2. Which of the following is an equation of the line tangent to the graph of f(x) = x 4 + 2x 2 at the point where

Transcription

1 AP Review Chapter Name: Date: Per: 1. The radius of a circle is decreasing at a constant rate of 0.1 centimeter per second. In terms of the circumference C, what is the rate of change of the area of the circle, in square centimeters per second? (A) (0.)πC (B) (0.1)C (C) (0.1)C π (D) (0.1) C (E) (0.1) πc. Which of the following is an equation of the line tangent to the graph of f(x) = x 4 + x at the point where f (x) = 1? (A) y = 8x 5 (B) y = x + 7 (C) y = x (D) y = x 0.1 (E) y = x If the base b of a triangle is increasing at a rate of 3 inches per minute while its height h is decreasing at a rate of 3 inches per minute, which of the following must be true about the area A of the triangle? (A) A is always increasing (B) A is always decreasing (C) A is decreasing only when b < h (D) A is decreasing only when b > h (E) A remains constant 4. Let f be a function that is differentiable on the open interval (1, 10). If f() = 5, f(5) = 5, and f(9) = 5, which of the following must be true? I. f has at least zeros II. The graph of f has at least one horizontal tangent. III. For some c, < c < 5, f(c) = 3. (A) None (B) I only (C) I and II only (D) I and III only (E) I, II and III 5. If x + xy = 10, then when x =, dy (A) 7 (B) (C) 7 (D) 3 (E) 7

2 6. Let f and g be differentiable functions with the following properties: (i) g(x) > 0 for all x (ii) f(0) = 1 If h(x) = f(x)g(x) and h (x) = f(x)g (x), then f(x) = (A) f (x) (B) g(x) (C) e x (D) 0 (E) 1 7. What is the instantaneous rate of change at x = of the function f given by f(x) = x x 1? (A) (B) 1 6 (C) 1 (D) (E) 6 8. The graph of the function f shown in the figure above has a vertical tangent at the point (,0) and horizontal tangents at the points (1, 1) and (3,1). For what values of x, < x < 4, is f not differentiable? (A) 0 only (B) 0 and only (C) 1 and 3 only (D) 0, 1, and 3 only (E) 0,1, and 3 9. An equation of the line tangent to the graph of y = x + cos x at the point (0, 1) is (A) y = x + 1 (B) y = x + 1 (C) y = x (D) y = x 1 (E) y = If f(x) = tan(x), then f ( π 6 ) = (A) 3 (B) 3 (C) 4 (D) 4 3 (E) 8

3 11. The radius of a circle is increasing at a constant rate of 0. meters per second. What is the rate of increase in the area of the circle at the instant when the circumference of the circle is 0π meters? (A) 0.04π m /sec (B) 0.4π m /sec (C) 4π m /sec (D) 0π m /sec (E) 100π m /sec 1. The function f is continuous for x 1 and differentiable for < x < 1. If f( ) = 5 and f(1) = 4, which of the following statements could be false? (A) There exists c, where < c < 1, such that f(c) = 0 (B) There exists c, where < c < 1, such that f (c) = 0 (C) There exists c, where < c < 1, such that f(c) = 3 (D) There exists c, where < c < 1, such that f (c) = 3 (E) There exists c, where c 1, such that f(c) f(x) for all x on the closed interval x Let f be a differentiable function with f() = 3 and f () = 5, and let g be the function defined by g(x) = xf(x). Which of the following is an equation of the line tangent to the graph of g at the point where x =? (A) y = 3x (B) y 3 = 5(x ) (C) y 6 = 5(x ) (D) y 6 = 7(x ) (E) y 6 = 10(x ) 14. If y = (x 3 + 1), then dy (A) (3x ) (B) (x 3 + 1) (C) (3x + 1) (D) 3x (x 3 + 1) (E) 6x (x 3 + 1) 15. If y = x+3 dy, then = 3x+ dx (A) 1x+13 (3x+) (B) 1x 13 (C) (D) (E) 3 (3x+) 5 (3x+) 5 (3x+)

