AB Calculus - Hardtke Chapter 5: Take Home Testfaculty.muhs.edu/hardtke/ABCalc_Notes2012-13/Assignment 5.8 Ch5... · Ch 5 Take Home – Part 2 - No ... For Problems 15-18: ... AB

AB Calculus - Hardtke Chapter 5: Take Home Test Name _______________________________

Date: Tuesday, 1/22

MAY USE YOUR CALCULATOR FOR THIS PAGE. Round answers to three places. Score: _____/50

Show diagrams and work to justify each answer.

1. Approximate the value of 3

3

12x x dx using a regular partition of MIDPOINT rectangles with n = 4. (Sketch the function &

show the rectangles you are using.) Write your final answer in the blank.

1.

2. The velocity v (in cm/s) of a rolling ball at time t (in seconds is recorded below.

Find an upper and lower estimate for the distance the ball has traveled during the ten-second interval.

Time (seconds) 0 2 3 5 8 12

Velocity (cm/s) 1 6.6 12.1 28.5 66.6 101

*Show what you are plugging into your calculator for each part

2. Upper:

2. Lower:

3. Calculate the sum to the nearest thousandth:

50

1 4 2k

k

k

3.

AB Calculus - Hardtke Ch 5 Take Home – Part 2 - No Calculator Name __________________________

4-16. Multiple Choice: Select the BEST answer for each statement.

4. For a constantly decreasing function the value of a Left-Endpoint Riemann Sum will be _?_. 4. _______

A. higher than the actual area B. exactly equal to the actual area C. lower than the actual area

D. lower than the midpoint area E. none of these

5. For a constantly increasing non-linear function, the value of a Midpoint Riemann Sum will be _?_. 5. _______

A. equal to the definite integral B. equal to half of the area C. a regular partition

D. lower than the sum of circumscribing rectangles E. all of these

6. For a Riemann Sum with partition x-coordinates at {8, 11, 15, 17, 19}, n = _?_. 6. _______

A. 7 B. 6 C. 5 D. 4 E.3

7. For a Riemann Sum with partition x-coordinates at {1, 4, 7, 10, 13}, ∆x = _?_. 7. _______

A. 5 B. 4 C. 3 D. 2 E. 1

8. Given partition x-coordinates at {8, 11, 15, 17, 19}, for a Midpoint Riemann Sum, x2 = _?_. 8. _______

A. 8 B. 11 C. 17 D. 13 E .16

9. For any Reimann Sum over [2, 16] with a regular partition and n = 7, ∆x= _?_ . 9. _______

A. 7 B. 6 C. 3 D. 2 E. 1

10. For a function defined over [a, b], in

n

i 1

lim f (x ) x

is _?_ , provided the limit exists. 10. ______

A. the derivative of f(x) B. the definition of b

af (x)dx C. an approximate value of the area under the curve

D. equal to E. none of these

11. If you estimate the area under the curve f(x) = x between x = 1 and x = 2 using 2 subintervals of equal length, 11. ______

what is the largest value the approximation could have?

A. 3 B. 3

4 C.

5

4 D.

7

4 E.

5

16

12. Estimate the area under the curve f(x) = x2 for 0 x 2. What is the value of the estimate using four 12. ______

rectangles at the left hand endpoints?

A. 1

2 B.

7

4 C. 3 D.

15

4 E.

29

8

13. Let 1

( )f xx

on the interval [1, 2]. Let the interval be divided into two equal subintervals. Find the value 13. ______

of the Riemann sum 2

*

1i

i

f x x

if each *ix is the midpoint of its interval.

A. 3

2 B.

3

4 C. 3 D.

24

35 E.

6

35

14. 3

15dx

14. ______

A. 5 B. 10 C. 15 D. 20 E. none of these

For Problems 15-18: Given f is integrable over a closed interval

containing a, b and c (in any order); c

af (x)dx = n and

b

cf (x)dx = m.

