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Math 2012 Calculus II Final Practice Spring 2008
Name: Last ___________________, First ______________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find the length of the curve.
y = (16 - x2/3) 3/2^ from x = 1 to x = 64 A) 96 B) 90 C) 180 D) 60
Find the area of the shaded region.
2 4 6 8 10 x
y 8 6 4 2
2 4 6 8 10 x
y 8 6 4 2
y = 2x y = x - 4
A) 32 B) 32 3 C) 128 3 D) 64 3
Find the volume of the solid generated by revolving the shaded region about the given axis.
- About the x-axis
π^ x 2
y 4
2
π^ x 2
y 4
2
y = 3 sin x
A) 9 2 π2^ - 3π B) 9 2 π2^ - 9π C) 9 2 π2^ D) 9 2 π2^ + 9π
Solve the problem.
A rescue cable attached to a helicopter weighs 2 lb/ft. A 160 - lb man grabs the end of the rope and is pulled from the ocean into the helicopter. How much work is done in lifting the man if the helicopter is 50 ft above the water? A) 13,000 ft (^) · lb B) 2660 ft (^) · lb C) 8100 ft (^) · lb D) 10,500 ft (^) · lb
1
Integrate the function.
- dx
∫ (x2 + 16)3/
A) 4 x 16 - x^
C B) x 16 16 - x^
(^16) x^ - x2+ C
C) x 4 16 + x^
Express the integrand as a sum of partial fractions and evaluate the integral.
- x^ +^8 x2^ + 6x
∫ dx
A) 4 3 ln x8(x (^) + 6)2^ + C B) 1 6 ln x8(x (^) + 6)2^ + C
C) 1 6 ln x
8 (x (^) + 6)^
Evaluate the integral.
π/
0
∫ x3 cos 5x dx
A) 1 5 x3 sin 5x - 3 25 x2 cos 5x + 6 125 x sin 5x + 6 625 cos 5x + C
B) 1 5 x3 sin 5x (^) + 3 5 x2 cos 5x (^) - 6 5 x sin 5x (^) - 6 5 cos 5x (^) + C
C) 1 5 x3 cos 5x + 3 25 x2 sin 5x - 6 125 x cos 5x - 6 625 sin 5x + C
D) 1 5 x3 sin 5x (^) + 3 25 x2 cos 5x (^) - 6 125 x sin 5x (^) - 6 625 cos 5x (^) + C
A recursion formula and the initial term(s) of a sequence are given. Write out the first five terms of the sequence.
a 1 = 1, a 2 = 3 , an+ 2 = an+ 1 - an A) 1, - 3 , 4 , - 5 , 6 B) 1, 3 , 2 , 1 , 0 C) 1, - 1, 2, - 3, 5 D) 1, 3 , 2 , - 1, - 3
Determine if the series converges or diverges; if the series converges, find its sum.
∞
n= 1
4n+^1
∑ 5n- 1
A) Converges; 100 B) Converges; 20 C) Converges; 80 D) Diverges
Find the Taylor polynomial of order 3 generated by f at a.
f(x) = x^3 , a = 7 A) 343 + 49(x - 49) + 49(x - 49)^2 + (x - 49)^3 B) 6 + 3(x - 49) + (x - 49)^2 + (x - 49)^3 C) 343 + 147(x - 49) + 21(x - 49)^2 + (x - 49)^3 D) 1372 + 147(x - 49) + 14(x - 49)^2 + (x - 49)^3
2
- Use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region about the x-axis.
y = x , y = 0, x= 4 A) 2 0
V = π∫ y + y dy= π
B) 2
0
V = π∫ y + y dy= π
C) 2
0
V = π∫ y − y dy= π
D) 4
0
V = π∫ y + y dy= π
E) 4
0
V = π∫ y − y dy= π
Find the arc length of the graph of the function
3 (^2 ) 3
y = x + over the interval [0,7].
A) 2 512 3 B)
C)
D) 2
E)
- Find the definite integral. (^2 )
0
x e −^ x dx ∫
A) (^1) – 1 2
(^) −e
B) (^) 2 1 (^) −e–4 C) (^) 1 −e–4 D) (^1) – 1 2
− ^ +e
E) (^) –2 1 (^) −e–4
- Find the indefinite integral.
sin^3 7
x ∫ dx A) (^2) 7 3 cos cos 7 7 3
x x
C
− ^ +
B) 2
7 2 cos cos 7 7 3
x x
C
− ^ +
C) 2
7 3 cos cos 7 7 3
x x
C
D) 2
7 1 cos cos 7 7 3
x x
C
E) 2
7 2 cos cos 7 7 3
x x
C
- Determine the convergence or divergence of the series.
0
n^9 n
∞
=
∑
A) Diverges B) Converges C) Cannot be determined from the methods in the chapter
- Determine the convergence or divergence of the series.
5 (^1 )
n (^) n
∞
=
∑
A) Inconclusive B) Converges C) Diverges
- Determine the convergence or divergence of the series.
0
n
n
∞
=
∑
A) Inconclusive B) Converges C) Diverges
- Use the Ratio Test to determine the convergence or divergence of the series.
5
1 6
n n
∞ n
=
∑
A) Diverges B) Converges C) Ratio Test is inconclusive
- Use the Root Test to determine the convergence or divergence of the series.
2 2 1
n
n
n n
∞
=
∑
A) Root Test is inconclusive B) Converges C) Diverges
- Determine the convergence or divergence of the series using any appropriate test from this chapter. Identify the test used.
1
n^5
n n
∞
= +
∑
A) Diverges; Ratio Test B) Diverges; Integral Test C) Converges; p-series D) Converges; Integral Test E) Both A and B F) Both C and D
- Find the interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval.)
n
n
∞ x
=
∑
A) (^) [ −2, 2) B) ( 2, 2)− C) (^) [ −2, 2] D) (^1 ) , 2 2
^ −^
E) 1
- Find the area of the surface generated by revolving the curve about the x-axis.
(^1 3) , 0 3 3
y = x ≤ x≤
A) 3 1 82 2 1
18
π
B) 3
π
C) 3
π
D) 3
E) 3
π
Answer Key 2 nd 10
11. E
12. E
13. A
14. B
15. A
16. A
17. A
18. B
19. C
20. C
21. A
22. C
23. B
24. B
25. A
26. A
27. B
