AP Calculus BC Fall Final Part IA. Calculator NOT Allowed. Name:

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1 AP Calculus BC Fall Final Part IA Calculator NOT Allowed Name:

2 3π cos + h 1. lim cos 3π h 0 = h 1 (a) 1 (b) (c) 0 (d) -1 (e) DNE dy. At which of the five points on the graph in the figure below are and dx both negative? d y dx (a) A (b) B (c) C (d) D (e) E 3. The slope of the tangent to the curve y 3 x + y x = 6 at, 1 is (a) 3 (b) 1 (c) 5 (d) 3 (e) Which of the following statements must be true? d I. II. dx ex + 3 = e x + 3 e x d dx lncosx = tan x

3 III. d dx 6x3 π + 3 x 8 x 3 =18x x5 + 6 x 4 (a) I only (b) II only (c) III only (d) I and III only (e) I, II, and III 5. Which of the following statements about the function given by f ( x) = x 4 x 3 is true? (a) The function has no relative extremum. (b) The graph of the function has one point of inflection and the function has two relative extrema. (c) The graph of the function has two points of inflection and the function has one relative extremum. (d) The graph of the function has two points of inflection and the function has two relative extrema. (e) The graph of the function has two points of inflection and the function has three relative extrema.

4 = sin ( 3 x) f ( 0) = 6. If f x, then (a) cos3 (b) sin 3cos3 (c) 6cos3 (d) sin 3cos3 (e) 6sin 3cos3 7. What is the average rate of change of the function f x on the closed interval [0, 3]? (a) 8.5 (b) 8.7 (c) (d) 33 (e) 66 = x 4 5x 8. The position of a particle moving along a line is given by s( t) = t 3 4t + 90t + 7 for t 0. For what values of t is the speed of the particle increasing? (a) 3 < t < 4 only (b) t > 4 only (c) t > 5 only (d) 0 < t < 3 and t > 5 (e) 3 < t < 4 and t > 5

9. ( x 1) x dx = 3 (a) (b) (c) x 1 x + c 3 x3 + 1 x1 + c 1 x x + c (d) 5 x5 3 x3 + c (e) 1 x + x 3 + c 10. What is x lim 4 x + x 4x (a) (b) 1 1 (c) (d) 1 (e) DNE 4 11.

5 9. ( x 1) x dx = 3 (a) (b) (c) x 1 x + c 3 x3 + 1 x1 + c 1 x x + c (d) 5 x5 3 x3 + c (e) 1 x + x 3 + c 10. What is x lim 4 x + x 4x (a) (b) 1 1 (c) (d) 1 (e) DNE The graph of the derivative of f x is shown below. Which of the following is true about the function f x? x = 0 I. f x is decreasing at II. f ( x) has a local minimum at x = III. f x is concave up at x = 1 (a) I only (b) II only (c) III only (d) II and III only (e) I, II, and III

6 x e t 0 1. If y = 5 + dt, which of the following is true? (a) (b) (c) (d) (e) dy and dx = e x y 0 dy and dx = e x y 0 dy and dx = e 4x y dy and dx = e 4x y 0 dy and dx = e 4x y = 5 = 5 = 5 = 5 = 5 = te t 13. A particle is moving along the x-axis and its position is given by x t. For what values of t is the particle at rest? 1 (a) No values (b) 0 only (c) only (d) 1 only (e) 0 and If the function y = x 3 has an average value of 9 on x 0, k, then k = (a) 3 (b) 3 (c) 18 (d) 36 (e) 36

7 The table above gives selected values for the derivative of a function g on the interval 1 x. If g 1 and Euler s method with a step-size of 1.5 is used to approximate g = x g x, what is the resulting approximation? (a) 6.5 (b) 1.5 (c) 1.5 (d).5 (e) 3

8 16. Three graphs labeled I, II, and III are shown above. One is the graph of, one is the graph of f x, and one is the graph of. Which of the following correctly identifies each of the three graphs? f ( x) f ( x) f x (a) I II III (b) I III II (c) II I III (d) II III I (e) III II I f ( x) f x

9 17. The region R is bounded by the lines y = x 4, x = 3, and y = 0. Which of these expressions gives the volume of the solid formed by revolving R around the line x = 5? (a) (b) (c) (d) (e) 3 0 (( x 4) 3 ) dx ( x 9) dx y + 4 y 6 y dy dy 3 dy = 1 x 18. If f x dx, then 1 dx is x 0 (a) convergent. (b) divergent because lim f x does not exist. x 0 + (c) divergent because lim does not exist. x f x (d) divergent because neither lim f x nor f x exist. x x 0 + lim (e) none of the above.

