CalculusCalculusCalculusCalculus Name:__________________________________Name:__________________________________Name:__________________________________Name:__________________________________ AP WorksheetAP WorksheetAP WorksheetAP Worksheet #1#1#1#11111 MitchellMitchellMitchellMitchell

o Show your work and circle your final answers.Show your work and circle your final answers.Show your work and circle your final answers.Show your work and circle your final answers.

1) 1) 1) 1) 2222

xxxxdxdxdxdx

3x 53x 53x 53x 5====

++++∫∫∫∫

(A) (A) (A) (A) 3333222222221111

9999 (3x 5) C(3x 5) C(3x 5) C(3x 5) C+ ++ ++ ++ + (B) (B) (B) (B) 3333222222221111

4444 (3x 5) C(3x 5) C(3x 5) C(3x 5) C+ ++ ++ ++ + (C) (C) (C) (C) 1111222222221111

12121212 (3x 5) C(3x 5) C(3x 5) C(3x 5) C+ ++ ++ ++ +

(D) (D) (D) (D) 1111222222221111

3333 (3x 5) C(3x 5) C(3x 5) C(3x 5) C+ ++ ++ ++ + (E) (E) (E) (E) 3333222222223333

2222 (3x 5) C(3x 5) C(3x 5) C(3x 5) C+ ++ ++ ++ +

2) 2) 2) 2) 2 22 22 22 2(x 1) dx(x 1) dx(x 1) dx(x 1) dx+ =+ =+ =+ =∫∫∫∫

(A) (A) (A) (A) 2 32 32 32 3(x 1)(x 1)(x 1)(x 1)

CCCC3333++++

++++ (B) (B) (B) (B) 22223333xxxx

x Cx Cx Cx C3333

+ ++ ++ ++ +

(C) (C) (C) (C) 5 35 35 35 3x 2xx 2xx 2xx 2x

x Cx Cx Cx C5 35 35 35 3+ + ++ + ++ + ++ + +

(D) (D) (D) (D) 2 32 32 32 3(x 1)(x 1)(x 1)(x 1)

CCCC6x6x6x6x++++

++++ (E) (E) (E) (E) 2 32 32 32 32x(x 1)2x(x 1)2x(x 1)2x(x 1)

CCCC3333++++

++++

3) 3) 3) 3) 2 32 32 32 3x cos(x )dxx cos(x )dxx cos(x )dxx cos(x )dx ====∫∫∫∫

(A) (A) (A) (A) 33331111sin(x ) Csin(x ) Csin(x ) Csin(x ) C

3333− +− +− +− + (B) (B) (B) (B)

33333333xxxx

sin(x ) Csin(x ) Csin(x ) Csin(x ) C3333

− +− +− +− + (C) (C) (C) (C) 3 43 43 43 4x xx xx xx xsin Csin Csin Csin C

3 43 43 43 4

++++

(D) (D) (D) (D) 33331111sin(x ) Csin(x ) Csin(x ) Csin(x ) C

3333++++ (E) (E) (E) (E)

33333333xxxx

sin(x ) Csin(x ) Csin(x ) Csin(x ) C3333

++++

4) 4) 4) 4) In the figure below, PQ represents a 40In the figure below, PQ represents a 40In the figure below, PQ represents a 40In the figure below, PQ represents a 40----foot ladder with end P against a vertical wall and end Q on foot ladder with end P against a vertical wall and end Q on foot ladder with end P against a vertical wall and end Q on foot ladder with end P against a vertical wall and end Q on level ground. If the ladder is slipping down the wall, what is the distance RQ at the instant when Q is level ground. If the ladder is slipping down the wall, what is the distance RQ at the instant when Q is level ground. If the ladder is slipping down the wall, what is the distance RQ at the instant when Q is level ground. If the ladder is slipping down the wall, what is the distance RQ at the instant when Q is moving along the groundmoving along the groundmoving along the groundmoving along the ground 3333

4444 as fast as P is moving down the wall?as fast as P is moving down the wall?as fast as P is moving down the wall?as fast as P is moving down the wall?