15. c

cf (x)dx = _?_. 15. ______

A. m B. m C. n D. n E. none of these

16. a

cf (x)dx = _?_. 16. ______

A. 2n B. m C. mn D. n E. n

17. b

af (x)dx = _?_ 17. ______

A. m + n B. m - n C. mn D. 2m E. none of these

18. a

c3f (x)dx = _?_. 18. ______

A. 3n B. – n C. – 3n D. 3 E. none of these

19. 7

4

7x dx

= _?_. 19. ______

A. twice the value of 7

4

0x dx B. negative C. one-half the value of

74

0x dx

D. zero E. none of these

20. If f(x) is an odd function, then 7

7f (x)dx

is _?_. 20. ______

A. twice the value of 7

0f (x)dx B. negative C. one-half the value of

7

0f (x)dx

D. zero E. none of these

21. sin x dx 21. _____

A. cos x B. – cos x C. cos x + C D. – ½ sin2

x + C E. none of these

22.

42

6

sec x dx

22. _____

A. 1 B. 3

12

C. 3

13

D. 2 3

2 3 E. none of these

23.23

1sin

2d

23. _____

A. 1 2 B. 1 C. 3

13

D. 1 2

2 2 E. none of these

24.6

02x dx 24. _____

A. 4 B. 8 C. 9 D. 10 E. none of these

25.

1

21

2

1

1dx

x 25. _____

A. 1 B. 0 C. 3

D.

2

3

E. none of these

26.2

5

1dx

x 26. _____

A. 5(x2 + 1)

2 + C B. arc tan x + C C. 5 arc tan x

D. 1

tan5

x C E. none of these

27.1

2

0

xe dx 27. _____

A. e2 – 1 B. ½ e

2 – 1 C. ½ (e

2 – 1) D. 2(e

2 – 1) E. none of these

28.1

dxx

= 28. _____

A. x -1

+ C B. ½ x -1

+ C C. arc tan x + c D. ln | x | + C E. none of these

29.

4

6

csc cotx x dx

= 29. _____

A. 0 B. 1 C. 2 D. 2 2 E. none of these

30. For n – 1, nx dx = 30. _____

A. x n – 1

+ C B.

1

1

nxC

n

C.

1nxC

n

D.1

nxC

n

E. none of these

31. 3

3

8

0 2

11x dx

x

= 31. _____

A. 18 B. 14 C. 6 D.5

2 E. none of these

32. If 2

33 37

bx dx , find the value of b. 32. _____

A. 5 B. 4 C. 7 D. 6 E. none of these

33. 2

1

2 1xdx

x

= 33. _____

A. ½ B. 1 C. 3

ln 22 D. 2 E. none of these

34. Find g’(

) given g(x) = ∫

35. Find w’ (t) given w(t) = ∫ √

36. Find s’(2) given s(t) = ∫ ( ) √

37. The graph of a car’s velocity function v(t) in mi/h is shown below. 37.A.___________

B. ____________

C. ____________

38. Given the graph of a function f below. Label the following quantities from smallest (#1) to largest (#5)

0 8 8 9 10

2 0 4 0 0

( ) ( ) ( ) ( ) ( )

#____ #____ #____ #____ #____

f x dx f x dx f x dx f x dx f x dx

39. 39.A. f(-3)= ____ f(-1)=_____

f(0)= ____ f(1) =_____

f(3)= _____ f(5)=_____

B. _______________________

C. ______________________

0 4 8 10 2 6

f(x)

For Problem 40: Given n

2

k 1

n(n 1)(2n 1)k

6

and

2n3

k 1

n(n 1)k

2

40. Simplify 20

2

k 1

10k 1

(No TI-89! Can you do it yourself?) 40. _______________________

41. Write the expression in Sigma notation: 10 13 16 19 22 25 28

9 16 25 36 49 64 81 41. _________________________

(No need to simplify, just rewrite the notation.)

42. Given 1

2

2x 1 dx 6

.

