Josephood7

 

Match each of the Maclaurin series with the function it represents. D 1. ∑n=0∞xnn! A 2. ∑n=0∞(−1)nx2n+1(2n+1)! C 3. ∑n=0∞(−1)nx2n(2n)! B 4. ∑n=0∞(−1)nx2n+12n+1 A. sin(x) B. arctan(x) C. cos(x) D. ex Write the Taylor series for f(x)=ex about x=−1 as ∑n=0∞cn(x+1)n. Find the first five coefficients. Find the first three nonzero terms of the Taylor series for the function f(x)=2x−x2−−−−−−√ about the point a=1. (Your answers should include the variable x when appropriate.) 2x−x2−−−−−−√= Let f(x)=2+4xx. Compute f(x) = (2+4x)/x f(1) = 6 f′(x) = -2/x^2 f′(1) = -2 f′′(x) = 4/x^3 f′′(1) = 4 f′′′(x) = -12/x^4 f′′′(1) = -12 f(iv)(x) = 48/x^5 f(iv)(1) = 48 f(v)(x) = -240/x^6 f(v)(1) = -240 We see that the first term does not fit a pattern, but we also see that f(k)(1) = (-1)^k*2k! for k≥1. Hence we see that the Taylor series for f centered at 1 is given by f(x)=6+∑k=1∞ 2*(-1)^k (x−1)k.