- 1. Solve for the integral : ∫xe-x2 dx
A) None of these
B) 1/ex2 + C
C) -1/(ex2) + C
D) x/(e-x2) - 1/(ex2) + C
E) -x/(e-x2) + 1/(ex2) - C
- 2. Solve for the integral on the interval 0 to 1 (give exact answer): ∫xe-x2 dx
A) (3/4)(1-1/e)
B) (1/2)(1 - 1/e)
C) 0.2638
D) (1-1/e)
E) None of these
- 3. Solve for the integral: ∫sin6(x)cos2(x)dx
A) None of these
B) cos7(x)/7 + cos9(x)/9
C) sin7(x)/7 - sin9(x)/9 + C
D) sin7(x)/9 + sin9(x)/7
E) -cos7(x)/9 - cos9(x)/7
- 4. Solve for the integral: ∫e-xcos(x)dx
A) (sin(x)-cos(x))/2ex
B) (cos(x)-sin(x))/2ex
C) -1/2ex
D) None of these
E) 1/2ex
- 5. Determine the convergence of the sum on the interval k=0 to ∞: ∑ (3lnk)/k
A) Diverges by Nth Term Test
B) Converges by P-Series Test
C) Converges by Integral Test
D) Converges by Alternating Series Test
E) Diverges
- 6. Does the ∑ (3lnk) (0 to ∞) meet the requirements for the integral test?
A) I don't know dude
B) YES
C) NO
D) Cannot be determined
E) Who cares!
- 7. In order for the integral test to be applied to a series, the series must be
A) continuous only
B) positive only
C) None of these
D) positive, decreasing, and continuous
E) decreasing only
- 8. Solve for the sum on the interval 0 to ∞: ∑ 1/kp when p = 987.7
A) Divergent by Alternating Series Test
B) Cannot be determined
C) Convergent by Integral Test
D) Convergent by Ratio Test
E) Convergent by P-series
- 9. Determine the inteval of convergence of the sum on the interval n=1 to ∞: ∑ xn / n
A) |x|
B) 4
C) x + 7
D) None of these
E) x2
- 10. Determine the inteval of convergence of the sum on the interval n=1 to ∞: ∑ xn / n
A) 1/2
B) None of these
C) |(x+1)/2|
D) 2
E) |(x+7)/3|
- 11. Evaluate the integral on the inteval π/6 to π/3 (calculator allowed) : ∫cos3(x)/sin1/2(x)dx
A) 0.208
B) 0.475
C) None of these
D) 0.239
E) 0.362
- 12. Find the arc length of bounded by the points A and B. The EXACT arc length is most nearly
A) 0.881
B) ln(sqrt(2) + 1) + ln(1)
C) 0.5575
D) ln(sqrt(2) + 1)
E) None of these
- 13. For the image above, the area is most nearly (calculator recommended; to 4 decimal places))
A) 0.4820
B) None of these
C) 0.9640
D) 0.2410
E) 0.1205
- 14. The x-coordinate of the centroid is most nearly
A) 0.1528
B) 0.1067
C) None of these
D) .2503
E) 0.4429
- 15. Solve for the integral: ∫ sec x / tan2 x dx
A) tan x + C
B) None of these
C) sec2 x + C
D) sec x + C
E) -csc x + C
- 16. Find the second Taylor polynomial for f(x) = sin x , expanded about c = π/6 .
A) P(x) = 1/2 + (sqrt(3)/2)(x-π/6) - 1/(2(2!))(x - π/6)2
B) P(x) = 1/2x - π/6
C) P(x) = 1/2 + (sqrt(3)/2)(x-π/6)
D) Cannot be determined
E) P = π/6
- 17. The convergence of a sequence guaratees convergence for the sum of that sequence.
A) True
B) False
- 18. The image shows the area bounded by y = x2 + 1, y=0, x = 0 and x =1. In order to find the volume of the solid formed by revolving the region about the y-axis, which method would be the simplest to use?
A) Shell Method
B) Disk Method
C) None of these
D) Newton-Raphson Method
E) Washer Method
- 19. The image shows the area bounded by y = x2 + 1, y=0, x = 0 and x =1. Determine the volume of the solid generated by revolving the region about the y-axis using either method.
A) None of these
B) 6π/9
C) 3π/10
D) Gauss-Jordan= 10
E) 3π/2
- 20. The image presented is a nozzle facing up (the fluid would come through the top). If you would look at it from the top, you would see two circles. An outer and an inner circle. The dimensions are r(inner) = 1 and r(outer) = 2 . Note, image not to scale. [Hint: x2 + y2 = r2: A = ∫ r2 dr]. Which integral would be used to calculate the difference in area?
A) (-1 to 1) ∫ sec2 x
B) None of these
C) (-1 to 1) ∫ sqrt(4-x2)
D) (-1 to 1) ∫ 3/ (sqrt(4-x2) + sqrt(1-x2))
E) (-1 to 1) ∫ 3/ sqrt(4-x2)
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