Chapter 3

A lighthouse is located on a small island, 3 \(\mathrm{km}\) away from the nearest point \(P\) on a straight shoreline, and its light makes four revolutions per minute. How fast is the beam of light moving along the shoreline when it is 1 \(\mathrm{km}\) from \(P\) ?

\(39-40=\) Use a graph to estimate the value of the limit. Then use l'Hospital's Rule to find the exact value. $$\lim _{x \rightarrow 0} \frac{5^{x}-4^{x}}{3^{x}-2^{x}}$$

Suppose \(f^{-1}\) is the inverse function of a differentiable function \(f\) and let \(G(x)=1 / f^{-1}(x)\) . If \(f(3)=2\) and \(f^{\prime}(3)=\frac{1}{y}\) , find \(G(2)\) .

$$41-44=\( Find \)y^{\prime}\( and \)y^{\prime \prime}$$ $$y=e^{\alpha x} \sin \beta x$$

Find the derivative. Simplify where possible. $$ y=\operatorname{coth}^{-1}(\sec x) $$

Prove that $$\quad \lim _{x \rightarrow \infty} \frac{e^{x}}{x^{n}}=\infty$$ for any positive integer \(n .\) This shows that the exponential function approaches infinity faster than any power of \(x .\)

(a) How is the logarithmic function \(y=\log _{a} x\) defined? (b) What is the domain of this function? (c) What is the range of this function? (d) Sketch the general shape of the graph of the function y \(=\log _{a} x\) if \(a>1\)

(a) Sketch the graph of the function \(f(x)=\sin \left(\sin ^{-1} x\right)\) (b) Sketch the graph of the function \(g(x)=\sin ^{-1}(\sin x)\) \(x \in \mathbb{R} .\) (c) Show that \(g^{\prime}(x)=\frac{\cos x}{|\cos x|}\) (d) Sketch the graph of \(h(x)=\cos ^{-1}(\sin x), x \in \mathbb{R},\) and find its derivative.

$$41-44=\( Find \)y^{\prime}\( and \)y^{\prime \prime}$$ $$y=\frac{\ln x}{r^{2}}$$

Prove that $$\lim _{x \rightarrow \infty} \frac{\ln x}{x^{p}}=0$$ for any number \(p>0 .\) This shows that the logarithmic function approaches \(\infty\) more slowly than any power of \(x .\)

(a) What is the natural logarithm? (b) What is the common logarithm? (c) Sketch the graphs of the natural logarithm function and the natural exponential function with a common set of axes.

Show that \(\frac{d}{d x} \arctan (\tanh x)=\operatorname{sech} 2 x\)

$$41-44=\( Find \)y^{\prime}\( and \)y^{\prime \prime}$$ $$y=x \ln x$$

\(43-44=\) What happens if you try to use l'Hospital's Rule to find the limit? Evaluate the limit using another method. $$\lim _{x \rightarrow \infty} \frac{x}{\sqrt{x^{2}+1}}$$

Find the exact value of each expression (without a calculator.) $$ \text { (a) }log _{5} 125 \quad \text { (b) } \log _{3}\left(\frac{1}{27}\right) $$

The Gateway Arch in St. Louis was designed by Eero Saarinen and was constructed using the equation $$y=211.49-20.96 \cosh 0.03291765 x$$ $$\begin{array}{l}{\text { for the central curve of the arch, where } x \text { and } y \text { are mea- }} \\ {\text { sured in meters and }|x| \leqslant 91.20} \\ {\text { (a) Graph the central curve. }} \\ {\text { (b) What is the height of the arch at its center? }} \\ {\text { (c) At what points is the height } 100 \mathrm{m} \text { ? }} \\ {\text { (d) What is the slope of the arch at the points in part (c)? }}\end{array}$$

$$41-44=\( Find \)y^{\prime}\( and \)y^{\prime \prime}$$ $$y=\ln (\sec x+\tan x)$$

