91Ó°ÊÓ

Find any relative extrema of the function. Use a graphing utility to confirm your result. $$ f(x)=x \sinh (x-1)-\cosh (x-1) $$

Find the integral. $$ \int \frac{2}{x \sqrt{1+4 x^{2}}} d x $$

Show that \(f\) is strictly monotonic on the given interval and therefore has an inverse function on that interval. \(f(x)=\cot x, \quad(0, \pi)\)

Find the derivative of the function. $$ y=\sinh ^{-1}(\tan x) $$

Find the derivative of the function. $$ y=x \tanh ^{-1} x+\ln \sqrt{1-x^{2}} $$

Find the limit. $$ \lim _{x \rightarrow \infty} \sinh x $$

Find the limit. $$ \lim _{x \rightarrow 0^{-}} \operatorname{coth} x $$

A differential equation, a point, and a slope field are given. (a) Sketch two approximate solutions of the differential equation on the slope field, one of which passes through the given point. (b) Use integration to find the particular solution of the differential equation and use a graphing utility to graph the solution. Compare the result with the sketches in part (a). To print an enlarged copy of the graph, select the MathGraph button. $$ \frac{d y}{d x}=e^{\sin x} \cos x, \quad(\pi, 2) $$

Find any relative extrema of the function. \(h(x)=\arcsin x-2 \arctan x\)

The table of values below was obtained by evaluating a function. Determine which of the statements may be true and which must be false, and explain why. (a) \(y\) is an exponential function of \(x\). (b) \(y\) is a logarithmic function of \(x\). (c) \(x\) is an exponential function of \(y\). (d) \(y\) is a linear function of \(x\). $$ \begin{array}{|l|l|l|l|} \hline x & 1 & 2 & 8 \\ \hline y & 0 & 1 & 3 \\ \hline \end{array} $$