MTelabdiscussion14

In Exercises 25 – 30, use the Taylor series given in Key Idea 8.8.1 to create the Taylor series of the given funct�ons. \[f(x) = \sum_{n=0}^\infty\frac{f^{n}(x)}{n!}(x-c)^n \]

\[f(x) = e^x sin x \\ f'(x) = e^xsinx + e^xcosx \\ f''(x) = e^xsinx + e^xcosx + (e^xcosx - e^xsinx) = 2e^xcosx\\ f'''(x) = 2e^xcosx + (-e^xsinx + e^xcosx) + (-e^xcosx - e^xsinx) = 2e^xcosx - 2e^xsinx \\ f''''(x) = 2e^xcosx - 2e^xsinx + (-2e^xsinx + -2e^xcosx) = -4sinx \\ p_4(x)= \frac{e^xsinx + e^xcosx(x-c)}1 +\frac{2e^xcosx(x-c)}2 +\frac{(2e^xcosx - 2e^xsinx)(x-c)}6 +\frac{-4e^xsinx(x-c)}{24}\]