Math 150B Exam 3 Solution

Exam (3) Math 150B. Calculus II Section # 18056 Fall 2022 Instructor: Henrik Sarkssian TUESDAY NOVEMBER 15, 12:30 PM - 13:35 PM Name: Answer key (Please print) - Show all of your work and make the final answers clear. Full credit will NOT be given for merely stating the right answer without any appropriate justification. - Cell phones must not be visible or audible at any time during the exam. This applies to all smart technology including watches, cameras, and pens. No Calculators! Your grade will be posted on Canvas once you submit a copy of the graded test there (no redoes, no changes). If you fail to submit, you will get a zero. Due Date to submit this test on canvas: Problem 1 2 3 4 5 Total Maximum [30] [30] [30] [30] [30] 150 Score 1

Problem 1. (30 pts.) Find the Taylor series for f(x) at X = a, up to the first four nonzero terms. f(x)=3-x at a=0. G- y 2 f(x)53 -X -X fix)613.3 3.3 -X fixs-(133 flo)s I f/10) -ln3 I fiolstin's I fio)==(n3 4.5 4.5 2 (4.5) MCX) 1 X- - in3 -X + m3.2 3! 63 o! 3 Mixis \ -((n3)x+ m3322 2 + in'3 6 5.5 0.5 orJ 0.5 2

Problem 2. (30 pts.) Find the radius of convergence and interval of convergence of following series. 2"(x-3)" n=1 2ne'(x-3)"41 5 eim/ now and an nixo I 2"(x-3)" Frei s lim VI 2nel 1x-31 200 Tree 2n in h lim / 17 . 2 1x-31 in 21x-31

Problem 3. (30 pts.) Find f(x) by evaluating the series. State the interval of convergence. 00 (i) s 5 1+ 'ex e+1 l XYO I=(0,00) 2 3 (ii) 5 N (x(x2))" E(x) no 3" no 5 1 3 = = 1- x2.2x 3 3-x2-2x 5 1 XCX2/1 3 , 3 -3

Problem 4. (30 pts.) Find the equation of the tangent line to ellipse at the 2 point where x = 1+ 2cost, y=1+sint 0

Problem 5. (30 pts.) Find the area that is inside the cardioid r=1+cos0 and the circle r = 3cos0 above the polar axis. 1+Cost 3CosO 1526000 + cos 9 Cost= 0= 2 . cos 6 4 I I 1+2COSV + 2/2 losodv 022 0 Costo f 2 I & 6