Donnie Smith Assignment Review for final due 05/02/2017 at 11:59pm MST 1. (1 point) Evaluate the indefinite integral. Z x4 p Sharma MAT 266 ONLINE B Spring 2017 5. (1 point) Find the slope of the tangent line to the polar curve r = sin(6) at = 12 . slope = 9 + x5 dx Answer(s) submitted: -3.732 (correct) +C Answer(s) submitted: 6. (1 point) The function f (x) = power series (2)/(15)(9+x5)((3)/(2)) 1 1+25x2 is represented as a (correct) f (x) = cn x n . n=0 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 Find the first few coefficients in the power series. c0 = c1 = 2ex dx 0 c2 = c3 = Answer(s) submitted: c4 = 2 Find the radius of convergence R of the series. R= . (correct) 3. (1 point) The equation r = 3 sin represents a circle. Find the cartesian coordinates of the center: x = y = and the radius: r = Answer(s) submitted: Answer(s) submitted: 0 (3)/(2) (3)/(2) (incorrect) (correct) 7. (1 point) Use integration by parts to evaluate the integral. 4. (1 point) Find the area inside one leaf of the rose: Z 2x cos(2x)dx r = 5 sin(4) +C The area is Answer(s) submitted: Answer(s) submitted: (cos 2x)/(4) (25pi)/(16) (incorrect) (correct) 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 + 14 x 12. (1 point) Consider the series an where n=1 en+1 an = n + 6(n + 8)! In this problem you must attempt to use the Ratio Test to decide whether the series converges. Answer(s) submitted: (incorrect) Compute an+1 L = lim n 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= 5 9. (1 point) The region bounded by y = , y = 0, x = 4, x and x = 6 is rotated about the x-axis. Find the volume of the resulting solid. Volume = Answer(s) submitted: (incorrect) 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: 10. (1 point) Which of the following integrals represents the length of the curve y = 2x , 0 x 3? Z 3q A. 1 + 2(ln 2)2 2x dx Z03 q 1 + (ln 2)2 22 dx B. C. D. E. F. Z0 3 1 + 2x dx Z0 3 p 1 + 22x dx Answer(s) submitted: Z 03 q 1 + (ln 2)2 2x dx Z 03 q 1 + (ln 2)2 22x dx (incorrect) 0 13. (1 point) Use the binomial series to expand the function f (x) = (1 + x)1/3 as a power series. Answer(s) submitted: cn x n . (incorrect) n=0 11. (1 point) Find the values of x so that the series below converges. Compute the following coefficients: c0 = (x 3)n 3n n=0 c1 = Give your answer in interval notation. ( , ) c2 = Answer(s) submitted: c3 = c4 = (incorrect) 2 Answer(s) submitted: b= , c= , d= . Answer(s) submitted: (incorrect) 14. (1 point) Find the Maclaurin series of the function f (x) = (3x2 )e6x ( f (x) = cn xn ) n=0 (incorrect) c1 = 20. (1 point) Suppose that c2 = 8x = cn xn . (11 + x) n=0 Find the first few coefficients. c0 = c3 = c4 = c1 = c5 = c2 = Answer(s) submitted: c3 = c4 = Find the radius of convergence R of the power series. R= . (incorrect) Answer(s) submitted: 15. (1 point) Find the area inside one leaf of the rose: r = 6 sin(6) The area is (incorrect) Answer(s) submitted: 21. (1 point) The Taylor series for f (x) = cos(x) at a = cn (x 4 )n . n=0 Find the first few coefficients. c0 = (incorrect) 16. (1 point) Find the slope of the tangent line to the polar curve r = sin(3) at = 6 . slope = Answer(s) submitted: c1 = (incorrect) c2 = 19. (1 point) Suppose parametric equations for the line segment between (8, 4) and (1, 1) have the form: c3 = x = a + bt y = c + dt c4 = 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= , Answer(s) submitted: 3 4 is 24. (1 point) Evaluate the indefinite integral. Z sin3 (7x) cos10 (7x)dx (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. R 1. x sin x dx +C Answer(s) submitted: (incorrect) 25. (1 point) The form of the partial fraction decomposition of a rational function is given below. A. substitution B. integration by parts C. neither 2. R x4 1+x5 dx \u0001 5x2 + 6x + 18 A Bx +C = + 2 2 (x + 2)(x + 9) x+2 x +9 A. neither B. integration by parts C. substitution A= R 5 3. x4 ex dx Answer(s) submitted: R 4 x cos(x5 ) dx A. substitution B. integration by parts C. neither 5. R 1 3x+1 (incorrect) dx 26. (1 point) Consider the area between the graphs x + 1y = 24 and x + 6 = y2 . This area can be computed in two different ways using integrals. First of all it can be computed as a sum of two integrals A. substitution B. neither C. integration by parts Z b (Note that because this is multiple choice, you will not be able to see which parts of the problem you got correct.) Z c f (x) dx + a Answer(s) submitted: g(x) dx b where a = ,b= ,c= and f (x) = g(x) = Alternatively this area can be computed as a single integral Z h(y) dy (incorrect) where = ,= h(y) = Either way we find that the area is 23. (1 point) Suppose a particle moves along a straight line with velocity v(t) = t 2 e3t meters per second after t seconds. It travels onds. C= Now evaluate the indefinite integral. \u0001 Z 5x2 + 6x + 18 dx = (x + 2)(x2 + 9) A. neither B. integration by parts C. substitution 4. B= Answer(s) submitted: meters during the first t sec- Answer(s) submitted: (incorrect) 4 and . 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. (incorrect) 30. (1 point) Use the ratio test to determine whether n+5 n! converges or diverges. n=23 y = x2 , y = 0, x = 1, about the y-axis (a) Find the ratio of successive terms. Write your answer as a fully simplified fraction. For n 23, an+1 = lim lim n an n Answer(s) submitted: (incorrect) (b) Evaluate the limit in the previous part. Enter as infinity and as\f-infinity. If the limit does not exist, enter DNE. an+1 = lim n an 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? Z 5 2 t +4 A. dt 1 2t t + 1 Z 5 1 B. dt 1 2t t 2 + 1 Z 5 t2 + 4 dt C. 1 4t 2 t 2 + 1 Z 5 t +2 D. dt 1 t t +1 Z 5 t +2 dt E. 1 2t t + 1 Z 5 t +2 F. dt 1 2t 4t + 4 (c) By the ratio test, does the series converge, diverge, or is the test inconclusive? Choose Converges Diverges Inconclusive Answer(s) submitted: (incorrect) c Generated by WeBWorK, http://webwork.maa.org, Mathematical Association of America 5