AP Calculus AB Name___________________________ Review for 5.1 – 5.5 Test Date__________________ Pd_______ For #1 – 11: Multiple Choice (No Calculator)

1. If x > 0 , x!

13 " dx =

(A) 23x23 +C (B) ! 1

3x43 +C (C) ! 3

2x23 +C (D) 3

2x23 +C (E) ! 3

4x!

43 +C

2. If (x2 ! 2x + 2) dx

0

6

" is approximated by three rectangles of equal width on the x-axis using a Right

Riemann Sum, then the approximation is: (A) 52 (B) 48 (C) 28 (D) 46 (E) 76

3. Find (2x3 + 3) dx

0

2

! .

(A) 8 (B) 11 (C) 14 (D) 20 (E) 24 4. The average value of a continuous function f (x) on the closed interval 3,7[ ] is 12. What is the value of

f (x) dx

3

7

! ?

(A) 3 (B) 4 (C) 12 (D) 36 (E) 48

5. Suppose that f (x) is an even function and let

f (x) dx0

1

! = 5 and

f (x) dx0

7

! = 1 . What is

f (x) dx!7

!1

" ?

(A) !5 (B) !4 (C) 0 (D) 4 (E) 5

6. If

2x3 ! kx2 + 2k( )0

2

" dx = 12 , then k must be:

(A) 1 (B) 2 (C) 3 (D) !2 (E) !3

7. The average value of the function f (x) = sin x on the closed interval 0,!2

"#$

%&'

is:

(A) !4

(B) !2

(C) 1 (D) 3!2

(E) 2!

8. ddx

du1+ u2x

0

! =

(A) 1x2 +1

(B) !1x2 +1

(C) x2 +1 (D) !x2 +1 (E) arctan x

9. If F(x) = t2 + 31

x2

! dt , then F ' (2)=

(A) 4 19 (B) 2 19 (C) 4 7 (D) 2 7 (E) 7

10.

!ecot x

sin2 x"4

"2# dx =

(A) !e (B) 1! e (C) !1 (D) e!1 (E) 1+ e

11. If g(x)dxa

b

! = 4a + b , then g(x)+ 7( )dxa

b

! =

(A) 8b !11a (B) 8b +11a (C) 8b ! 3a (D) 7b ! 7a (E) 4a + b + 7

For #12 – 22: Free Response 12. Evaluate each integral without a calculator.

a) dx!3

4

" b) t dta

a2

!

c) (x3 + 3x2 ! 2)dx!1

4

" 13. Evaluate each integral with a calculator.

a) 25 ! x2 dx!5

5

" b) x(x ! 4)(x + 2)dx!3

5

" c) cos(2x)cos(6x)dx32

3

!

14. Find the average value and the value of c for the function f (x) = 2sin x over 0, 3!2

"#$

%&'

.

15. If ddx

f (t)dt1

x

! = 5x " 2x + 3

, then:

a) Find f (x) b) Evaluate f (6)

16. Find F ' (x) if F(x) = t6

x3

! sin t dt .

17. Find F ' (x) if F(x) = sec2 tx

x4

! dt .

18. a) Use the Fundamental Theorem of Calculus (FTC) to evaluate

ddx

t + t( )1

cos x

! dt

b) Evaluate

t + t( )1

cos x

! dt

c) Find the derivative of the answer from part b. d) Does your work in part b and c support the FTC (part a)? 19. Evaluate each indefinite integral.

a) 2x3 + e!2x ! sec2 x +1( )" dx b)

x +1( )ex2+2x! dx

c)

2ex ! 2e! x

ex + e! x( )2" dx

20. Evaluate each definite integral without a calculator.

a)

x 4 + x23

0

2

! dx b) (x + cos x

!3

!2

" ) dx

c) xe!x22

0

2

" dx

21. Let g(x) = f (t)dt0

x

! , where f is the function whose graph is shown. Let g represent the position of a

particle moving up and down. a) Evaluate the following: g(0) = ____ g(1) = ____ g(2) = ____ g(3) = ____ g(6) = ____ g(7) = ____ For parts b through h, justify your answers. b) On what interval(s) is g increasing? c) Where does g have a maximum? d) When does g move towards the origin? e) When does g move away from the origin?

  f) At , which side of the origin is the particle? g) What is the particle velocity at ? h) Is the acceleration negative or positive at ?  

22. The table below gives the values of a function obtained from an experiment.

x 0 1 2 3 4 5 6 f(x) 9.3 9.0 8.3 6.5 2.3 !7.6 !10.5

Use the table to estimate f (x)dx0

6

! using three subintervals with:

a) right endpoints b) left endpoints c) midpoints d) trapezoids

Use the table to estimate f (x)dx0

6

! using six subintervals with:

a) right endpoints b) left endpoints c) trapezoids REMEMBER: This review packet is not exhaustive. The test will cover sections 5.1 – 5.5. To prepare for the test you should complete this review packet AND study all your notes/worksheets/previous HWs/previous quizzes from 5.1 through 5.5. Here is an additional review assignment to prepare for the test: Pg. 382 – 383 (9 – 17 odd, 53 – 69 odd)