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Donnie Smith Sharma MAT 266 ONLINE B Spring 2017 Assignment Review for final due 05/02/2017 at 11:59pm MST 1. (1 point) Evaluate the indefinite integral. Z x 4 p 9+x 5dx +C Answer(s) submitted: • (2)/(15)(9+xˆ5)ˆ((3)/(2)) (correct)

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Donnie Smith Sharma MAT 266 ONLINE B Spring 2017 Assignment Review for final due 05/02/2017 at 11:59pm MST 1. (1 point) Evaluate the indefinite integral. Z x 4 p 9+x 5dx +C Answer(s) submitted: • (2)/(15)(9+xˆ5)ˆ((3)/(2)) (correct) 2. (1 point) Determine whether the integral is divergent or convergent. If it is convergent, evaluate it. If not, state your answer as divergent . Z ∞ 0 2e −x dx Answer(s) submitted: • 2 (correct) 3. (1 point) The equation r = 3 sinθ represents a circle. Find the cartesian coordinates of the center: x = y = and the radius: r = Answer(s) submitted: • 0 • (3)/(2) • (3)/(2) (correct) 4. (1 point) Find the area inside one leaf of the rose: r = 5 sin(4θ) The area is Answer(s) submitted: • (25pi)/(16) (correct) 5. (1 point) Find the slope of the tangent line to the polar curve r = sin(6θ) at θ = π 12 . slope = Answer(s) submitted: • -3.732 (correct) 6. (1 point) The function f(x) = 1 1+25x 2 is represented as a power series f(x) = ∞ ∑ n=0 cnx n . Find the first few coefficients in the power series. c0 = c1 = c2 = c3 = c4 = Find the radius of convergence R of the series. R = . Answer(s) submitted: • • • • • • (incorrect) 7. (1 point) Use integration by parts to evaluate the integral. Z 2x cos(2x)dx +C Answer(s) submitted: • (cos 2x)/(4) (incorrect) 1 8. (1 point) Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Then find the area of the region. y = 2+ √ x, y = 2+ 1 4 x Answer(s) submitted: • (incorrect) 9. (1 point) The region bounded by y = 5 x , y = 0, x = 4, and x = 6 is rotated about the x-axis. Find the volume of the resulting solid. Volume = Answer(s) submitted: • (incorrect) 10. (1 point) Which of the following integrals represents the length of the curve y = 2 x , 0 ≤ x ≤ 3? • A. Z 3 0 q 1+2(ln 2) 22 x dx • B. Z 3 0 q 1+ (ln 2) 22 2 dx • C. Z 3 0 √ 1+2 x dx • D. Z 3 0 p 1+2 2x dx • E. Z 3 0 q 1+ (ln 2) 22 x dx • F. Z 3 0 q 1+ (ln 2) 22 2x dx Answer(s) submitted: • (incorrect) 11. (1 point) Find the values of x so that the series below converges. ∞ ∑ n=0 (x−3) n 3 n Give your answer in interval notation. ( , ) Answer(s) submitted: • • (incorrect) 12. (1 point) Consider the series ∞ ∑ n=1 an where an = e n+1 √ n+6(n+8)! In this problem you must attempt to use the Ratio Test to decide whether the series converges. Compute L = lim n→∞ an+1 an Enter the numerical value of the limit L if it converges, INF if it diverges to infinity, MINF if it diverges to negative infinity, or DIV if it diverges but not to infinity or negative infinity. L = Which of the following statements is true? A. The Ratio Test says that the series converges absolutely. B. The Ratio Test says that the series diverges. C. The Ratio Test says that the series converges conditionally. D. The Ratio Test is inconclusive, but the series converges absolutely by another test or tests. E. The Ratio Test is inconclusive, but the series diverges by another test or tests. F. The Ratio Test is inconclusive, but the series converges conditionally by another test or tests. Enter the letter for your choice here: Answer(s) submitted: • • (incorrect) 13. (1 point) Use the binomial series to expand the function f(x) = (1+x) 1/3 as a power series. ∞ ∑ n=0 cnx n . Compute the following coefficients: c0 = c1 = c2 = c3 = c4 = 2 Answer(s) submitted: • • • • • (incorrect) 14. (1 point) Find the Maclaurin series of the function f(x) = (3x 2 )e −6x (f(x) = ∞ ∑ n=0 cnx n ) c1 = c2 = c3 = c4 = c5 = Answer(s) submitted: • • • • • (incorrect) 15. (1 point) Find the area inside one leaf of the rose: r = 6 