91Ó°ÊÓ

Evaluate each expression without using a calculator, or state that the value does not exist. Simplify answers to the extent possible. a. \(\mathrm{cosh 0}\) b. \(\mathrm{tanh 0}\) c. \(\mathrm{csch 0}\) d. \(\mathrm{sech}(sinh 0)\) e. \(\operatorname{coth}(\ln 5) \quad\) f. \(\sinh (2 \ln 3)\) g. \(\cosh ^{2} 1 \quad\) h. \(\operatorname{sech}^{-1}(\ln 3)\) i. \(\cosh ^{-1}(17 / 8)\) j. \(\sinh ^{-1}\left(\frac{e^{2}-1}{2 e}\right)\)

The harmonic sum is \(1+\frac{1}{2}+\frac{1}{3}+\dots+\frac{1}{n} .\) Use a right Riemann sum to approximate \(\int_{1}^{n} \frac{d x}{x}(\) with unit spacing between the grid points) to show that \(1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}>\ln (n+1)\) Use this fact to conclude that \(\lim _{n \rightarrow \infty}\left(1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}\right)\) does not exist.

Find the critical points of the function \(f(x)=\sinh ^{2} x \cosh x\).

a. Show that the critical points of \(f(x)=\frac{\cosh x}{x}\) satisfy \(x=\operatorname{coth} x\). b. Use a root finder to approximate the critical points of \(f\).

Find the \(x\) -coordinate of the point(s) of inflection of \(f(x)=\tanh ^{2} x\).

Find the \(x\) -coordinate of the point(s) of inflection of \(f(x)=\operatorname{sech} x .\) Report exact answers in terms of logarithms (use Theorem 10).

Find the area of the region bounded by \(y=\operatorname{sech} x, x=1,\) and the unit circle.

Explain why l'Hôpital's Rule fails when applied to the limit \(\lim _{x \rightarrow \infty} \frac{\sinh x}{\cosh x}\), and then find the limit another way.

Use l'Hôpital's Rule to evaluate the following limits. \(\lim _{x \rightarrow \infty} \frac{1-\operatorname{coth} x}{1-\tanh x}\)

Use l'Hôpital's Rule to evaluate the following limits. \(\lim _{x \rightarrow 0} \frac{\tanh ^{-1} x}{\tan (\pi x / 2)}\)