Chapter 6

Consider the parabola \(y=x^{2} .\) Let \(P, Q,\) and \(R\) be points on the parabola with \(R\) between \(P\) and \(Q\) on the curve. Let \(\ell_{P}, \ell_{Q},\) and \(\ell_{R}\) be the lines tangent to the parabola at \(P, Q,\) and \(R,\) respectively (see figure). Let \(P^{\prime}\) be the intersection point of \(\ell_{Q}\) and \(\ell_{R} ;\) let \(Q^{\prime}\) be the intersection point of \(\ell_{P}\) and \(\ell_{R} ;\) and let \(R^{\prime}\) be the intersection point of \(\ell_{P}\) and \(\ell_{Q} .\) Prove that Area \(\Delta P Q R=2 \cdot\) Area \(\Delta P^{\prime} Q^{\prime} R^{\prime}\) in the following cases. a. \(P\left(-a, a^{2}\right), Q\left(a, a^{2}\right),\) and \(R(0,0),\) where \(a\) is a positive real number b. \(P\left(-a, a^{2}\right), Q\left(b, b^{2}\right),\) and \(R(0,0),\) where \(a\) and \(b\) are positive real numbers c. \(P\left(-a, a^{2}\right), Q\left(b, b^{2}\right),\) and \(R\) is any point between \(P\) and \(Q\) on the curve

Evaluate the following integrals. $$\int_{1}^{2 e} \frac{3^{\ln x}}{x} d x$$

Evaluate the following definite integrals. Use Theorem 10 to express your answer in terms of logarithms. \(\int_{1 / 8}^{1} \frac{d x}{x \sqrt{1+x^{2 / 3}}}\)

A reasonable model (with different parameters for different people) for the flow of air in and out of the lungs is $$V^{\prime}(t)=-\frac{\pi V_{0}}{10} \sin \left(\frac{\pi t}{5}\right),$$ where \(V(t)\) is the volume of air in the lungs at time \(t \geq 0,\) measured in liters, \(t\) is measured in seconds, and \(V_{0}\) is the capacity of the lungs. The time \(t=0\) corresponds to a time at which the lungs are full and exhalation begins. a. Graph the flow rate function with \(V_{0}=10 \mathrm{L}\). b. Find and graph the function \(V\), assuming that \(V(0)=V_{0}=10 \mathrm{L}\). c. What is the breathing rate in breaths/minute?

Consider the region \(R\) bounded by the curves \(y=a x^{2}+1, y=0, x=0,\) and \(x=1,\) for \(a \geq-1 .\) Let \(S_{1}\) and \(S_{2}\) be solids generated when \(R\) is revolved about the \(x\) - and \(y\) -axes, respectively. a. Find \(V_{1}\) and \(V_{2},\) the volumes of \(S_{1}\) and \(S_{2},\) as functions of \(a\). b. Are there values of \(a \geq-1\) for which \(V_{1}(a)=V_{2}(a) ?\)

Evaluate the following definite integrals. Use Theorem 10 to express your answer in terms of logarithms. \(\int_{\ln 5}^{\ln 9} \frac{\cosh x}{4-\sinh ^{2} x} d x\)

Some species have growth rates that oscillate with an (approximately) constant period \(P .\) Consider the growth rate function $$N^{\prime}(t)=A \sin \left(\frac{2 \pi t}{P}\right)+r,$$ where \(A\) and \(r\) are constants with units of individuals/year. A species becomes extinct if its population ever reaches 0 after \(t=0\). a. Suppose \(P=10, A=20,\) and \(r=0 .\) If the initial population is \(N(0)=10,\) does the population ever become extinct? Explain. b. Suppose \(P=10, A=20,\) and \(r=0 .\) If the initial population is \(N(0)=100,\) does the population ever become extinct? Explain. c. Suppose \(P=10, A=50,\) and \(r=5 .\) If the initial population is \(N(0)=10,\) does the population ever become extinct? Explain. d. Suppose \(P=10, A=50,\) and \(r=-5 .\) Find the initial population \(N(0)\) needed to ensure that the population never becomes extinct.

