Use the first and second derivatives to explain the properties of the graph of $\cosh x$.

Find all extrema and inflection points. $x e^{-2 x}$

Evaluate the integral. $\int_0^1 \frac{x^2}{x^3+1} d x$

Graph the indicated function. $e^{-2 /\left(x^3-x\right)}$

Use the first and second derivatives to explain the properties of the graph of $\tanh x$..

Evaluate the integral. $\int 2^{-5 x} d x$

In this exercise, we give a different derivation from the text for the derivative of $\sin ^{-1} x$. Use Theorem $2.3$ of to show that $\frac{d}{d x} \sin ^{-1} x=\frac{1}{\cos \left(\sin ^{-1} x\right)}$. Then, evaluate $\cos \left(\sin ^{-1} x\right)$

Find all solutions. $\sin (x / 4)=-\frac{1}{2}$

Use a graph to determine whether the function is one-to-one. If it is, graph the inverse function. $f(x)=\frac{4}{x^{2}+1}$

State whether the statement is true or false and briefly explain why. If the statement is false, try to "fix it" by modifying the given statement to a new statement that is true. If $f$ has a derivative that exists for all $x$, then $f$ has an inverse.

Find the derivative of the given function. $\frac{e^{4 x}}{x}$

Radar works by bouncing a high-frequency electromagnetic pulse off of a moving object, then measuring the disturbance in the pulse as it is bounced back. Explain why this is an inverse problem by identifying the input and output.

Use the integral definition of ln x (interpreted as area) to explain why it is reasonable that $\lim _{x \rightarrow 0^{+}} \ln x=-\infty$ and $\lim _{x \rightarrow \infty} \ln x=\infty$.

Briefly describe the graphs of sinh x, cosh x and tanh x. Which simple polynomials do the graphs of sinh x and cosh x resemble?