10 dx is 1 x 0 (a) 3 (b) (c) (d) (e) divergent 5 0. If the average value of the function f ( x) = x a on 1, 1 is, what 8 is/are the values of a? (a) ±1 (b) ± 1 (c) ± 1 (d) 0 (e) None of these 4 1. Let f be a polynomial function with degree greater than. If a b and f ( a) = f ( b) =1, which of the following must be true of at least one value of x between a and b? = 0 f ( x) = 0 f ( x) = 0 I. f x II. III. (a) I only (b) II only (c) III only (d) II and III only (e) I, II, and III

11 = sin x f. If f x, then π 3 = 1 1 (a) (b) (c) (d) (e) 3 3 t H t A small plant is purchased from a nursery and the change in height of the plant is measured at the end of each day for four days. The data, where H ( t) is measured in inches per day and t is measured in days, are listed above. Using the trapezoidal rule, which of the following represents an estimate of the average rate of growth of the plant over the four-day period? (a) (b) (c) (d) (e) 1 ( ) ( ) ( ( 1.5)+ (.1)+.6) ( 1.3 )+ ( 1.5)+ (.1) ( ( 1.5)+ (.1)+.6) ( )

12 4. Which of the following is a slope field for the differential equation dy dx = x y

13 f ( x) 5. The graph of a function f is shown above. If lim x b f x exists and is not continuous at b, then b = (a) 1 (b) 0 (c) 1 (d) (e) 3

14 End of AP Calculus BC Fall Final Part IA

15 AP Calculus BC Fall Final Part IB Calculator Allowed Name:

16 1. A particle travels along a straight line with a velocity of v( t) = 3e t sin t meters per second. What is the total displacement, in meters, traveled by the particle during the time interval 0 t seconds? (a) (b) (c).055 (d).61 (e) 7.05 π π T 4 6. Let T be the region above bounded by y = π ( sin 1 x 1), y = π x, and y = π The area of T is (a) 5.36 (b) (c) (d) 3.09 (e)

17 3. Which of the following is the solution to the differential equation = where y? dy dx = 4x y (a) (b) (c) (d) (e) y = x for x > 0 y = x 6 for x 3 y = 4x 1 for x > 3 y = 4x 1 for x > 3 y = 4x 6 for x > If f ( t) = sin πex and, the f 0 =1 f = (a) (b) (c) (d) (e) x f x < 0 5. Let f be a function such that f x for all x in the closed interval [1, ]. Selected values of f are shown in the table above. Which of the following must be true about f 1.? (a) (b) (c) (d) (e) < 0 <1.6 <1.8 < f 1. 0 < f < f < f 1. < f 1.

18 = 3 f ( 5) = 4 x = 5 6. A differentiable function f has the property that f 5 and. What is the estimate for f 4.8 using the local linear approximation for f at (a). (b).8 (c) 3.4 (d) 3.8 (e) 4.6 = 000e 0.t 7. Oil is leaking from a tanker at the rate of R t gallons per hour, where t is measured in hours. How much oil leaks out of the tanker from time t = 0 to t = 10? (a) (b) (c) (d) (e) 54 gallons 71 gallons 865 gallons 8,647 gallons 14,778 gallons 8. The graph of f is shown above. If f x dx =.3 and F x, then 1 F ( 3) F ( 0) = (a) 1.3 (b).3 (c) 3.3 (d) 4.3 (e) = f ( x)

19 9. The graphs of five functions are shown below. Which function has a nonzero average value over the closed interval x π, π?

20 10. The function defined on all the Reals such that f ( x) = x + kx 3 for x 1. For which of the following values of k and b will 3x + b for x >1 the function f be both continuous and differentiable on its entire domain? (a) (b) (c) (d) (e) k = 1, b = 3 k =1, b = 3 k =1, b = 4 k =1, b = 4 k = 1, b = 6 = e 4x 11. Let f be the function given by f x. For what value of x is the slope of the line tangent to the graph of f at x, f x equal to 3? (a) (b) 0.74 (c) (d) 0.34 (e) 0.551

21 End of AP Calculus BC Fall Final Part IB

AP Calculus BC Spring Final Part IA. Calculator NOT Allowed. Name:

AP Calculus BC Spring Final Part IA. Calculator NOT Allowed. Name: AP Calculus BC 6-7 Spring Final Part IA Calculator NOT Allowed Name: . Find the derivative if the function if f ( x) = x 5 8 2x a) f b) f c) f d) f ( ) ( x) = x4 40 x 8 2x ( ) ( x) = x4 40 +x 8 2x ( )