Chapter 7

In Exercises \(13-24,\) find the derivative of \(y\) with respect to the appropriate variable. $$y=\ln (\cosh z)$$

Solve the differential equations in Exercises \(9-22\) $$\frac{d y}{d x}=\frac{e^{2 x-y}}{e^{x+y}}$$

Use l'Hopital's rule to find the limits in Exercises \(7-50\) . $$ \lim _{\theta \rightarrow-\pi / 3} \frac{3 \theta+\pi}{\sin (\theta+(\pi / 3))} $$

Find the derivative of \(y\) with respect to \(x, t,\) or \(\theta,\) as appropriate. \begin{equation}y=\theta^{3} e^{-2 \theta} \cos 5 \theta\end{equation}

In Exercises \(7-38,\) find the derivative of \(y\) with respect to \(x, t,\) or \(\theta,\) as appropriate. $$ y=t \sqrt{\ln t} $$

\begin{equation} \begin{array}{l}{\text { a. Graph the function } f(x)=1 / x . \text { What symmetry does the }} \\ {\text { graph have? }} \\ {\text { b. Show that } f \text { is its own inverse. }}\end{array} \end{equation}

Show that \(e^{x}\) grows faster as \(x \rightarrow \infty\) than \(x^{n}\) for any positive integer \(n,\) even \(x^{1,000,000} .\) (Hint: What is the \(n\) th derivative of \(x^{n} ? )\)

In Exercises \(13-24,\) find the derivative of \(y\) with respect to the appropriate variable. $$y=(\operatorname{sech} \theta)(1-\ln \operatorname{sech} \theta)$$

Find the derivative of \(y\) with respect to \(x, t,\) or \(\theta,\) as appropriate. \begin{equation}y=\ln \left(3 t e^{-t}\right)\end{equation}

In Exercises \(7-38,\) find the derivative of \(y\) with respect to \(x, t,\) or \(\theta,\) as appropriate. $$ y=\frac{x^{4}}{4} \ln x-\frac{x^{4}}{16} $$

Find the limits in Exercises \(13-20 .\) (If in doubt, look at the function's graph.) $$ \lim _{x \rightarrow-\infty} \csc ^{-1} x $$

The function \(e^{x}\) outgrows any polynomial Show that \(e^{x}\) grows faster as \(x \rightarrow \infty\) than any polynomial $$ a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0} $$

In Exercises \(13-24,\) find the derivative of \(y\) with respect to the appropriate variable. $$y=(\operatorname{csch} \theta)(1-\ln \operatorname{csch} \theta)$$

Use l'Hopital's rule to find the limits in Exercises \(7-50\) . $$ \lim _{x \rightarrow 1} \frac{x-1}{\ln x-\sin \pi x} $$

Find the derivative of \(y\) with respect to \(x, t,\) or \(\theta,\) as appropriate. \begin{equation}y=\ln \left(2 e^{-t} \sin t\right)\end{equation}

In Exercises \(7-38,\) find the derivative of \(y\) with respect to \(x, t,\) or \(\theta,\) as appropriate. $$ y=\left(x^{2} \ln x\right)^{4} $$

Each of Exercises \(19-24\) gives a formula for a function \(y=f(x)\) and shows the graphs of \(f\) and \(f^{-1} .\) Find a formula for \(f^{-1}\) in each case. $$f(x)=x^{2}, \quad x \leq 0$$

In Exercises \(21-42,\) find the derivative of \(y\) with respect to the appropriate variable. $$ y=\cos ^{-1}\left(x^{2}\right) $$