A. Find a number c that satisfies the conclusion of the mean value theorem for integrals 42.A

B. Find the average value of the given function on the interval. 42B.

43. Solve the differential equation given f " (x) = 6x – 4; f ' (2) = 5; and f(2) = 4.

43. __________________________

AB Calculus - Hardtke Chapter 5: Take Home Test SOLUTION KEY

Date: Tuesday, 1/22

MAY USE YOUR CALCULATOR FOR THIS PAGE. Round answers to three places. Score: _____/50

Show diagrams and work to justify each answer.

1. Approximate the value of 3

3

12x x dx using a regular partition of MIDPOINT rectangles with n = 4. (Sketch the function &

show the rectangles you are using.) Write your final answer in the blank.

1.

2. The velocity v (in cm/s) of a rolling ball at time t (in seconds is recorded below.

Find an upper and lower estimate for the distance the ball has traveled during the ten-second interval.

Time (seconds) 0 2 3 5 8 12

Velocity (cm/s) 1 6.6 12.1 28.5 66.6 101

*Show what you are plugging into your calculator for each part

2. Upper:

2. Lower:

3. Calculate the sum to the nearest thousandth:

50

1 4 2k

k

k

3.

AB Calculus - Hardtke Ch 5 Review – Part 2 - No Calculator SOLUTION KEY

4-16. Multiple Choice: Select the BEST answer for each statement.

4. For a constantly decreasing function the value of a Left-Endpoint Riemann Sum will be _?_. 4. _______

A. higher than the actual area B. exactly equal to the actual area C. lower than the actual area

D. lower than the midpoint area E. none of these

5. For a constantly increasing non-linear function, the value of a Midpoint Riemann Sum will be _?_. 5. _______

A. equal to the definite integral B. equal to half of the area C. a regular partition

D. lower than the sum of circumscribing rectangles E. all of these

6. For a Riemann Sum with partition x-coordinates at {8, 11, 15, 17, 19}, n = _?_. 6. _______

A. 7 B. 6 C. 5 D. 4 E.3

7. For a Riemann Sum with partition x-coordinates at {1, 4, 7, 10, 13}, ∆x = _?_. 7. _______

A. 5 B. 4 C. 3 D. 2 E. 1

8. Given partition x-coordinates at {8, 11, 15, 17, 19}, for a Midpoint Riemann Sum, x2 = _?_. 8. _______

A. 8 B. 11 C. 17 D. 13 E .16

9. For any Reimann Sum over [2, 16] with a regular partition and n = 7, ∆x= _?_ . 9. _______

A. 7 B. 6 C. 3 D. 2 E. 1

10. For a function defined over [a, b], in

n

i 1

lim f (x ) x

is _?_ , provided the limit exists. 10. ______

A. the derivative of f(x) B. the definition of b

af (x)dx C. an approximate value of the area under the curve

D. equal to E. none of these

11. If you estimate the area under the curve f(x) = x between x = 1 and x = 2 using 2 subintervals of equal length, 11. ______

what is the largest value the approximation could have?

A. 3 B. 3

4 C.

5

4 D.

7

4 E.

5

16

12. Estimate the area under the curve f(x) = x2 for 0 x 2. What is the value of the estimate using four 12. ______

rectangles at the left hand endpoints?

A. 1

2 B.

7

4 C. 3 D.

15

4 E.

29

8

13. Let 1

( )f xx

on the interval [1, 2]. Let the interval be divided into two equal subintervals. Find the value 13. ______

of the Riemann sum 2

*

1i

i

f x x

if each *ix is the midpoint of its interval.

A. 3

2 B.

3

4 C. 3 D.

24

35 E.

6

35

14. 3

15dx

14. ______

A. 5 B. 10 C. 15 D. 20 E. none of these

For Problems 15-18: Given f is integrable over a closed interval

containing a, b and c (in any order); c

af (x)dx = n and

b

cf (x)dx = m.