(A) (A) (A) (A) 66665555 10101010 (B) (B) (B) (B) 8888

5555 10101010

(C) (C) (C) (C) 808080807777

(D) 24(D) 24(D) 24(D) 24

(E) 32(E) 32(E) 32(E) 32

PPPP

RRRR QQQQ

40404040

5) 5) 5) 5) If If If If 2 32 32 32 3f '(x) (x 2)(x 3) (x 4)f '(x) (x 2)(x 3) (x 4)f '(x) (x 2)(x 3) (x 4)f '(x) (x 2)(x 3) (x 4)= − − −= − − −= − − −= − − − , then f has which of the following relative extrema?, then f has which of the following relative extrema?, then f has which of the following relative extrema?, then f has which of the following relative extrema?

I.I.I.I. A relative maximum at A relative maximum at A relative maximum at A relative maximum at x 2x 2x 2x 2==== II.II.II.II. A relative minimum at A relative minimum at A relative minimum at A relative minimum at x 3x 3x 3x 3==== III.III.III.III. A relative maximum at A relative maximum at A relative maximum at A relative maximum at x 4x 4x 4x 4====

(A) I only(A) I only(A) I only(A) I only (B)(B)(B)(B) III onlyIII onlyIII onlyIII only (C) I and III only(C) I and III only(C) I and III only(C) I and III only (D) II and III only(D) II and III only(D) II and III only(D) II and III only (E) I, II, and III(E) I, II, and III(E) I, II, and III(E) I, II, and III

6) 6) 6) 6) The figure below shows the graph of f’, the derivative of the function f, on the open interval The figure below shows the graph of f’, the derivative of the function f, on the open interval The figure below shows the graph of f’, the derivative of the function f, on the open interval The figure below shows the graph of f’, the derivative of the function f, on the open interval 7 x 77 x 77 x 77 x 7− < <− < <− < <− < < . If f’ has four zeros on . If f’ has four zeros on . If f’ has four zeros on . If f’ has four zeros on 7 x 77 x 77 x 77 x 7− < <− < <− < <− < < , h, h, h, how many relative maxima does f have on ow many relative maxima does f have on ow many relative maxima does f have on ow many relative maxima does f have on 7 x 77 x 77 x 77 x 7− < <− < <− < <− < < ????

(A) one (A) one (A) one (A) one (B) two(B) two(B) two(B) two (C) three(C) three(C) three(C) three (D) four(D) four(D) four(D) four (E) five(E) five(E) five(E) five

7) 7) 7) 7) If the tangent to the graph of the function f at the point (1,7) passes through the point (If the tangent to the graph of the function f at the point (1,7) passes through the point (If the tangent to the graph of the function f at the point (1,7) passes through the point (If the tangent to the graph of the function f at the point (1,7) passes through the point (----2,2,2,2,----2), then 2), then 2), then 2), then f’(1) isf’(1) isf’(1) isf’(1) is

((((A) A) A) A) ----5555 (B) 1(B) 1(B) 1(B) 1 (C) 3(C) 3(C) 3(C) 3 (D) 7(D) 7(D) 7(D) 7 (E) undefined(E) undefined(E) undefined(E) undefined

8) 8) 8) 8) An equation of the line tangent to the graph of An equation of the line tangent to the graph of An equation of the line tangent to the graph of An equation of the line tangent to the graph of y cos(2x)y cos(2x)y cos(2x)y cos(2x)==== at at at at 4444xxxx ππππ==== isisisis

(A) (A) (A) (A) 4444y 1 (x )y 1 (x )y 1 (x )y 1 (x )ππππ− = − −− = − −− = − −− = − − (B) (B) (B) (B) 4444y 2(x )y 2(x )y 2(x )y 2(x )ππππ= −= −= −= − (C) (C) (C) (C) 4444y 2(x )y 2(x )y 2(x )y 2(x )ππππ= − −= − −= − −= − − (D) (D) (D) (D) 4444y 1 2(x )y 1 2(x )y 1 2(x )y 1 2(x )ππππ− = − −− = − −− = − −− = − − (E) (E) (E) (E) 4444y (x )y (x )y (x )y (x )ππππ= − −= − −= − −= − −

