Math 121 Final Exam May 12, 2014

Math 121 Final Exam May 12, 2014 • This is a closed-book examination. No books, notes, calculators, cell phones, communication devices of any sort, or other aids are permitted. • You However, numerical answers such as  π need3 not simplify algebraically complicated answers. √ ln 4 7 − ln 5 3 ln 3 , e , arctan( 3), or cosh(ln 3) should be simplified. , 4 2 , e , ln(e ), e sin 6 • Please show all of your work and justify all of your answers. (You may use the backs of pages for additional work space.) 1. [15 Points] Evaluate each of the following limits. Please justify your answers. Be clear if the limit equals a value, +∞ or −∞, or Does Not Exist. "r  #x 1 cosh(4x) − 1 − arctan(4x) + 4x 1 (b) lim + sinh (a) lim 1− x→∞ x→0 ln(1 − x) + arcsin x x x 2. [20 Points] Evaluate each of the following integrals. Z 4 Z 3√3 x + 5x2 − x + 3 1 √ (a) dx (b) dx 3 x + 3x 36 − x2 3 Z 0 p (c) x3 1 − x2 dx using a trigonometric substitution −1 3. [30 Points] For each of the following improper integrals, determine whether it converges or diverges. If it converges, find its value. Z e5 Z ∞ 1 1 dx (b) dx (a) 2 x − 8x + 19 x[25 + (ln x)2 ] 0 7 Z ∞ Z 1 (c) cosh x dx (d) ln x dx 0 −∞ 4. (a) [15 Points] ∞ X n=1 (d) (−1)n 3n+2 24n−1 ∞ X (−1)n n=0 Find the sum of each of the following series (which do converge): n+1 =1− (b) ∞ X (−1)n (ln 8)n 3n+1 n! (c) n=0 ∞ X (−1)n π 2n+1 n=0 1 1 1 1 1 + − + − + ... 2 3 4 5 6 (e) 9n (2n + 1)! ∞ X (−1)n π 2n+1 n=0 42n+1 (2n)! 5. [35 Points] In each case determine whether the given series is absolutely convergent, conditionally convergent, or divergent. Justify your answers. (a) (d) ∞ X (−1)n √ n+ n n=1 ∞ X 7 arctan(7n) + 7 n 7 n +1 n=1 (b) ∞ X ln n n2 (c) n=1 (e) ∞ X ∞ X n=1 (−1)n n=1 1 n3 n7 + 5 (f) n+3 ln(n + 3) ∞ X (−1)n π n (2n)! nn (4)n n! n=1 6. [15 Points] Find the Interval and Radius of Convergence for the following power series ∞ X (−1)n (3x + 2)n . Analyze carefully and with full justification. (n + 1) 4n n=0 7. [10 Points] (a) Write the first 6 non-zero terms of the MacLaurin Series for f (x) = sin(x3 ) + cos(x3 ). (b) Use this series to determine the sixth, seventh, eighth and ninth derivatives of f (x) = sin(x3 ) + cos(x3 ) evaluated at x = 0. (Hint: Do not compute out those derivatives manually.) (Hint: Write out the definition of the MacLaurin Series for any f (x).) 8. [15 Points] Please analyze with detail and justify carefully.
(a) Write the MacLaurin series representation for f (x) = x arctan x2 . Your answer should ∞ X . be in sigma notation n=0
(b) Use the MacLaurin series representation for f (x) = x arctan x2 from Part(a) to Z 1
1 x arctan x2 dx with error less than Estimate . Justify in words that your error is 50 0 1 . indeed less than 50 9. [15 Points] (a) Consider the region bounded by y = ex + 2, y = sin x, x = 0 and x = π. Rotate the region about the vertical line x = −2 . Set-Up but DO NOT EVALUATE the integral representing the volume of the resulting solid using the Cylindrical Shells Method. Sketch the solid, along with one of the approximating cylindrical shells. (b) Consider the region bounded by y = ln x, y = 1, and x = 4. Rotate the region about the vertical line x = 5 . Set-Up but DO NOT EVALUATE the integral representing the volume of the resulting solid using the Cylindrical Shells Method. Sketch the solid, along with one of the approximating cylindrical shells. (c) Consider the region bounded by y = arctan x, y = 0, x = 0 and x = 1. Rotate the region about the y-axis . COMPUTE the volume of the resulting solid using the Cylindrical Shells Method. Sketch the solid, along with one of the approximating cylindrical shells. 10. t [15 Points] Consider the Parametric Curve represented by x = t − et and y = 1 − 4e 2 . (a) COMPUTE the arclength of this parametric curve for 0 ≤ t ≤ ln 5. (b) Set-Up but DO NOT EVALUATE the surface area obtained by rotating this same curve about the y-axis, for 0 ≤ t ≤ 1. 11. [15 Points] Compute the area bounded outside the polar curve r = 2 + 2 sin θ and inside the polar curve r = 6 sin θ. Sketch the Polar curves. 2