1. AP Calculus AB Exam 3 Multiple Choice Section Name:_____________ 2. An equation of the line tangent to the graph of f( x ) = x ( 1 – 2x) 3 at the point ( 1,-1) is 1. Find the value of the derivative of f(x) = x 3 – x 2 at x = 1. (A) -1(B) 1(C) 0(D) 6(E) It does not exist 3. If f ( x ) = 2 + |x – 3| for all x, then the value of the derivative f  (x) at x = 3 is (A) -1 (B) 0 (C) 1 (D) 2 (E) nonexistent

2. 5.If the position of an object dropped from the top of a cliff is given as s(t) = -16t 2 + 225 where s is measured in feet and t is measured in seconds, find the velocity of this object after it has been falling for one seconds. A) –32 ft/s B) –64 ft/s C) –128 ft/s D) –225 ft/sec E) –16 ft/sec 8. If f ( x ) = x + sin x, then f  ( x ) = (A) 1 + cos x (B) 1  cos x (C) cos x (D) sin x + x cos x (E) sin x + x cos x 6. A particle moves along the x-axis so that at any time t > 0 its position is given by x( t) = t 3 - 3t 2 – 9t + 1. For what values of t is the particle at rest? (A) It is never at rest (B) t =1 only (C) t =3 only (D) t = 5 only (e) t = 1 and 3 7. At what point on the graph y =  x 2 is the tangent line parallel to the line 2x – 4y=3? (A) ( ,-  ) (B) ( ,  ) (C) (1,-  ) (D) (1,  ) (E) (2,2)

3. 10. If is differentiable at x = 2, then a + b = 12. Suppose that f is an odd function and that f  (c) exists. Which of the following must necessarily be equal to f  (-c)?

4. The function f given by f ( x ) = x 3 + 12x  24 is (A) increasing for x 2 (B) decreasing for x 0 (C) increasing for all x (D) decreasing for all x (E) decreasing for x 2 An equation of the line tangent to the graph of at the point (1,5) is 13. 16. 15. A particle moves along a line so that at time t, where 0 < t < , its position is given by s ( t ) = -2cos t   t 2 + 10. What is the velocity when the acceleration is zero? 14.

5. 18.Which of the following will calculate the derivative of f(x) if f(x) = x 2 + x? AP Calculus AB Exam 3 Calculator Active Section Name:_____________ 20. Find any point(s) on where there is a horizontal tangent line (A) (0.816, 1.089) & (-0.816, -1.089) (B) (1.414,0) & (-1.414,0) only (C)(1.414,0), (-1.414,0) & (0,0) (D) (0.816, 1.089) only (E) (0.816, 0) & (-0.816, 0)

6. 22. At x = 3, the function given by (A)Undefined (B)Continuous but not differentiable (C)Differentiable but not continuous (D)Neither continuous nor differentiable (E)Both continuous and differentiable 21. If f is a differentiable function, then f  (a) is given by which of the following? (A) I only (B) II only (C) I and II only (D) I and III only (E) I, II and III 23 If the second derivative of f is given by f  = 2x – cos x, which of the following could be f(x)? (A)  x 3 + cos x – x + 1 (B)  x 3  cos x – x + 1 (C) x 3 + cos x – x + 1 (D) x 2  sin x + 1 (E) x 2 + sin x + 1

7. 24. 25. If the line 3x – 4y = 0 is tangent in the first quadrant to the curve y = x 3 + k, then k is

8. 27. The position function of a particle that is moving along the x-axis for 0 < t < 6 is given as a. Find the average velocity for t = 1 to t = 3 b. Find any the interval(s) where the particle is traveling to the right. Justify your answer(s) c. Find the total distance traveled by the particle for 0 < t < 6 Let f be a function such that Which of the following must be true? I. f is continuous at x = 2. II. f is differentiable at x = 2. III. The derivative of f is 5. (A) I only(B) II only (C) III only (D) I and II only (E) I, II and III 26. Free Response