Chapter 6

Consider the functions \(f(x)=x^{n}\) and \(g(x)=x^{1 / n},\) where \(n \geq 2\) is a positive integer. a. Graph \(f\) and \(g\) for \(n=2,3,\) and \(4,\) for \(x \geq 0\) b. Give a geometric interpretation of the area function \(A_{n}(x)=\int_{0}^{x}(f(s)-g(s)) d s,\) for \(n=2,3,4, \ldots\) and \(x>0\) c. Find the positive root of \(A_{n}(x)=0\) in terms of \(n\). Does the root increase or decrease with \(n\) ?

Determine whether the following statements are true and give an explanation or counterexample. a. \(\frac{d}{d x}(\sinh \ln 3)=\frac{\cosh \ln 3}{3}.\) b. \(\frac{d}{d x}(\sinh x)=\cosh x\) and \(\frac{d}{d x}(\cosh x)=-\sinh x.\) c. Differentiating the velocity equation for an ocean wave \(v=\sqrt{\frac{g \lambda}{2 \pi} \tanh \left(\frac{2 \pi d}{\lambda}\right)}\) results in the acceleration of the wave. d. \(\ln (1+\sqrt{2})=-\ln (-1+\sqrt{2}).\) e. \(\int_{0}^{1} \frac{d x}{4-x^{2}}=\frac{1}{2}\left(\operatorname{coth}^{-1} \frac{1}{2}-\cot ^{-1} 0\right).\)

Use a left Riemann sum with at least \(n=2\) sub-intervals of equal length to approximate \(\ln 2=\int_{1}^{2} \frac{d t}{t}\) and show that \(\ln 2<1 .\) Use a right Riemann sum with \(n=7\) sub-intervals of equal length to approximate \(\ln 3=\int_{1}^{3} \frac{d t}{t}\) and show that \(\ln 3>1\).

Use a calculator to evaluate each expression or state that the value does not exist. Report answers accurate to four decimal places. a. \(\coth 4\) b. \(\tanh ^{-1} 2\) c. \(\operatorname{csch}^{-1} 5\) d. \(\left.\operatorname{csch} x\right|_{1 / 2} ^{2}\) e. \(\left.\ln \left|\tanh \left(\frac{x}{2}\right)\right|\right|_{1} ^{10}\) f. \(\left.\tan ^{-1}(\sinh x)\right|_{-3} ^{3}\) g. \(\left.\frac{1}{4} \operatorname{coth}^{-1} \frac{x}{4}\right|_{20} ^{36}\)

Consider the functions \(f(x)=a \sin 2 x\) and \(g(x)=(\sin x) / a,\) where \(a>0\) is a real number. a. Graph the two functions on the interval \([0, \pi / 2],\) for \(a=\frac{1}{2}, 1\) and 2. b. Show that the curves have an intersection point \(x^{*}\) (other than \(x=0)\) on \([0, \pi / 2]\) that satisfies \(\cos x^{*}=1 /\left(2 a^{2}\right),\) provided \(a>1 / \sqrt{2}\) c. Find the area of the region between the two curves on \(\left[0, x^{*}\right]\) when \(a=1\) d. Show that as \(a \rightarrow 1 / \sqrt{2}^{+}\). the area of the region between the two curves on \(\left[0, x^{*}\right]\) approaches zero.

Assume that \(y>0\) is fixed and that \(x>0 .\) Show that \(\frac{d}{d x}(\ln x y)=\frac{d}{d x}(\ln x) .\) Recall that if two functions have the same derivative, then they differ by an additive constant. Set \(x=1\) to evaluate the constant and prove that \(\ln x y=\ln x+\ln y\).