4 16. The rate of change of the volume, V, of water in a tank with respect to time, t, is directly proportional to the square root of the volume. Which of the following is a differential equation that describes the relationship? (A) V(t) = k t (B) V(t) = k V (C) dv dt = k t dv (D) = k dt V (E) dv dt = k V 17. The graph of a function f is shown above. At which value of x is f continuous, but not differentiable? (A) a (B) b (C) c (D) d (E) e 18. If y = x sin(x), then dy (A) xcos(x) (B) 4xcos(x) (C) x(sin x + cos x ) (D) x(sin x x cos x ) (E) x(sin x + x cos x ) 19. If the line tangent to the graph of the function f at the point (1, 7) passes through the point (, ), then f (1) is (A) 5 (B) 1 (C) 3 (D) 7 (E) undefined 0. A curve has slope x + 3 at each point ( x, y ) on the curve. Which of the following is an equation for this curve if it passes through the point (1, )? (A) y = 5x 3 (B) y = x + 1 (C) y = x + 3x (D) y = x + 3x (E) y = x + 3x 3 1. Let f be the function given above. Which of the following statements are true about f? I. lim x 3 f(x) exists. II. f is continuous at x = 3. III. f is differentiable at x = 3. (A) None (B) I only (C) II only (D) I and II only (E) I, II and III

5 . Let f be the function defined by f(x) = 4x 3 5x + 3. Which of the following is an equation of the line tangent to the graph of f at the point where x = 1? (A) y = 7x 3 (B) y = 7x + 7 (C) y = 7x + 11 (D) y = 5x 1 (E) y = 5x 5 3. What is the slope of the line tangent to the curve 3y x = 6 xy at the point (3, )? (A) 0 (B) 4 9 (C) 7 9 (D) 6 7 (E) The radius of a sphere is decreasing at a rate of centimeters per second. At the instant when the radius of the sphere is 3 centimeters, what is the rate of change, in square centimeters per second, of the surface area of the sphere? (The surface area S of a sphere with radius r is S = 4πr ) (A) 108π (B) 7π (C) 48π (D) 4π (E) 16π 5. If f(x) = (x 1)(x + ) 3, then f (x)= (A) 6x(x + ) (B) 6x(x 1)(x + ) (C) (x + ) (x + 3x 1) (D) (x + ) (7x 6x + ) (E) 3(x 1)(x + ) 6. If f(x) = cos(3x), then f ( π 9 ) = (A) 3 3 (B) 3 (C) 3 (D) 3 (E) If sin(xy) = x, then dy (A) 1 cos(xy) 1 (B) xcos(xy) (C) 1 cos(xy) cos(xy) (D) 1 ycos(xy) xcos(xy) (E) y(1 cos(xy)) x

6 8. In the xy-plane, the line x + y = k, where k is a constant, is tangent to the graph of y = x + 3x + 1. What is the value of k? (A) 3 (B) (C) 1 (D) 0 (E) 1 9. The function f is twice differentiable with f() = 1, f () = 4, and f"() = 3. What is the value of the approximation of f(1.9) using the line tangent to the graph of f at x =? (A) 0.4 (B) 0.6 (C) 0.7 (D) 1.3 (E) Let f be the function defined above, where c and d are constants. If f is differentiable at x =, what is the value of c + d? (A) 4 (B) (C) 0 (D) (E) The graph of a function f is shown above. Which of the following statements about f is false? (A) f is continuous at x = a. (B) f has a relative maximum at x = a. (C) x = a is in the domain of f. (D) lim f(x) is equal to lim f(x). x a + x a (E) lim x a f(x) exists.

7 3. The function f is continuous on the closed interval [0,] and has values that are given in the table above. The equation f(x) = 1 must have at least two solutions in the interval [0, ] if k = (A) 0 (B) ½ (C) 1 (D) (E) For which of the following does lim x 4 f(x) exist? (A) I only (B) II only (C) III only (D) I and II only (E) I and III only FRQ Practice No calculator is allowed on this problem. The function f is defined by f(x) = 5 x for 5 x 5. a) Find f (x). b) Write an equation for the line tangent to the graph of f at x = 3. f(x) for 5 x 3 c) Let g be the function defined by g(x) = x + 7 for 3 < x 5 Is g continuous at x = 3? Use the definition of continuity to explain your answer.