Math 121, Section 1 Final Exam

Math 121, section 1 Final Exam 1. [10 points] Evaluate each limit. Justify your answer. x3 sin x − arctan x (a) lim x→0 (b) lim (1 + ln(2x))tan(πx) x→ 12 2. [15 points] Evaluate each integral. Z π/2 cos4 t dt (a) 0 √ Z 2 1 dx + 2)5/2 0 Z arctan x (c) dx x2 (b) (x2 3. [15 points] For each improper integral, evaluate it if possible, or determine that it diverges. Z ∞ 1 √ √ dx (a) x x + x 1 Z ∞ x (b) dx 2 x + 3x + 2 0 Z ∞ 1 dx (c) x ((ln x)2 + ln x) e 4. [15 points] Evaluate each of the following sums (you may assume that each sum converges). ∞ X x2n (express your answer in terms of x). (a) n! n=0 4 (b) 4 − 43 + 45 − 47 + 94 − 11 + ··· 1 1 1 1 (c) − 2 + 3 − 4 + ··· 2 2 ·2 2 ·3 2 ·4 ∞ X 1 + 2n (d) n! (e) n=1 ∞ X n=1 (−1)n−1 n x , assuming that x > 0 (express your answer in terms of x). (2n + 1)! 5. [20 points] For each series, determine whether it is absolutely convergent, conditionally convergent, or divergent. Justify your answers. √ ∞ X n−1 n + n (a) (−1) n3 + 1 n=1 ∞ √ 2 X n +1 (b) n (c) n=1 ∞ X (−1)n−1 n=1 n! (2n)n Math 121, section 1 (d) (e) Final Exam ∞ X n2 + 100n n=0 ∞ X n=0 3n arctan(n) n2 6. [7 points] Find the interval of convergence for the following power series. Analyze carefully with full justification. ∞ X 1 xn 2 n +n+1 n=0 7. [4 points] Suppose that a patient takes a single dose of a certain medication every day for a long period of time. Each dose is 100mg of medication, which is slowly removed from the patient’s bloodstream according to the following rule: t days after the patient takes
the dose, t the number of milligrams left from that dose in the patient’s bloodstream is 100 · 45 . If the patient continues to take this medication once a day for a very long period of time, how many milligrams of medication will be in her bloodstream immediately after taking a dose? 8. [8 points] Consider the region bounded by the curves y = ex , y = e, and the y-axis. This region is revolved around the line x = −1 to obtain a solid. Compute the volume of this solid. 9. [10 points] Consider the paramteric curve given by the following equations. x(t) = 4 sinh(t) y(t) = 8 cosh(t) (a) Determine the tangent line to this curve when t = ln 2. (b) Set up but do not evaluate an integral computing the arc length of this curve between t = 0 and t = ln 2. (c) Set up but do not evaluate an integral for the surface area obtained by revolving this curve around the y-axis. 10. [12 points] For each region described, set up but do not evaluate an integral (or sum of integrals) for its area. (a) The area bounded inside the polar curve r = 2 − 2 cos θ. (b) The area outside the polar curve r = 1 and inside the polar curve r = 1 − sin θ. (c) The area inside both of the polar curves r = 1 and r = 2 − 2 cos θ. √ (d) The image below depicts the polar curve r = 1 − 2 sin θ. Set up but do not evaluate an integral computing the area of the shaded region (enclosed by the “inner loop” of the curve). Math 121, section 1 Final Exam