Without evaluating integrals, explain the following equalities. (Hint: Draw pictures.) a. \(\pi \int_{0}^{4}(8-2 x)^{2} d x=2 \pi \int_{0}^{8} y\left(4-\frac{y}{2}\right) d y\) b. \(\int_{0}^{2}\left(25-\left(x^{2}+1\right)^{2}\right) d x=2 \int_{1}^{5} y \sqrt{y-1} d y\)

Evaluate each expression without using a calculator or state that the value does not exist. Simplify answers to the extent possible. a. \(\cosh 0\) b. \(\tanh 0\) c. \(\operatorname{csch} 0\) d. \(\operatorname{sech}(\sinh 0)\) e. \(\operatorname{coth}(\ln 5)\) f. \(\sinh (2 \ln 3)\) g. \(\cosh ^{2} 1\) h. \(\operatorname{sech}^{-1}(\ln 3)\) i. \(\cosh ^{-1}(17 / 8)\) j. \(\sinh ^{-1}\left(\frac{e^{2}-1}{2 e}\right)\)

In Chapter \(8,\) we will encounter the harmonic \(\operatorname{sum} 1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n} .\) Use a left Riemann sum to approximate \(\int_{1}^{n+1} \frac{d x}{x}(\) with unit spacing between the grid points) to show that \(1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}>\ln (n+1) .\) Use this fact to conclude that \(\lim _{n \rightarrow \infty}\left(1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}\right)\) does not exist.

Solve the following problems with and without calculus. A good picture helps. a. A cube with side length \(r\) is inscribed in a sphere, which is inscribed in a right circular cone, which is inscribed in a right circular cylinder. The side length (slant height) of the cone is equal to its diameter. What is the volume of the cylinder? b. A cube is inscribed in a right circular cone with a radius of 1 and a height of \(3 .\) What is the volume of the cube? c. A cylindrical hole 10 in long is drilled symmetrically through the center of a sphere. How much material is left in the sphere? (Enough information is given.)

Find the critical points of the function \(f(x)=\sinh ^{2} x \cosh x.\)

a. Show that the critical points of \(f(x)=\frac{\cosh x}{x}\) satisfy \(x=\operatorname{coth} x.\) b. Use a root finder to approximate the critical points of \(f.\)

Find the \(x\) -coordinate of the point(s) of inflection of \(f(x)=\operatorname{sech} x .\) Report exact answers in terms of logarithms (use Theorem 6.10 ).

Find the area of the region bounded by \(y=\operatorname{sech} x, x=1,\) and the unit circle.

Explain why l'Hôpital's Rule fails when applied to the limit \(\lim _{x \rightarrow \infty} \frac{\sinh x}{\cosh x}\) and then find the limit another way.

Use l'Hôpital's Rule to evaluate the following limits. $$\lim _{x \rightarrow \infty} \frac{1-\operatorname{coth} x}{1-\tanh x}$$

Use l'Hôpital's Rule to evaluate the following limits. $$\lim _{x \rightarrow 0} \frac{\tanh ^{-1} x}{\tan (\pi x / 2)}$$

Use l'Hôpital's Rule to evaluate the following limits. $$\lim _{x \rightarrow 1^{-}} \frac{\tanh ^{-1} x}{\tan (\pi x / 2)}$$

Use l'Hôpital's Rule to evaluate the following limits. $$\lim _{x \rightarrow 0^{+}}(\tanh x)^{x}$$

Evaluate the following integrals. $$\int \frac{\cosh z}{\sinh ^{2} z} d z$$

Evaluate the following integrals. $$\int \frac{\cos \theta}{9-\sin ^{2} \theta} d \theta$$

Evaluate the following integrals. $$\int_{5 / 12}^{3 / 4} \frac{\sinh ^{-1} x}{\sqrt{x^{2}+1}} d x$$

Evaluate the following integrals. $$\int_{25}^{225} \frac{d x}{\sqrt{x^{2}+25 x}}(\text { Hint: } \sqrt{x^{2}+25 x}=\sqrt{x} \sqrt{x+25} .)$$

When an object falling from rest encounters air resistance proportional to the square of its velocity, the distance it falls (in meters) after \(t\) seconds is given by \(d(t)=\frac{m}{k} \ln (\cosh (\sqrt{\frac{k g}{m}} t)),\) where \(m\) is the mass of the object in kilograms, \(g=9.8 \mathrm{m} / \mathrm{s}^{2}\) is the acceleration due to gravity, and \(k\) is a physical constant. a. A \(\mathrm{BASE}\) jumper \((m=75 \mathrm{kg})\) leaps from a tall cliff and performs a ten-second delay (she free-falls for \(10 \mathrm{s}\) and then opens her chute). How far does she fall in \(10 \mathrm{s} ?\) Assume \(k=0.2\) b. How long does it take her to fall the first \(100 \mathrm{m} ?\) The second \(100 \mathrm{m} ?\) What is her average velocity over each of these intervals?

