (a) Estimate the resulting range in computed values for Q. (b) Estimate the resulting relative error in the computed value for Q. 42. Assume that ball bearings are to be made whose volume must be within a 1% error. Use differentials to determine the necessary tolerance of the radius of the given ball bearings. 43. Suppose that a sphere is expanding at a rate given by dr/dt = 5m/sec. If S denotes the surface area of this sphere and V denotes its volume, compute dv/dt and ds/dt when r = 2. Do the same for r = 4. Does this make sense?

11 44. Suppose that you are standing exactly 1km from the Pearl Tower in Shanghai. Now suppose that someone releases a heliumfilled balloon from the top of the Pearl Tower, and that this balloon is rising at a constant rate of 1m/sec. Let D be the distance between you (standing on the ground) and the balloon. Let h(t) be the height of the balloon as a function of time, where we assume that h(0) = 400m. (a) Compute dd/dt when h = 1000m. (b) Compute dd/dt when h = 2000m. (c) Compute lim h dd/dt. (Does this make sense?) (d) Show that dd/dt is a increasing function of both h and t. 45. Let h be a function defined for all x 0 such that h(4) = 3 and the derivative of h is given by h (x) = x2 2 for all x 0 x (a) Find all values of x for which the graph of h has a horizontal tangent, and determine whether h has a local maximum, a local minimum, or neither at each of these values. Justify your answers. (b) On what intervals, if any, is the graph of h concave up? Justify your answer. (c) Write an equation for the line tangent to the graph of h at x = 4. (d) Does the line tangent to the graph of h at x = 4 lie above or below the graph of h for x > 4. Why? 46. A cubic polynomial function g is defined by f(x) = x 3 + ax 2 + bx + c where a, b, and c are constants. The function has a critical value at x = 1, and the graph of f has a point of inflection at the point ( 2, 0). Find a, b, and c.

12 x 0 0 < x < < x < < x < < x < 4 f(x) 1 Negative 0 Positive 2 Positive 0 Negative f (x) 4 Positive 0 Positive DNE Negative 3 Negative f (x) 2 Negative 0 Positive DNE Negative 0 Positive 47. Let f be a function that is continuous on the interval [0, 4). The function f is twice differentiable except at x = 2. The function f and its derivatives have the properties indicated in the table above, where DNE indicates that the derivatives of f do not exist at x = 2. (a) For 0 < x < 4, find all values of x at which f has a relative extremum. Determine whether f has a relative maximum or a relative minimum at each of these values. Justify your answer. (b) On the axes provided, sketch the graph of a function that has all the characteristics of f.

13 48. An airplane is flying at a constant speed at a constant altitude of 3 km in a straight line that will take it directly over an observer at ground level. At a given instant the observer notes that the angle θ is π/3 and is increasing at 1/20 radians per second. Find the speed of the airplane, in kilometers per hour. 3 km θ Observer Airplane 49. Continue that we have, as in the above problem, an airplane flying toward the observer at an altitude of 3 km. This time, however, assume that angular rate of change, dθ dt = 1, and is 20 constant. (i) What is the speed of the airplane when it is directly overhead? (ii) Is the airplane slowing down or speeding up as it approaches the observer? Justify your conclusion.