Chapter 10: Infinite Sequences and Series - Section 10.8 - Taylor and Maclaurin Series - Exercises 10.8 - Page 619: 42

$L(x)=1+x$ and $Q(x)=1+x+\dfrac{x^2}{2!}$

Differentiate the given function $f(x)$ as follows: $f'(x)=-\cos x e^{\sin x} ; \\f''(x)=-(\sin x) e^{\sin x}+(\cos x)^2 e^{\sin x} $ Now, $ f(0)=1; \\ f'(0)=1;\\ f''(0)=1$ $L(x)=f(0)+xf'(0)=1+x; \\ Q(x) =f(0) x +xf'(0) +\dfrac{x^2 f''(0)}{2!}=1+x+\dfrac{x^2}{2!}$ Our result is: $L(x)=1+x \\ Q(x)=1+x+\dfrac{x^2}{2!}$