Math 121 Final Exam 2016

Math 121 Final Exam May 9, 2017 • 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. [12 Points] Evaluate the following limit. Please justify your answer. Be clear if the limit equals a value, +∞ or −∞, or Does Not Exist. Simplify.   x 5xex − arctan(5x) 5 (a) lim 1 − arcsin (b) Compute lim x→∞ x→0 sinh x + ln(1 − x) x 2. [16 Points] Evaluate each of the following integrals. Z Z 0 cos x (b) (a) x arcsin x dx
7 dx −1 1 + sin2 x 2 3. [40 Points] For each of the following improper integrals, determine whether it converges or diverges. If it converges, find its value. Simplify. Z e3 (a) 0 Z (c) 1 4. 2 (b) 0 Z 2 dx 2 x − 6x + 8 [20 Points] ln x √ dx x ∞ (d) 5 x2 1 dx − 6x + 13 Find the sum of each of the following series (which do converge). Simplify. ∞ X (−1)n 32n−1 (a) 42n+1 n=1 (d) 1 − e Z 1 dx x [9 + (ln x)2 ] 1 1 1 1 + − + −... 3 5 7 9 ∞ X (−1)n (ln 8)n (b) 3n+1 n! (c) n=0 (e) − π2 2! + π4 4! − ∞ X n=0 π6 6! + π8 8! −... (−1)n π 2n (36)n (2n + 1)! (f) −1 + 1 1 1 − + −... 2 3 4 5. [30 Points] In each case determine whether the given series is absolutely convergent, conditionally convergent, or divergent. Justify your answers. ∞ X (−1)n (n3 + 7) (a) n7 + 3 n=1 (c) ∞ X (−1)n arctan(n2 ) n2 + 1 n=1 ∞ X n+1 n2 n=1   ∞ X n2 (d) arctan n2 + 1 (b) (−1)n n=1 1 (e) ∞ X (−1)n (3n)! ln n (n!)2 24n nn n=1 6. [12 Points] Find the Interval and Radius of Convergence for the following power series ∞ X (−1)n (3x + 1)n . Analyze carefully and with full justification. (n + 7) · 7n n=1 Z 1
1 7. [10 Points] (a) Use MacLaurin series to Estimate x sin x2 dx with error less than . 1000 0 Please analyze with detail and justify carefully. Simplify. 1 1 1 . Justify in words that your error is indeed less than . (b) Estimate √ with error less than 100 100 e 8. [8 Points] For each of the following functions, find the MacLaurin Series and, then State the Radius of Convergence.   1 1 (a) f (x) = sinh x (b) f (x) = . Hint: Differentiate (1 − x)2 1−x 9. [18 Points] π , and x = 0. Rotate the region about 2 the vertical line x = 3 . Set-up, BUT DO NOT EVALUATE!!, the integral to compute the volume of the resulting solid using the Cylindrical Shells Method. Sketch the solid, along with one of the approximating shells. (a) Consider the region bounded by y = arcsin x, y = (b) Consider the region bounded by y = arctan x, y = 0, x = 0 and x = 1. Rotate the region about the vertical line y-axis . COMPUTE the volume of the resulting solid using the Cylindrical Shells Method. Sketch the solid, along with one of the approximating shells. 10. [14 Points] √ x = ln t + ln(1 − t2 ) and y = 8 arcsin t. 1 1 ≤ t ≤ . Show that the answer COMPUTE the arclength of this parametric curve for 4 2   5 simplifies to ln 2 Consider the Parametric Curve represented by 11. [18 Points] For each of the following problems, do the following two things: 1. Sketch the Polar curves and shade the described bounded region. 2. Set-Up but DO NOT EVALUATE the Integral representing the area of the described bounded region. (a) The area bounded outside the polar curve r = 1+cos θ and inside the polar curve r = 3 cos θ. (b) The area bounded outside the polar curve r = 2 and inside the polar curve r = 4 sin θ. (c) The area that lies inside both of the curves r = 1+sin θ and inside the polar curve r = 1−sin θ. 2