Verify the following identities. $$\sinh \left(\cosh ^{-1} x\right)=\sqrt{x^{2}-1}, \text { for } x \geq 1$$

Verify the following identities. $$\cosh \left(\sinh ^{-1} x\right)=\sqrt{x^{2}+1}, \text { for all } x$$

Verify the following identities. $$\cosh (x+y)=\cosh x \cosh y+\sinh x \sinh y$$

Verify the following identities. $$\sinh (x+y)=\sinh x \cosh y+\cosh x \sinh y$$

Show that \(\cosh ^{-1}(\cosh x)=|x|\) by using the formula \(\cosh ^{-1} t=\ln (t+\sqrt{t^{2}-1})\) and by considering the cases \(x \geq 0\) and \(x<0.\)

a. The definition of the inverse hyperbolic cosine is \(y=\cosh ^{-1} x \Leftrightarrow x=\cosh y,\) for \(x \geq 1,0 \leq y<\infty.\) Use implicit differentiation to show that \(\frac{d}{d x}\left(\cosh ^{-1} x\right)=\) \(1 / \sqrt{x^{2}-1}.\) b. Differentiate \(\sinh ^{-1} x=\ln (x+\sqrt{x^{2}+1})\) to show that \(\frac{d}{d x}\left(\sinh ^{-1} x\right)=1 / \sqrt{x^{2}+1}.\)

There are several ways to express the indefinite integral of \(\operatorname{sech} x\). a. Show that \(\left.\int \operatorname{sech} x d x=\tan ^{-1}(\sinh x)+C \text { (Theorem } 6.9\right)\) (Hint: Write sech \(x=\frac{1}{\cosh x}=\frac{\cosh x}{\cosh ^{2} x}=\frac{\cosh x}{1+\sinh ^{2} x}\) and then make a change of variables.) b. Show that \(\int \operatorname{sech} x d x=\sin ^{-1}(\tanh x)+C .\) (Hint: Show that \(\operatorname{sech} x=\frac{\operatorname{sech}^{2} x}{\sqrt{1-\tanh ^{2} x}}\) and then make a change of variables.) c. Verify that \(\int \operatorname{sech} x d x=2 \tan ^{-1} e^{x}+C\) by proving \(\frac{d}{d x}\left(2 \tan ^{-1} e^{x}\right)=\operatorname{sech} x.\)

Carry out the following steps to derive the formula \(\int \operatorname{csch} x d x=\ln |\tanh (x / 2)|+C(\text { Theorem } 6.9)\) a. Change variables with the substitution \(u=x / 2\) to show that $$\int \operatorname{csch} x d x=\int \frac{2 d u}{\sinh 2 u}.$$ b. Use the identity for sinh \(2 u\) to show that \(\frac{2}{\sinh 2 u}=\frac{\operatorname{sech}^{2} u}{\tanh u}.\) c. Change variables again to determine \(\int \frac{\operatorname{sech}^{2} u}{\tanh u} d u\) and then express your answer in terms of \(x.\)

Inverse hyperbolic tangent Recall that the inverse hyperbolic tangent is defined as \(y=\tanh ^{-1} x \Leftrightarrow x=\tanh y,\) for \(-1

Use the substitution \(u=x^{r}\) to show that \(\int \frac{d x}{x \sqrt{1-x^{2 r}}}=-\frac{1}{r} \operatorname{sech}^{-1} x^{r}+C,\) for \(r>0\) and \(0