My First Semester AP Calculus AB Final Exam ____________________________________ Part I – Non-calculator Directions: Solve each of the following problems, using the available space for scratch work. After examining the form of the choices, decide which is the best of the choices given and fill in the corresponding “bubble” on your Scantron form. Do not spend too much time on any one problem. Please be sure to place your name in the appropriate place on your Scantron form. 1.

c For what value of c will y = x 2 + have a relative minimum at x = − 1 ? x

c =− 4 c=−2 c=2 c=4

(A) (B) (C) (D) (E)

None of these

2.

Suppose that the domain of the function f is all real numbers and its derivative is given by

f ' ( x) =

(x − 1)(x − 4)2 1+ x2

Which of the following is true about the function f ? I II III

f is decreasing on (− ∞, 1 ) f has a relative minimum at x = 4 f is concave up at x = 8

(A) (B) (C) (D) (E)

I only I and II only I and III only II and III only I, II, and III

3.

2x 2 + x − 3 Find lim x →1 3 x 2 − x − 2

(A)

2 3

(B)

3 2

(C)

1

(D)

0

(E) Do not exist

4.

If the function f is defined by f ( x ) =

2x − a , where a is a positive constant, then which a−x

of the following statement(s) is/are true? I II III (A) (B) (C) (D) (E)

5.

6.

f has one zero The line x = a is a vertical asymptote 2 The line y = − 2 is a horizontal asymptote

I only II only I and III only II and III only I, II, and III

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

19.

20.

21.

If f ( x ) = 4 sin x + 2 , then f ' (0) =

(A) (B) (C)

0 1

(D) (E)

−2

2 2 2

22.

If the radius of a sphere is increasing at the rate of 2 inches per second, how fast, in cubic inches per second, is the volume increasing when the radius is 10 inches? [Note: The volume is sphere can be found with: V =

(A) (B) (C) (D) (E)

23.

4 π r3 3

40π 80π 800π 800

3200π

What is the equation of the line tangent to the graph of f ( x) = 7 x − x 2 at the point where

f ' ( x) = 3 ? (A) (B) (C) (D) (E)

y = 5 x − 10 y = 3x + 4 y = 3x + 8 y = 3 x − 10 y = 3 x − 16

BONUS Multiple Choice [but keep your eye on the time] B1 [Mark as #24] Suppose that f ( x) is a twice-differentiable function on the closed interval [a, b] . If there is a number c, a < c < b for which f ' (c) = 0 , which of the following could be true?

(A) (B) (C) (D) (E)

I f has a horizontal tangent somewhere on the open interval (a, b ) II f has a relative extreme value at x = c III f has a point of inflection at x = c I only II only I and II II and III None of these

B2 [Mark as #25] Let f be the function given by f ( x) = x 3 . What are all the values of c that satisfy the Mean Value Theorem on the closed interval [− 1, 2] ? (A) (B) (C) (D)

0 only 1 only 3 only

(E)

± 3

±1

End of Part I. Turn in this part of the test. [Be sure that your name is on it and hold onto your Scantron form.] Pick up Part II of the test. Once you have turned in this part, then you may NOT return to it.

My First Semester AP Calculus AB Final Exam ____________________________________ Part II –Calculator Directions: Solve each of the following problems, using the available space for scratch work. After examining the form of the choices, decide which is the best of the choices given and fill in the corresponding “bubble” on your Scantron form. Do not spend too much time on any one problem. Please be sure to place your name in the appropriate place on your Scantron form. Begin “bubbling” your Scantron at #26

26.

27.

If f is a differentiable function on the closed interval [0, 5] with f (0) = − 2 and f (5) = 12 , then which of the following statements must be FALSE/

(A)

There is some c, 0 < c < 5 , such that f (c) = 0

(B) (C) (D) (E)

There is a number c in the closed interval [0.5] such at f (c) ≥ f ( x) for all x in [0, 5] There is some c, 0 < c < 5 , such that f (c) = 2 .8 There is some c, 0 < c < 5 , such that f ' (c) = 2 .8 There is some c, 0 < c < 5 , such that f ' (c) = 0

28.

(A) (B) (C) (D) (E)

(0, 1) only (0, 10 and (4, 5) (0, 1), (3, 4), and (4, 5) (0, 2) and (4, 6) (2, 4) and (6, 7)

29.

The table below shows the rate in liters/minute at which water leaked out of a container.

0 1.2 2.3 3.8 5.4 t , [Time in minutes] R (t ), [Rate in liters/minute] 5.6 4.3 3.1 2.2 1.5 5 .4

Find an approximation of

∫ R (t ) dt with a right-hand Riemann Sum using four subintervals

0 indicated by the data given in the table. (A) (B) (C) (D) (E)

12.70 liters 14.27 liters 16.70 liters 16.95 liters 19.62 liters

30.

31.

32.

33. The first derivative of the function f is given by f ' ( x ) = does f have on the open interval (0, 10)? (A) (B) (C) (D) (E)

One Three Four Five Seven

cos 2 x 1 − . How many critical values x 5

34. Let f (x ) be the function whose graph is shown below.

A left Riemann sum, a right Riemann sum, and a midpoint Riemann sum are used to

b approximate the value of

∫ f ( x)dx . 0

b value of

∫ f ( x) dx ? 0

I. II III

(A) (B) (C) (D) (E)

Left sum Right sum Midpoint sum

I only II only III only I and II II and III

Which of the sums gives an under-estimate of the

35.

A rectangle is to be inscribed in a semicircle of radius 8, with one side lying on the diameter of the circle. What is the maximum possible area of the rectangle? Hint: Use f ( x ) =

(A) (B) (C) (D) (E)

36.

64 − x 2

4 2 8 2 32

32 2 64

Let f be a function that is everywhere differentiable. The value of f ' ( x ) is given for several values of x in the table below.

x f ' ( x)

− 10 −2

− 5 0 5 10 −1 0 1 2

If f ' ( x ) is always increasing, which statement about f (x ) must be true? (A) (B) (C) (D) (E)

f f f f f

(x) (x) (x) (x) (x)

has a relative minimum at x = 0 is concave down for all x has a point of inflection at (0, f (0) ) passes through the origin is an even function

37.

The graph of the derivative of a twice-differentiable function f is shown above. If f ( 2) = − 22 , then which of the following is TRUE? (A) (B) (C) (D) (E)

f (2 ) < f ' (2 ) < f ' ' (2) f " (2) < f ' (2) < f (2) f ' ( 2) < f (2) < f " ( 2) f (2) < f " (2) < f ' ( 2) f ' ( 2) < f " (2) < f ( 2)

This final was written by Ms. McCleary for the 2008-2009 school year. It contains some problems from previously released AP exams. [There might be some typos in it.]

My Free Response Part of AP Calculus AB First Semester Exam [December 2008] Name ______________________________________________________________________ FR1 If f (x ) and g (x ) are twice differentiable and if h ( x ) = f ( g ( x) ) , then find h ' ( x ) .

FR2 Ms. McCleary’s rocket ship has positive velocity v (t ) after being launched upward from an initial height of 0 feet at time t = 0 . The velocity of the rocket is recorded for selected values of t over the interval 0 ≤ t ≤ 70 seconds as shown in the table below.

t

0 10 20 30 40 50 60 70

Seconds 5 14 22 29 35 40 35 50

v (t ) Meters per second (a)

(b)

Show that there is at least one time during the recorded flight time that the acceleration of the rocket is equal to zero.

70 Using correct units, explain the meaning of ∫ v (t ) dt in terms of the rocket’s flight. Then 0 use a right Riemann sum to approximate the definite integral.