Chapter 8

Use Taylor's theorem to show that $$ \exp (1)=e=\sum_{n=0}^{\infty} \frac{1}{n !} $$

Show that for any rational number \(\alpha>0\) $$ \lim _{x \rightarrow+\infty} x^{\alpha}=+\infty $$

Show that \(\arctan (1)=\frac{\pi}{4}\) and \(\arctan (-1)=-\frac{\pi}{4}\).

Define \(f:(0,+\infty) \rightarrow \mathbb{R}\) by \(f(x)=x^{a},\) where \(a \in \mathbb{R}, a \neq 0\) Show that \(f^{\prime}(x)=a x^{a-1}\).

Show that $$ \lim _{x \rightarrow 0^{+}} x^{\alpha} \log (x)=0 $$ for any rational number \(\alpha>0\).

Show that for any \(x \in \mathbb{R}\) $$ \sin \left(\frac{\pi}{2}-x\right)=\cos (x) $$ and $$ \cos \left(\frac{\pi}{2}-x\right)=\sin (x) $$

Suppose \(a\) is a positive real number and \(f: \mathbb{R} \rightarrow \mathbb{R}\) is defined by \(f(x)=a^{x}\). Show that \(f^{\prime}(x)=a^{x} \log (a)\).

Show that for any \(x \in \mathbb{R}\) $$ \sin (2 x)=2 \sin (x) \cos (x) $$ and $$ \cos (2 x)=\cos ^{2}(x)-\sin ^{2}(x) $$

Show that for any real numbers \(x\) and \(y\), $$ \sinh (x+y)=\sinh (x) \cosh (y)+\sinh (y) \cosh (x) $$ and $$ \cosh (x+y)=\cosh (x) \cosh (y)+\sinh (x) \sinh (y) $$

Show that for any \(x \in \mathbb{R}\) $$ \sin ^{2}(x)=\frac{1-\cos (2 x)}{2} $$ and $$ \cos ^{2}(x)=\frac{1+\cos (2 x)}{2} $$

Show that for any real number \(x,\) $$ \cosh ^{2}(x)-\sinh ^{2}(x)=1 $$

Show that $$ \begin{array}{l} \sin \left(\frac{\pi}{4}\right)=\cos \left(\frac{\pi}{4}\right)=\frac{1}{\sqrt{2}} \\ \sin \left(\frac{\pi}{6}\right)=\cos \left(\frac{\pi}{3}\right)=\frac{1}{2} \end{array} $$ and $$ \sin \left(\frac{\pi}{3}\right)=\cos \left(\frac{\pi}{6}\right)=\frac{\sqrt{3}}{2} $$

If \(f(x)=\sinh (x)\) and \(g(x)=\cosh (x),\) show that $$ f^{\prime}(x)=\cosh (x) $$ and $$ g^{\prime}(x)=\sinh (x) $$

If \(f(x)=\tan (x)\) and \(g(x)=\cot (x),\) show that $$ f^{\prime}(x)=\sec ^{2}(x) $$ and $$ g^{\prime}(x)=-\csc ^{2}(x) $$

If \(f(x)=\sec (x)\) and \(g(x)=\csc (x)\), show that $$ f^{\prime}(x)=\sec (x) \tan (x) $$ and $$ g^{\prime}(x)=-\csc (x) \cot (x) $$

Find the Taylor polynomial \(P_{9}\) of order 9 for \(f(x)=\sin (x)\) at 0. Note that this is equal to the Taylor polynomial of order 10 for \(f\) at \(0 .\) Is \(P_{9}\left(\frac{1}{2}\right)\) an overestimate or an underestimate for \(\sin \left(\frac{1}{2}\right) ?\) Find an upper bound for the error in this approximation.