\(43-44\) . What happens if you try to use l'Hospital's Rule to find the limit? Evaluate the limit using another method. $$\lim _{x \rightarrow(\pi / 2)^{-}} \frac{\sec x}{\tan x}$$

Find the exact value of each expression (without a calculator.) $$ \text { (a) }ln (1 / e) \quad \text { (b) } \log _{10} \sqrt{10} $$

\(45-46=\) Find an equation of the tangent line to the curve at the given point. $$y=\ln \left(x^{2}-3 x+1\right), \quad(3,0)$$

Find the exact value of each expression (without a calculator.) $$ \begin{array}{l}{\text { (a) } \log _{2} 6-\log _{2} 15+\log _{2} 20} \\\ {\text { (b) } \log _{3} 100-\log _{3} 18-\log _{3} 50}\end{array} $$

If an initial amount \(A_{0}\) of money is invested at an interest rate \(r\) compounded \(n\) times a year, the value of the investment after \(t\) years is $$A=A_{0}\left(1+\frac{r}{n}\right)^{n t}$$ If we let \(n \rightarrow \infty,\) we refer to the continuous compounding of interest. Use l'Hospital's Rule to show that if interest is compounded continuously, then the amount after \(t\) years is $$A=A_{0} e^{r t}$$

A flexible cable always hangs in the shape of a catenary \(y=c+a \cosh (x / a),\) where \(c\) and \(a\) are constants and \(a>0\) (see Figure 4 and Exercise 48 ). Graph several members of the family of functions \(y=a \cosh (x / a) .\) How does the graph change as a varies?

\(45-46=\) Find an equation of the tangent line to the curve at the given point. $$y=e^{x} / x, \quad(1, e)$$

Find the exact value of each expression (without a calculator.) $$ { (a) }e^{-2 \ln 5} \quad \text { (b) } \ln \left(\ln e^{2 n}\right) $$

If an object with mass \(m\) is dropped from rest, one model for its speed \(v\) after \(t\) seconds, taking air resistance into account, is $$v=\frac{m g}{c}\left(1-e^{-c t / m}\right)$$ where \(g\) is the acceleration due to gravity and \(c\) is a positive constant. (a) Calculate \(\lim _{t \rightarrow \infty} v .\) What is the meaning of this limit? (b) For fixed \(t,\) use l'Hospital's Rule to calculate lim \(_{c \rightarrow 0^{+}} v\) What can you conclude about the velocity of a falling object in a vacuum?

A telephone line hangs between two poles 14 \(\mathrm{m}\) apart in the shape of the catenary \(y=20 \cosh (x / 20)-15,\) where \(x\) and \(y\) are measured in meters. (a) Find the slope of this curve where it meets the right pole. (b) Find the angle \(\theta\) between the line and the pole.

\(47-48=\) Differentiate \(f\) and find the domain of \(f\) $$f(x)=\frac{x}{1-\ln (x-1)}$$

Use the properties of logarithms to expand the quantity. $$ \ln \sqrt{a b} $$

If an electrostatic field \(E\) acts on a liquid or a gaseous polar dielectric, the net dipole moment \(P\) per unit volume is $$P(E)=\frac{e^{E}+e^{-E}}{e^{E}-e^{-E}}-\frac{1}{E}$$ Show that \(\lim _{E \rightarrow 0^{+}} P(E)=0\)

Using principles from physics it can be shown that when a cable is hung between two poles, it takes the shape of a curve \(y=f(x)\) that satisfies the differential equation $$\frac{d^{2} y}{d x^{2}}=\frac{\rho g}{T} \sqrt{1+\left(\frac{d y}{d x}\right)^{2}}$$ where \(\rho\) is the linear density of the cable, \(g\) is the acceleration due to gravity, \(T\) is the tension in the cable at its lowest point, and the coordinate system is chosen appro-priately. Verify that the function $$ y=f(x)=\frac{T}{\rho g} \cosh \left(\frac{\rho g x}{T}\right) $$ is a solution of this differential equation.

