1.
A. 0 |
B. 1/4 |
C. 1/2 |
D. 1 |
E. ¥ |
2. What is the average value of xe^x^{2} on the interval [2, 4]?
A. (e^{16} - e^{4})/4 |
B. (e^{16} - e^{4})/2 |
C. (2e^{16} - e^{4})/2 |
D. 2e^{16} - e^{4} |
E. 2e^{16} + e^{4} |
3. Let F(x) be a strictly decreasing continuously differentiable function on [a, b]. Then must equal to
A. |F(b)| - |F(a)| |
B. F(a) - F(b) |
C. F(b)-F(a) |
D. |F(a)| - |F(b)| |
E. F(-b) - F(-a) |
4. Let f(x) = 2x + 1 for 0 £ x £ 1. If the interval [0,1] is partitioned into 4 subintervals of equal length, then what is the smallest Riemann sum for f(x) and this partition?
A. 7/4 |
B. 15/8 |
C. 2 |
D. 7/2 |
E. 7 |
5.
A. -p ^{2}/2 |
B. -p |
C. 0 |
D. p |
E. p ^{2} |
6. What is the area of the closed region bounded by x = -1, x = 0, y = x^{2}, and y = x^{3} ?
A. 1/12 |
B. 1/6 |
C. 1/4 |
D. 5/12 |
E. 7/12 |
7. The closed region in the first quadrant bounded by the curves y = x^{3} and y = x^{(1/3)} is rotated about the x-axis. What is the volume of the resulting solid?
A. 1/2 |
B. 128p /455 |
C. 16p /35 |
D. p /2 |
E. 32p /35 |
8. What is the x-coordinate of the centroid of the closed region
R = {(x,y)Î R^{2}: 0 £ x £ 1, (1-x^{2})^{1/2} £ y £ 1} ?
A. 1/6 |
B. 1/3 |
C. 2/(3(4-p )) |
D. 4/3p |
E. 6/(4-p ) |
9.
A. -2 |
B. -1 |
C. 1 |
D. 2 |
E. does not exist |
10.
A. 5/6 - 3 ln 6 |
B. 61/6 + 3 ln 6 |
C. 5/6 + 3 ln 6 |
D. 17/6 + 3 ln 6 |
E. 17/6 - 3 ln 6 |
11.
A. -2 + (9/4)ln 3 |
B. -4 + (9/2)ln 3 |
C. -(1/4) + (9/4)ln 3 |
D. -(5/2) + (9/2)ln 3 |
E. -2 + (9/2)ln 3 |
12. What is the area of the closed region bounded by y = x^{2} - |x| and the x-axis, between x = -1 and x = 1?
A. 1/12 |
B. 1/6 |
C. 1/3 |
D. 2/3 |
E. 5/6 |
13. What is the average value of the function f(x) = x sin(x^{2}) over the interval [2,4]?
A. sin 4 + 2 sin 16 |
B. -sin 4 + 2 sin 16 |
C. (cos 4 - cos 16)/2 |
D. (cos 4 - cos 16)/4 |
E. (-cos 4 + cos 16)/4 |
14.
A. (1/12)ln 8 |
B. 3/8 |
C. û(5/12) |
D. ln 8 |
E. does not exist |
15.
A. ln (1 + e^{-1}) |
B. - ln (1 + e^{-1}) |
C. ln (1 + e) |
D. arctan (e^{1/2}) |
E. does not exist |
16.
A. 3/8 |
B. 2/3 |
C. 3/2 |
D. 9/4 |
E. 8/3 |
17. If and then
A. -3 |
B. -1 |
C. 0 |
D. 1 |
E. 3 |
18. Let
A. 0 |
B. 1/3 |
C. 2/3 |
D. 1 |
E. Does not exist. |
19.
A. (p ^{3}/256)cos(p /8) |
B. (Ö 2/32)(p ^{2}+8p -32) |
C. (p ^{2}/32)Ö 2 |
D. (Ö 2/32)(p ^{2}-8p +32) |
E. (p ^{3}/384)(2-Ö 2) |
20. Let S be the closed region in the first quadrant of the xy-plane bounded by y = 6x^{2}, y = 0, x = 0, and x = 1. What is the volume of the solid obtained by revolving S about the line x = -1?
A. 3p |
B. 7p |
C. 36p /5 |
D. 8p |
E. 56p /5 |
21. The region S is bounded by y = x^{2} - 2x + 3, y = 0, x = 0, and x = 9. Which of the following is the approximation to the area of S obtained by computing the sum of the areas of the 3 inscribed rectangles with bases [0,3], [3,6], and [6,9] (lower Riemann sum)?
A. 105 |
B. 108 |
C. 117 |
D. 189 |
E. 297 |
22.
A.ln (x^{2}+2x+3)+C |
B.(1/(2x+2))ln(x^{2}+2x+3)+C |
C.-(x+1)/(x^{2}+2x+3)^{2} +C |
D.Ö 2arctan(x+1)+C |
E.(1/Ö 2)arctan((x+1)/Ö 2)+C |
23. Let f(x) = x^{2}. For what value of x does f(x) equal the average value of f on the closed interval [2,5]?
A. Ö (107/9) |
B. Ö 13 |
C. Ö (125/9) |
D. Ö (133/9) |
E. 7 |
24. If and , then
A. -31 |
B. -19 |
C. 11 |
D. 30 |
E. 49 |
25.
A. -(47/576) |
B. 1/6 |
C. ln (8/9) |
D. ln 2 |
E. ln (32/9) |
26. Let S be the closed region in the first quadrant of the xy-plane bounded by y = sin(p x/2) and y = x for 0 £ x £ 1. What is the volume of the closed region in R^{3} obtained by revolving S about the x-axis?
A. 2 - (p /2) |
B. p /6 |
C. p /3 |
D. p /2 |
E. (2p )/3 |
27. Let f(x) = x^{2}(x^{3} + 1)^{4}. What is the area under the graph of f over the closed interval [-1,1]?
A. 0 |
B. 32/15 |
C. 62/15 |
D. 64/15 |
E. 32/5 |
28. What is the average value of y = (2x+1)^{1/2} over the interval [4,12]?
A. Ö 3 - 1/3 |
B. 2(Ö 3 - 1/3) |
C. 4 |
D. 49/12 |
E. Ö 17 |
29.
A. 2 - ln(2) |
B. ln(2) - 1/2 |
C.2 tan^{2}(2) + 4 ln[cos(2)] |
D. 2 ln[sec(2)]-sin^{2}(2) |
E. 2 - p /2 |
30.
A. 2/p |
B. 25/8 |
C. 2 |
D. (1+2p )/p ^{2} |
E. 2p |