Chapter 7

Find the derivative of \(y\) with respect to the appropriate variable. $$y=\operatorname{sech} \theta(1-\ln \operatorname{sech} \theta)$$

Use l'Hôpital's rule to find the limits. $$\lim _{\theta \rightarrow \pi / 2} \frac{1-\sin \theta}{1+\cos 2 \theta}$$

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

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. $$f(x)=x^{2}+1, \quad x \geq 0$$

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

Solve the differential equations. $$\frac{d y}{d x}=x y+3 x-2 y-6$$

Find the derivative of \(y\) with respect to the appropriate variable. $$y=\operatorname{csch} \theta(1-\ln \operatorname{csch} \theta)$$

Use l'Hôpital's rule to find the limits. $$\lim _{x \rightarrow 1} \frac{x-1}{\ln x-\sin \pi x}$$

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

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. $$f(x)=x^{2}, \quad x \leq 0$$

a. Show that \(\ln x\) grows slower as \(x \rightarrow \infty\) than \(x^{1 / n}\) for any positive integer \(n,\) even \(x^{1 / 1,000,000}\) b. Although the values of \(x^{1 / 1,000,000}\) eventually overtake the values of \(\ln x,\) you have to go way out on the \(x\) -axis before this happens. Find a value of \(x\) greater than 1 for which \(x^{1 / 1.000 .000}>\ln x .\) You might start by observing that when \(x>1\) the equation \(\ln x=x^{1 / 1,000,000}\) is equivalent to the equation \(\ln (\ln x)=(\ln x) / 1,000,000\). c. Even \(x^{1 / 10}\) takes a long time to overtake \(\ln x\). Experiment with a calculator to find the value of \(x\) at which the graphs of \(x^{1 / 10}\) and \(\ln x\) cross, or, equivalently, at which \(\ln x=10 \ln (\ln x)\) Bracket the crossing point between powers of 10 and then close in by successive halving. d. (Continuation of part \((c) .)\) The value of \(x\) at which \(\ln x=10 \ln (\ln x)\) is too far out for some graphers and root finders to identify. Try it on the equipment available to you and see what happens.

Solve the differential equations. $$\frac{1}{x} \frac{d y}{d x}=y e^{x^{2}}+2 \sqrt{y} e^{x^{2}}$$

Find the derivative of \(y\) with respect to the appropriate variable. $$y=\ln \cosh v-\frac{1}{2} \tanh ^{2} v$$

Use l'Hôpital's rule to find the limits. $$\lim _{x \rightarrow 0} \frac{x^{2}}{\ln (\sec x)}$$

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. $$f(x)=x^{3}-1$$

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

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

Solve the differential equations. $$\frac{d y}{d x}=e^{x-y}+e^{x}+e^{-y}+1$$

Find the derivative of \(y\) with respect to the appropriate variable. $$y=\ln \sinh v-\frac{1}{2} \operatorname{coth}^{2} v$$

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

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. $$f(x)=x^{2}-2 x+1, \quad x \geq 1$$

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

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

The analysis of tooth shrinkage by C. Loring Brace and colleagues at the University of Michigan's Museum of Anthropology indicates that human tooth size is continuing to decrease and that the evolutionary process did not come to a halt some 30,000 years ago, as many scientists contend. In northern Europeans, for example, tooth size reduction now has a rate of \(1 \%\) per 1000 years. a. If \(t\) represents time in years and \(y\) represents tooth size, use the condition that \(y=0.99 y_{0}\) when \(t=1000\) to find the value of \(k\) in the equation \(y=y_{0} e^{k t} .\) Then use this value of \(k\) to answer the following questions. b. In about how many years will human teeth be \(90 \%\) of their present size? c. What will be our descendants' tooth size 20,000 years from now (as a percentage of our present tooth size)?

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 exponentials and simplify.)

