Question

Please show your procedures to achieve your answers Find the first three terms of the Maclaurin series of function e* sinx. (15 points) 1. 3

Answer 1

"Let f(x) = e^x sin x The Maclaurin series is given by f(x) = f(0) + f'(0)/1! + f""(0)x^2/2! + f""'(0)x^3/3! + ... + f^n(0)/n! + ... f(0) = e^0 sin 0 = 0 differentiating w.r.t. x using the product rule d/dx (e^x sin x) = e^x cos x + e^x sin x Therefore f'(x) = e^x cos x + e^x sin x ---(1) f'(0) = e^0 cos 0 + e^0 sin 0 = 1 differentiating eqn (1) again w.r.t x we get \r\n f""(x) = d/dx (e^x cos x) + d/dx (e^x sin x) f""(x) = -e^x sin x + e^x cos x + e^x cosx + e^x sin x f""(x) = 2e^x cos x ---(2) f""(0) = 2e^0 cos 0 = 2 differentiating eqn (2) again w.r.t. x we get f""'(x) = d/dx(2e^x cos x) = 2(-e^x sin x + e^x cos x) = 2e^x cos x - 2e^x sin x Therefore f""'(0) = 2e^0 cos 0 - 2e^0 sin 0 = 2 "