Compute the Taylor polynomials $T_n$, $n=0,\dots,5$ of $f(x)=\sin x+\cos x$ at $x_0=0$.
Solution: By Definition 5.3, we have to compute $f(x_0)$, $f’(x_0)$, $f^{(2)}(x_0)$, $f^{(3)}(x_0)$, $f^{(4)}(x_0)$, $f^{(5)}(x_0)$. It is not hard to see that $$f’(x)=\cos x -\sin x,$$ $$f^{(2)}(x)=-\sin x-\cos x,$$ $$f^{(3)}(x)=-\cos x + \sin x,$$ $$f^{(4)}(x)=\sin x+\cos x,$$ $$f^{(5)}(x)=\cos x-\sin x.$$ Hence we get $$f(0)=f’(0)=f^{(4)}(0)=f^{(5)}(0)=1,$$ $$f^{(2)}(0)=f^{(3)}(0)=-1.$$Now it is easy to see that $$T_0=1,$$ $$T_1=x+1,$$ $$T_2=-\frac{1}{2}x^2+x+1,$$ $$T_3=-\frac{1}{6}x^3-\frac{1}{2}x^2+x+1$$ $$T_4=\frac{1}{24}x^4-\frac{1}{6}x^3-\frac{1}{2}x^2+x+1,$$ $$T_5=\frac{1}{120}x^5+\frac{1}{24}x^4-\frac{1}{6}x^3-\frac{1}{2}x^2+x+1.$$