$$ \begin{array}{l}{\text { a. Show that ln } x \text { grows slower as } x \rightarrow \infty \text { than } x^{1 / n} \text { for any positive }} \\\ {\text { integer } n, \text { even } x^{1 / 1,000,000} .}\\\\{\text { b. Although the values of } x^{1 / 1,000,000} \text { eventually overtake the }} \\\ {\text { values of ln } x, \text { you have to go way out on the } x \text { -axis before }} \\ {\text { this happens. Find a value of } x \text { greater than } 1 \text { for which }} \\ {x^{1 / 1,000,000}>\ln x . \text { You might start by observing that when }}\\\\{x>1 \text { the equation } \ln x=x^{1 / 1,000,000} \text { is equivalent to the }} \\ {\text { equation } \ln (\ln x)=(\ln x) / 1,000,000}\\\\{\text { c. Even } x^{1 / 10} \text { takes a long time to overtake } \ln x . \text { Experiment }} \\ {\text { with a calculator to find the value of } x \text { at which }} \\ {\text { the graphs of } x^{1 / 10} \text { and } \ln x \text { cross, or, equivalently, at }}\\\\{\text { which } \ln x=10 \ln (\ln x) . \text { Bracket the crossing point }} \\ {\text { between powers of } 10 \text { and then close in by successive }} \\ {\text { halving. }}\\\\{\text { d. (Continuation of part }(c) . ) \text { The value of } x \text { at which }} \\\ {\quad \text { ln } x=10 \ln (\ln x) \text { is too far out for some graphers and root }} \\ {\text { finders to identify. Try it on the equipment available to you }} \\ {\text { and see what happens. }}\end{array} $$

In Exercises \(13-24,\) find the derivative of \(y\) with respect to the appropriate variable. $$y=\ln \cosh v-\frac{1}{2} \tanh ^{2} v$$

Solve the differential equations in Exercises \(9-22\) $$\frac{1}{x} \frac{d y}{d x}=y e^{x^{2}}+2 \sqrt{y} e^{x^{2}}$$

Use l'Hopital's rule to find the limits in Exercises \(7-50\) . $$ \lim _{x \rightarrow 0} \frac{x^{2}}{\ln (\sec x)} $$

Find the derivative of \(y\) with respect to \(x, t,\) or \(\theta,\) as appropriate. \begin{equation}y=\ln \left(\frac{e^{\theta}}{1+e^{\theta}}\right)\end{equation}

In Exercises \(7-38,\) find the derivative of \(y\) with respect to \(x, t,\) or \(\theta,\) as appropriate. $$ y=\frac{\ln t}{t} $$

Each of Exercises \(19-24\) gives a formula for a function \(y=f(x)\) and shows the graphs of \(f\) and \(f^{-1} .\) Find a formula for \(f^{-1}\) in each case. $$f(x)=x^{3}-1$$

In Exercises \(21-42,\) find the derivative of \(y\) with respect to the appropriate variable. $$ y=\cos ^{-1}(1 / x) $$

The function In \(x\) grows slower than any polynomial Show that ln \(x\) grows slower as \(x \rightarrow \infty\) than any nonconstant polynomial.

In Exercises \(13-24,\) find the derivative of \(y\) with respect to the appropriate variable. $$y=\ln \sinh v-\frac{1}{2} \operatorname{coth}^{2} v$$

Use l'Hopital's rule to find the limits in Exercises \(7-50\) . $$ \lim _{x \rightarrow \pi / 2} \frac{\ln (\csc x)}{(x-(\pi / 2))^{2}} $$

Find the derivative of \(y\) with respect to \(x, t,\) or \(\theta,\) as appropriate. \begin{equation}y=\ln \left(\frac{\sqrt{\theta}}{1+\sqrt{\theta}}\right)\end{equation}

In Exercises \(7-38,\) find the derivative of \(y\) with respect to \(x, t,\) or \(\theta,\) as appropriate. $$ y=\frac{1+\ln t}{t} $$

Each of Exercises \(19-24\) gives a formula for a function \(y=f(x)\) and shows the graphs of \(f\) and \(f^{-1} .\) Find a formula for \(f^{-1}\) in each case. $$f(x)=x^{2}-2 x+1, \quad x \geq 1$$

In Exercises \(21-42,\) find the derivative of \(y\) with respect to the appropriate variable. $$ y=\arcsin \sqrt{2} t $$

$$\begin{array}{l}{\text { a. Suppose you have three different algorithms for solving the }} \\ {\text { same problem and each algorithm takes a number of steps that }} \\ {\text { is of the order of one of the functions listed here: }}\end{array}$$ $$n \log _{2} n, \quad n^{3 / 2}, \quad n\left(\log _{2} n\right)^{2}$$ $$\begin{array}{l}{\text { Which of the algorithms is the most efficient in the long run? }} \\ {\text { Give reasons for your answer. }}\end{array}$$ $$\begin{array}{l}{\text { b. Graph the functions in part (a) together to get a sense of how }} \\ {\text { rapidly each one grows. }}\end{array}$$

In Exercises \(13-24,\) find the derivative of \(y\) with respect to the appropriate variable. $$y=\left(x^{2}+1\right) \operatorname{sech}(\ln x)$$ (Hint: Before differentiating, express in terms of exponential and simplify.)