15. c

cf (x)dx = _?_. 15. ______

A. m B. m C. n D. n E. none of these

16. a

cf (x)dx = _?_. 16. ______

A. 2n B. m C. mn D. n E. n

17. b

af (x)dx = _?_ 17. ______

A. m + n B. m - n C. mn D. 2m E. none of these

18. a

c3f (x)dx = _?_. 18. ______

A. 3n B. – n C. – 3n D. 3 E. none of these

19. 7

4

7x dx

= _?_. 19. ______

A. twice the value of 7

4

0x dx B. negative C. one-half the value of

74

0x dx

D. zero E. none of these

20. If f(x) is an odd function, then 7

7f (x)dx

is _?_. 20. ______

A. twice the value of 7

0f (x)dx B. negative C. one-half the value of

7

0f (x)dx

D. zero E. none of these

21. sin x dx 21. _____

A. cos x B. – cos x C. cos x + C D. – ½ sin2

x + C E. none of these

22.

42

6

sec x dx

22. _____

A. 1 B. 3

12

C. 3

13

D. 2 3

2 3 E. none of these

23.23

1sin

2d

23. _____

A. 1 2 B. 1 C. 3

13

D. 1 2

2 2 E. none of these

24.6

02x dx 24. _____

A. 4 B. 8 C. 9 D. 10 E. none of these

25.

1

21

2

1

1dx

x 25. _____

A. 1 B. 0 C. 3

D.

2

3

E. none of these

26.2

5

1dx

x 26. _____

A. 5(x2 + 1)

2 + C B. arc tan x + C C. 5 arc tan x

D. 1

tan5

x C E. none of these

27.1

2

0

xe dx 27. _____

A. e2 – 1 B. ½ e

2 – 1 C. ½ (e

2 – 1) D. 2(e

2 – 1) E. none of these

28.1

dxx

= 28. _____

A. x -1

+ C B. ½ x -1

+ C C. arc tan x + c D. ln | x | + C E. none of these

29.

4

6

csc cotx x dx

= 29. _____

A. 0 B. 1 C. 2 D. 2 2 E. none of these

30. For n – 1, nx dx = 30. _____

A. x n – 1

+ C B.

1

1

nxC

n

C.

1nxC

n

D.1

nxC

n

E. none of these

31. 3

3

8

0 2

11x dx

x

= 31. _____

A. 18 B. 14 C. 6 D.5

2 E. none of these

32. If 2

33 37

bx dx , find the value of b. 32. _____

A. 5 B. 4 C. 7 D. 6 E. none of these

33. 2

1

2 1xdx

x

= 33. _____

A. ½ B. 1 C. 3

ln 22 D. 2 E. none of these

34. Find g’(

) given g(x) = ∫

35. Find w’ (t) given w(t) = ∫ √

36. Find s’(2) given s(t) = ∫ ( ) √

37. The graph of a car’s velocity function v(t) in mi/h is shown below. 37.A.___________

B. ____________

C. ____________

38. Given the graph of a function f below. Label the following quantities from smallest (#1) to largest (#5)

0 8 8 9 10

2 0 4 0 0

( ) ( ) ( ) ( ) ( )

#____ #____ #____ #____ #____

f x dx f x dx f x dx f x dx f x dx

39. 39.A. f(-3)= ____ f(-1)=_____

f(0)= ____ f(1) =_____

f(3)= _____ f(5)=_____

B. _______________________

C. ______________________

0 4 8 10 2 6

f(x)

For Problem 40: Given n

2

k 1

n(n 1)(2n 1)k

6

and

2n3

k 1

n(n 1)k

2

40. Simplify 20

2

k 1

10k 1

(No TI-89! Can you do it yourself?) 40. _______________________

41. Write the expression in Sigma notation: 10 13 16 19 22 25 28

9 16 25 36 49 64 81 41. _________________________

(No need to simplify, just rewrite the notation.)

42. Given 1

2

2x 1 dx 6

.

A. Find a number c that satisfies the conclusion of the mean value theorem for integrals 42.A

B. Find the average value of the given function on the interval. 42B.

43. Solve the differential equation given f " (x) = 6x – 4; f ' (2) = 5; and f(2) = 4.

43. __________________________