\(47-48=\) Differentiate \(f\) and find the domain of \(f\) $$f(x)=\ln \ln \ln x$$

Use the properties of logarithms to expand the quantity. $$ \log _{10} \sqrt{\frac{x-1}{x+1}} $$

A metal cable has radius \(r\) and is covered by insulation, so that the distance from the center of the cable to the exterior of the insulation is \(R .\) The velocity \(v\) of an electrical impulse in the cable is $$v=-c\left(\frac{r}{R}\right)^{2} \ln \left(\frac{r}{R}\right)$$ where \(c\) is a positive constant. Find the following limits and interpret your answers. $$\text {(a)}\lim _{R \rightarrow r^{+}} v \quad \text { (b) } \lim _{r \rightarrow 0^{+}} v$$

Let $$f(x)=c x+\ln (\cos x) .\( For what value of \)c\( is \)f^{\prime}(\pi / 4)=6 ?$$

Use the properties of logarithms to expand the quantity. $$ \ln \frac{x^{2}}{y^{3} z^{4}} $$

The first appearance in print of 1 'Hospital's Rule was in the book Analyse des Infiniment Petits published by the Marquis de l'Hospital in \(1696 .\) This was the first calculus textbook ever published and the example that the Marquis used in that book to illustrate his rule was to find the limit of the function $$y=\frac{\sqrt{2 a^{3} x-x^{4}}-a \sqrt[3]{a a x}}{a-\sqrt[4]{a x^{3}}}$$ as \(x\) approaches \(a,\) where \(a>0 .\) (At that time it was common to write \(a a\) instead of \(a^{2} .\) ) Solve this problem.

Evaluate \(\lim _{x \rightarrow \infty} \frac{\sinh x}{e^{x}}\)

Let $$f(x)=\log _{a}\left(3 x^{2}-2\right) .\( For what value of \)a\( is \)f^{\prime}(1)=3 ?$$

Use the properties of logarithms to expand the quantity. $$ \ln \left(s^{4} \sqrt{t \sqrt{u}}\right) $$

(a) Show that any function of the form $$y=A \sinh m x+B \cosh m x$$ satisfies the differential equation \(y^{\prime \prime}=m^{2} y\) . (b) Find \(y=y(x)\) such that \(y^{\prime \prime}=9 y, y(0)=-4\) , and \(y^{\prime}(0)=6\)

\(51-60=\) Use logarithmic differentiation or an alternative method to find the derivative of the function. $$y=\left(x^{2}+2\right)^{2}\left(x^{4}+4\right)^{4}$$

Express the given quantity as a single logarithm. $$ \ln 5+5 \ln 3 $$

Evaluate $$\lim _{x \rightarrow \infty}\left[x-x^{2} \ln \left(\frac{1+x}{x}\right)\right]$$

If \(x=\ln (\sec \theta+\tan \theta),\) show that \(\sec \theta=\cosh x\)

\(51-60=\) Use logarithmic differentiation or an alternative method to find the derivative of the function. $$ y=\frac{e^{-x} \cos ^{2} x}{x^{2}+x+1} $$

Express the given quantity as a single logarithm. $$ \ln (a+b)+\ln (a-b)-2 \ln c $$

Suppose \(f\) is a positive function. If \(\lim _{x \rightarrow a} f(x)=0\) and \(\lim _{x \rightarrow a} g(x)=\infty,\) show that $$\quad \lim _{x \rightarrow a}[f(x)]^{g(x)}=0$$ This shows that \(0^{\infty}\) is not an indeterminate form.

At what point of the curve \(y=\cosh x\) does the tangent have slope 1 ?

\(51-60=\) Use logarithmic differentiation or an alternative method to find the derivative of the function. $$ y=\sqrt{\frac{x-1}{x^{4}+1}} $$