Use l'Hopital's rule to find the limits in Exercises \(7-50\) . $$ \lim _{t \rightarrow 0} \frac{t(1-\cos t)}{t-\sin t} $$

Find the derivative of \(y\) with respect to \(x, t,\) or \(\theta,\) as appropriate. \begin{equation}y=e^{(\cos t+\ln t)}\end{equation}

Each of Exercises \(19-24\) gives a formula for a function \(y=f(x)\) and shows the graphs of \(f\) and \(f^{-1} .\) Find a formula for \(f^{-1}\) in each case. $$f(x)=(x+1)^{2}, \quad x \geq-1$$

In Exercises \(21-42,\) find the derivative of \(y\) with respect to the appropriate variable. $$ y=\sin ^{-1}(1-t) $$

Atmospheric pressure The earth's atmospheric pressure \(p\) is often modeled by assuming that the rate \(d p / d h\) at which \(p\) changes with the altitude \(h\) above sea level is proportional to \(p .\) Suppose that the pressure at sea level is 1013 millibars (about 14.7 pounds per square inch) and that the pressure at an altitude of 20 \(\mathrm{km}\) is 90 millibars. \begin{equation} \begin{array}{l}{\text { a. Solve the initial value problem }} \\ {\text { Differential equation: } d p / d h=k p \quad(k \text { a constant) }} \\\ {\text { Initial condition: } \quad p=p_{0} \quad \text { when } h=0} \\\ {\text { to express } p \text { in terms of } h \text { . Determine the values of } p_{0} \text { and } k} \\ {\text { from the given altitude-pressure data. }}\\\\{\text { b. What is the atmospheric pressure at } h=50 \mathrm{km} ?} \\ {\text { c. At what altitude does the pressure equal } 900 \text { millibars? }}\end{array} \end{equation}

In Exercises \(13-24,\) find the derivative of \(y\) with respect to the appropriate variable. $$y=\left(4 x^{2}-1\right) \operatorname{csch}(\ln 2 x)$$

Use l'Hopital's rule to find the limits in Exercises \(7-50\) . $$ \lim _{t \rightarrow 0} \frac{t \sin t}{1-\cos t} $$

Find the derivative of \(y\) with respect to \(x, t,\) or \(\theta,\) as appropriate. \begin{equation}y=e^{\sin t}\left(\ln t^{2}+1\right)\end{equation}

In Exercises \(7-38,\) find the derivative of \(y\) with respect to \(x, t,\) or \(\theta,\) as appropriate. $$ y=\frac{x \ln x}{1+\ln x} $$

Each of Exercises \(19-24\) gives a formula for a function \(y=f(x)\) and shows the graphs of \(f\) and \(f^{-1} .\) Find a formula for \(f^{-1}\) in each case. $$f(x)=x^{2 / 3}, \quad x \geq 0$$

In Exercises \(21-42,\) find the derivative of \(y\) with respect to the appropriate variable. $$ y=\sec ^{-1}(2 s+1) $$

Suppose you are looking for an item in an ordered list one million items long. How many steps might it take to find that item with a sequential search? A binary search?

First-order chemical reactions In some chemical reactions, the rate at which the amount of a substance changes with time is proportional to the amount present. For the change of \(\delta\) -glucono lactone into gluconic acid, for example, $$\frac{d y}{d t}=-0.6 y$$ when \(t\) is measured in hours. If there are 100 grams of \(\delta\) -glucono lactone present when \(t=0,\) how many grams will be left after the first hour?

In Exercises \(25-36,\) find the derivative of \(y\) with respect to the appropriate variable. $$y=\sinh ^{-1} \sqrt{x}$$

Find the derivative of \(y\) with respect to \(x, t,\) or \(\theta,\) as appropriate. \(y=\int_{0}^{\ln x} \sin e^{t} d t\)