----7777 7777 0000 xxxx

yyyy

9) 9) 9) 9) If c is the number that satisfies the conclusion of the Mean Value Theorem for If c is the number that satisfies the conclusion of the Mean Value Theorem for If c is the number that satisfies the conclusion of the Mean Value Theorem for If c is the number that satisfies the conclusion of the Mean Value Theorem for 3 23 23 23 2f (x) x 2xf (x) x 2xf (x) x 2xf (x) x 2x= −= −= −= − on on on on the interval the interval the interval the interval 0 x 20 x 20 x 20 x 2≤ ≤≤ ≤≤ ≤≤ ≤ , then , then , then , then cccc ====

(A) 0(A) 0(A) 0(A) 0 (B) (B) (B) (B) 11112222 (C) 1(C) 1(C) 1(C) 1 (D) (D) (D) (D) 4444

3333 (E) 2(E) 2(E) 2(E) 2

10) 10) 10) 10) A curve C is defined by the parametric equations A curve C is defined by the parametric equations A curve C is defined by the parametric equations A curve C is defined by the parametric equations 2222x t 4t 1x t 4t 1x t 4t 1x t 4t 1= − += − += − += − + and and and and 3333y ty ty ty t==== . Which of the . Which of the . Which of the . Which of the followifollowifollowifollowing is an equation of the line tangent to the graph of C at the point (ng is an equation of the line tangent to the graph of C at the point (ng is an equation of the line tangent to the graph of C at the point (ng is an equation of the line tangent to the graph of C at the point (----3,8)?3,8)?3,8)?3,8)?

(A) (A) (A) (A) x 3x 3x 3x 3= −= −= −= − (B) (B) (B) (B) y 8y 8y 8y 8==== (C) (C) (C) (C) y 12(x 3) 8y 12(x 3) 8y 12(x 3) 8y 12(x 3) 8= + += + += + += + + (D) (D) (D) (D) x 2x 2x 2x 2==== (E) (E) (E) (E) 27272727

10101010y (x 3) 8y (x 3) 8y (x 3) 8y (x 3) 8= − + += − + += − + += − + +

11) 11) 11) 11) For what value of k will For what value of k will For what value of k will For what value of k will kkkk

xxxxxxxx

++++ have a relative maximum at have a relative maximum at have a relative maximum at have a relative maximum at x 2x 2x 2x 2= −= −= −= − ????

(A) (A) (A) (A) ----4444 (B) (B) (B) (B) ----2222 (C) 2(C) 2(C) 2(C) 2 (D) 4(D) 4(D) 4(D) 4 (E) None of these(E) None of these(E) None of these(E) None of these

12) 12) 12) 12) Let f be the function with the derivative given by Let f be the function with the derivative given by Let f be the function with the derivative given by Let f be the function with the derivative given by 2222 2222f '(x) xf '(x) xf '(x) xf '(x) x

xxxx= −= −= −= − . On which of the following . On which of the following . On which of the following . On which of the following

intervals is f decreasing?intervals is f decreasing?intervals is f decreasing?intervals is f decreasing?

(A) (A) (A) (A) ( , 1]( , 1]( , 1]( , 1]−∞ −−∞ −−∞ −−∞ − (B) [(B) [(B) [(B) [----1,0)1,0)1,0)1,0) (C) (C) (C) (C) 3333[ 2, )[ 2, )[ 2, )[ 2, )∞∞∞∞ (D) (D) (D) (D) ( ,0)( ,0)( ,0)( ,0)−∞−∞−∞−∞ (E) (E) (E) (E) 3333(0, 2](0, 2](0, 2](0, 2]