Application of Taylor's Theorem

Example 1. Consider the function \(f(x)=\sin(x)\). We want to determine the Taylor polynomial of degree 6 with expansion point \(x_0=\frac\pi2\). Since we have \[\sin'(x)=\cos(x),\quad\sin''(x)=-\sin(x),\quad\sin^{(3)}(x)=-\cos(x),\] \[\sin^{(4)}(x)=\sin(x),\quad\sin^{(5)}(x)=\cos(x),\quad\sin^{(6)}(x)=-\sin(x)\] and \[\sin(x_0)=\sin\left(\frac\pi2\right)=1,\quad \cos(x_0)=\cos\left(\frac\pi2\right)=0,\] the Taylor polynomial of degree 6 is given by \[T_6(x)=1-\frac12\left(x-\frac{\pi}{2}\right)^2+\frac{1}{24}\left(x-\frac{\pi}{2}\right)^4-\frac1{720}\left(x-\frac{\pi}{2}\right)^6.\] The remainder term reads \[R_6(x,x_0)=\frac{\sin^{(7)}(\hat{x})}{7!}\left(x-\frac{\pi}{2}\right)^{7} =\frac{-\cos(\hat{x})}{5040}\left(x-\frac{\pi}{2}\right)^{7}.\] Taking into account that \(|\cos(x)|\leq1\) for all \(x\in\mathbb{R}\), we have that \[|R_6(x,x_0)|\leq\frac{\left|x-\frac{\pi}{2}\right|^{7}}{5040}.\] This leads to the estimate \[|\sin(x)-T_6(x)|=|R_6(x,x_0)|\leq\frac{\left|x-\frac{\pi}{2}\right|^{7}}{5040}.\]

Example 2. We want to compute \(\log(1.2)\) up to 3 digit precision. A nice expansion point is \(x_0=1\) since we know the precise values of \(\log^{(k)}(1)\). Consider \[\log'(x)=\frac1x,\quad \log''(x)=-\frac{1!}{x^2},\] \[\log^{(3)}(x)=\frac{2!}{x^3},\quad\log^{(4)}(x)=-\frac{3!}{x^4}\] and therefore, the Taylor polynomial of degree 3 is given by \[T_3(x)=(x-1)-\frac12(x-1)^2+\frac13(x-1)^3.\] In particular, we have \[T_3(1.2)=0.2-\frac12\cdot 0.2^2+\frac13 \cdot 0.2^3=\frac{137}{750}.\] Now we estimate \(|\log(1.2)-\frac{137}{750}|\): The remainder term is given by \[R_3(x,x_0)=-\frac{3!}{\hat{x}^4}\frac{(x-x_0)^{4}}{4!}=-\frac{(x-x_0)^{4}}{4\hat{x}^4}\] for some \(\hat{x}\) between \(x\) and \(x_0\). For \(x=1.2\), \(x_0=1\) we have \(1<\hat{x}<1.2\) and therefore \[| R_3(1.2,1) | =\frac{(0.2)^{4}}{4\hat{x}^4}=4\cdot10^{-4}\frac{1}{\hat{x}^4}\leq4\cdot10^{-4}.\] This leads to \[|\log(1.2)-0.182\overline{6}|=|R_3(1.2,1)|\leq4\cdot10^{-4},\] so we have determined \(\log(1.2)\) up to three digits.