Intermediate Calculus Final Exam

Math 121 Final Exam December 20, 2015 • 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 sin , e , arctan( 3), or cosh(ln 3) should be simplified. , 4 2 , e , ln(e ), e 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.   x 3 − ex ln(1 − x) + arctan x 6 (a) lim (b) lim (c) lim 1 − arcsin x − sinh x 1 x→∞ x→0 x→ln 3 xe x e−2x − 9 2. [30 Points] Evaluate each of the following integrals. Z 3 Z 1 x5 √ √ dx dx (using a trigonometric substitution) (b) (a) x (x + 3) 4 − x2 1 Z e√5 Z 1 (c) dx (d) x arcsin x dx 3 e x(4 + (ln x)2 ) 2 3. [24 Points] For each of the following improper integrals, determine whether it converges or diverges. If it converges, find its value. Z 2 Z ∞ 4 1 (a) dx (b) dx 2 2 1 x − 8x + 12 −∞ x − 8x + 19 Z 1 Z 1 1 ln x √ dx = (c) x− 2 ln x dx x 0 0 4. (a) [18 Points] ∞ X n=1 Find the sum of each of the following series (which do converge): (−1)n 42n+1 33n−1 ∞ X (−1)n+1 2n+1 (ln 6)n (b) n! (c) n=0 ∞ X (−1)n π 2n n=0 24n (2n)! ∞ ∞ X X 1 1 1 1 1 1 1 1 1 (−1)n π 2n (d) − + − + − . . . (e) 1 − + − + − . . . (f) (g) 5 2 · 52 3 · 53 4 · 54 3 5 7 9 en (36)n (2n + 1)! n=0 n=0 5. [35 Points] In each case determine whether the given series is absolutely convergent, conditionally convergent, or divergent. Justify your answers.   ∞ ∞ ∞ X X X (−1)n (n4 + 7) (−1)n arctan(7n) 1 (a) (b) (c) n · arctan 7 n n +4 e +7 n n=1 n=1 n=1 √ ∞ ∞ X X (−1)n n (−1)n+1 e3n (3n)! (d) (e) n+3 nn 42n (n!)2 n=1 n=1 1 6. [15 Points] Find the Interval and Radius of Convergence for the power series ∞ X (−1)n (ln n) (4x − 1)n . n2 · 5n Analyze carefully and with full justification. n=1 7. [8 Points] (a) Write the MacLaurin Series for the hyperbolic cosine f (x) = cosh x. (b) Write the MacLaurin Series for f (x) = cosh(2x3 ). (c) Use this series to determine the twelfth, and thirteenth, derivatives of f (x) = cosh(2x3 ) evaluated at x = 0. That is, compute f (12) (0) and f 13 (0). Do not simplify your answers here. 8. [12 Points] Please analyze with detail and justify carefully. Simplify your answers. Z 1

2 (a) Use the MacLaurin series representation for f (x) = x sin x to Estimate x sin x2 dx 1 1 0 with error less than . Justify in words that your error is less than . 100 100   1 1 with error less than . Justify in words that your error is indeed less (b) Estimate cos 2 100 1 than . 100 9. [10 Points] π . 2 Rotate the region about the vertical line x = 3 . COMPUTE the volume of the resulting solid using the Cylindrical Shells Method. Sketch the solid, along with one of the approximating cylindrical shells. 10. Consider the region bounded by y = cos x, y = x + 1, x = 0 and x = [18 Points] (a) Consider the Parametric Curve represented by x = t + 1 and y = 2 ln(1 + t). 1+t COMPUTE the arclength of this parametric curve for 0 ≤ t ≤ 4. √ (b) Consider a different Parametric Curve represented by x = t − e2t and y = 1 − 8 et . COMPUTE the surface area obtained by rotating this curve about the y-axis , for 0 ≤ t ≤ 3. 11. [15 Points] Compute the area bounded outside the polar curve r = 1 + sin θ and inside the polar curve r = 3 sin θ. Sketch the Polar curves and shade the bounded area. 2