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

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. $$f(x)=(x+1)^{2}, \quad x \geq-1$$

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

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 hectopascals and that the pressure at an altitude of \(20 \mathrm{km}\) is 90 hectopascals. a. Solve the initial value problem Differential equation: \(\quad d p / d h=k p \quad(k\) a constant) Initial condition: \(\quad p=p_{0} \quad\) when \(\quad h=0\) to express \(p\) in terms of \(h\). Determine the values of \(p_{0}\) and \(k\) from the given altitude-pressure data. b. What is the atmospheric pressure at \(h=50 \mathrm{km} ?\) c. At what altitude does the pressure equal 900 hectopascals?

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'Hôpital's rule to find the limits. $$\lim _{t \rightarrow 0} \frac{t \sin t}{1-\cos t}$$

Find the derivative of \(y\) with respect to \(x, t,\) or \(\theta,\) as appropriate. $$y=\int_{e^{4 \sqrt{x}}}^{e^{2 x}} \ln t d t$$

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. $$f(x)=x^{2 / 3}, \quad x \geq 0$$

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?

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?

Find the derivative of \(y\) with respect to the appropriate variable. $$y=\sinh ^{-1} \sqrt{x}$$

Use l'Hôpital's rule to find the limits. $$\lim _{x \rightarrow(\pi / 2)^{-}}\left(x-\frac{\pi}{2}\right) \sec x$$

Find \(d y / d x.\) $$\ln y=e^{y} \sin x$$

Gives a formula for a function \(y=f(x) .\) In each case, find \(f^{-1}(x)\) and identify the domain and range of \(f^{-1} .\) As a check, show that \(f\left(f^{-1}(x)\right)=f^{-1}(f(x))=x\). $$f(x)=x^{5}$$

Find the derivative of \(y\) with respect to \(x, t,\) or \(\theta,\) as appropriate. $$y=\theta(\sin (\ln \theta)+\cos (\ln \theta))$$

You are looking for an item in an ordered list 450,000 items long (the length of Webster's Third New International Dictionary). How many steps might it take to find the item with a sequential search? A binary search?

The processing of raw sugar has a step called "inversion" that changes the sugar's molecular structure. Once the process has begun, the rate of change of the amount of raw sugar is proportional to the amount of raw sugar remaining. If \(1000 \mathrm{kg}\) of raw sugar reduces to \(800 \mathrm{kg}\) of raw sugar during the first 10 hours, how much raw sugar will remain after another 14 hours?

Find the derivative of \(y\) with respect to the appropriate variable. $$y=\cosh ^{-1} 2 \sqrt{x+1}$$

Use l'Hôpital's rule to find the limits. $$\lim _{x \rightarrow(\pi / 2)^{-}}\left(\frac{\pi}{2}-x\right) \tan x$$

Find \(d y / d x.\) $$\ln x y=e^{x+y}$$

Gives a formula for a function \(y=f(x) .\) In each case, find \(f^{-1}(x)\) and identify the domain and range of \(f^{-1} .\) As a check, show that \(f\left(f^{-1}(x)\right)=f^{-1}(f(x))=x\). $$f(x)=x^{4}, \quad x \geq 0$$

Find the derivative of \(y\) with respect to \(x, t,\) or \(\theta,\) as appropriate. $$y=\ln (\sec \theta+\tan \theta)$$

The intensity \(L(x)\) of light \(x\) meters beneath the surface of the ocean satisfies the differential equation $$\frac{d L}{d x}=-k L$$ As a diver, you know from experience that diving to 6 meters in the Caribbean Sea cuts the intensity in half. You cannot work without artificial light when the intensity falls below one-tenth of the surface value. About how deep can you expect to work without artificial light?

Find the derivative of \(y\) with respect to the appropriate variable. $$y=(1-\theta) \tanh ^{-1} \theta$$

Use l'Hôpital's rule to find the limits. $$\lim _{\theta \rightarrow 0} \frac{3^{\sin \theta}-1}{\theta}$$