sin(6θ) The area is Answer(s) submitted: • (incorrect) 16. (1 point) Find the slope of the tangent line to the polar curve r = sin(3θ) at θ = π 6 . slope = Answer(s) submitted: • (incorrect) 19. (1 point) Suppose parametric equations for the line segment between (8,−4) and (−1,−1) have the form: x=a+bt y=c+dt If the parametric curve starts at (8,−4) when t = 0 and ends at (−1,−1) at t = 1, then find a, b, c, and d. a = , b = , c = , d = . Answer(s) submitted: • • • • (incorrect) 20. (1 point) Suppose that 8x (11+x) = ∞ ∑ n=0 cnx n . Find the first few coefficients. c0 = c1 = c2 = c3 = c4 = Find the radius of convergence R of the power series. R = . Answer(s) submitted: • • • • • • (incorrect) 21. (1 point) The Taylor series for f(x) = cos(x) at a = π 4 is ∞ ∑ n=0 cn(x− π 4 ) n . Find the first few coefficients. c0 = c1 = c2 = c3 = c4 = Answer(s) submitted: • • 3 • • • (incorrect) 22. (1 point) For each of the following integrals, indicate whether integration by substitution or integration by parts is more appropriate, or if neither method is appropriate. Do not evaluate the integrals. 1. R x sinx dx • A. substitution • B. integration by parts • C. neither 2. R x 4 1+x 5 dx • A. neither • B. integration by parts • C. substitution 3. R x 4 e x 5 dx • A. neither • B. integration by parts • C. substitution 4. R x 4 cos(x 5 )dx • A. substitution • B. integration by parts • C. neither 5. R √ 1 3x+1 dx • A. substitution • B. neither • C. integration by parts (Note that because this is multiple choice, you will not be able to see which parts of the problem you got correct.) Answer(s) submitted: • • • • • (incorrect) 23. (1 point) Suppose a particle moves along a straight line with velocity v(t) = t 2 e −3t meters per second after t seconds. It travels meters during the first t seconds. Answer(s) submitted: • (incorrect) 24. (1 point) Evaluate the indefinite integral. Z sin3 (7x) cos10(7x)dx +C Answer(s) submitted: • (incorrect) 25. (1 point) The form of the partial fraction decomposition of a rational function is given below. − 5x 2 +6x+18 (x+2)(x 2 +9) = A x+2 + Bx+C x 2 +9 A = B = C = Now evaluate the indefinite integral. Z − 5x 2 +6x+18 (x+2)(x 2 +9) dx = Answer(s) submitted: • • • • (incorrect) 26. (1 point) Consider the area between the graphs x+1y = 24 and x + 6 = y 2 . This area can be computed in two different ways using integrals. First of all it can be computed as a sum of two integrals Z b a f(x)dx+ Z c b g(x)dx where a = , b = , c = and f(x) = g(x) = Alternatively this area can be computed as a single integral Z β α h(y)dy where α = , β = and h(y) = Either way we find that the area is . Answer(s) submitted: • • • • • • • • • (incorrect) 28. (1 point) Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. y = x 2 , y = 0, x = 1, about the y-axis Answer(s) submitted: • (incorrect) 29. (1 point) Which of the following integrals represents the length of the parametric curve x = ln(t), y = √ t +1, 1 ≤ t ≤ 5, about the x-axis? • A. Z 5 1 t 2 +4 2t √ t +1 dt • B. Z 5 1 1 2t √ t 2 +1 dt • C. Z 5 1 t 2 +4 4t 2 √ t 2 +1 dt • D. Z 5 1 t +2 t √ t +1 dt • E. Z 5 1 t +2 2t √ t +1 dt • F. Z 5 1 t +2 2t √ 4t +4 dt Answer(s) submitted: • (incorrect) 30. (1 point) Use the ratio test to determine whether ∞ ∑ n=23 n+5 n! converges or diverges. (a) Find the ratio of successive terms. Write your answer as a fully simplified fraction. For n ≥ 23, lim n→∞ an+1 an = lim n→∞ (b) Evaluate the limit in the previous part. Enter ∞ as infinity and −∞ as -infinity. If the limit does not exist, enter DNE. lim n→∞ an+1 an = (c) By the ratio test, does the series converge, diverge, or is the test inconclusive? • Choose • Converges • Diverges • Inconclusive Answer(s) submitted: • • • • (incorrect) Generated by c WeBWorK, http://webwork.maa.org, Mathematical Association of America 5

 
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