MATH-2212: Calculus of One Variable II Worksheet for sections 11.10, 11.11 . et =1+x+ 21 31 + ... 1, which converges for all r E R. x' x2n+1 . sinc = x - 3! 51 71 +...= (-1) (2n + 1)!, which converges for all r E R. n=0 . COST = 1 - 2! 4! 61 7 ... (-1)2 2n (2n)!' which converges for all r ER. n=0 Practice problems 1. Using the definition, find the second degree Maclaurin polynomial T2(x) for the given function. (a) f(x) = et cosx (b) f(x) = tan 2x + sec 3x (c) f(x) = Vx +4 2. Using the definition, find the third order Taylor polynomial T3(x) for the given function centered at the given point. (a) f(x) = sin 2x at a = -, i.c. in powers of ( x - " ). (b) f(x) = Inx at a = 2, i.c. in powers of (x - 2). (c) f(x) = 23/2 at a = 4, i.c. in powers of (x - 4). 3. Find the sum of the series. (Hint: you must find an exact numerical value. ) 22n+1 (a) > (-1)- (c) ) (-1)2 37 n-0 (2n + 1)! n= 0 n ! 00 (b) > (-1) "- (d) (-1) 72n 22n+1(2n + 1)! (f) > (-1)n+1 30 n-0 n-0 (2n)! n=1 n5n 4. Find the Maclaurin series representation of the given function. (Hint: in each problem, start with a known Maclaurin series of a basic elementary function.) (a) f(x) = e-2x (b) f(x) = In(1 + 5x) (c) f(x) = xcos(7x) 5. Using the Maclaurin series expansions found in the previous problem, evaluate the following indefinite integrals as power series. (a) e-2 dx (b) / In ( 1 + 5x) dr (c) / x cos (7x) da 2