Let \(R\) be the region in the first quadrant bounded by the circle \(x^{2}+y^{2}=r^{2}\) and the coordinate axes. Find the volume of a hemisphere of radius \(r\) in the following ways. a. Revolve \(R\) about the \(x\) -axis and use the disk method. b. Revolve \(R\) about the \(x\) -axis and use the shell method. c. Assume the base of the hemisphere is in the \(x y\) -plane and use the general slicing method with slices perpendicular to the \(x y\) -plane and parallel to the \(x\) -axis.

Evaluate the following integrals. $$\int_{1}^{e^{2}} \frac{(\ln x)^{5}}{x} d x$$

Verify that the volume of a right circular cone with a base radius of \(r\) and a height of \(h\) is \(\pi r^{2} h / 3 .\) Use the region bounded by the line \(y=r x / h,\) the \(x\) -axis, and the line \(x=h,\) where the region is rotated around the \(x\) -axis. Then (a) use the disk method and integrate with respect to \(x,\) and (b) use the shell method and integrate with respect to \(y\).

The portion of the curve \(y=\frac{17}{15}-\cosh x\) that lies above the \(x\) -axis forms a catenary arch. Find the average height of the arch above the \(x\) -axis.

Evaluate the following integrals. $$\int \frac{\ln ^{2} x+2 \ln x-1}{x} d x$$

Show that the arc length of the catenary \(y=\cosh x\) over the interval \([0, a]\) is \(L=\sinh a\).

At Earth's surface the acceleration due to gravity is approximately \(g=9.8 \mathrm{m} / \mathrm{s}^{2}\) (with local variations). However, the acceleration decreases with distance from the surface according to Newton's law of gravitation. At a distance of \(y\) meters from Earth's surface, the acceleration is given by $$a(y)=-\frac{g}{(1+y / R)^{2}},$$ where \(R=6.4 \times 10^{6} \mathrm{m}\) is the radius of Earth. a. Suppose a projectile is launched upward with an initial velocity of \(v_{0} \mathrm{m} / \mathrm{s} .\) Let \(v(t)\) be its velocity and \(y(t)\) its height (in meters) above the surface \(t\) seconds after the launch. Neglecting forces such as air resistance, explain why \(\frac{d v}{d t}=a(y)\) and \(\frac{d y}{d t}=v(t).\) b. Use the Chain Rule to show that \(\frac{d v}{d t}=\frac{1}{2} \frac{d}{d y}\left(v^{2}\right)\). c. Show that the equation of motion for the projectile is \(\frac{1}{2} \frac{d}{d y}\left(v^{2}\right)=a(y),\) where \(a(y)\) is given previously. d. Integrate both sides of the equation in part (c) with respect to \(y\) using the fact that when \(y=0, v=v_{0} .\) Show that $$\frac{1}{2}\left(v^{2}-v_{0}^{2}\right)=g R\left(\frac{1}{1+y / R}-1\right).$$ e. When the projectile reaches its maximum height, \(v=0\) Use this fact to determine that the maximum height is $$y_{\max }=\frac{R v_{0}^{2}}{2 g R-v_{0}^{2}}.$$ f. Graph \(y_{\max }\) as a function of \(v_{0} .\) What is the maximum height when \(v_{0}=500 \mathrm{m} / \mathrm{s}, 1500 \mathrm{m} / \mathrm{s},\) and \(5 \mathrm{km} / \mathrm{s} ?\) g. Show that the value of \(v_{0}\) needed to put the projectile into orbit (called the escape velocity) is \(\sqrt{2 g R}\).

Find the area of the region bounded by the curve \(x=\frac{1}{2 y}-\sqrt{\frac{1}{4 y^{2}}-1}\) and the line \(x=1\) in the first quadrant.

Evaluate the following integrals. $$\int_{0}^{\ln 2} \frac{e^{3 x}-e^{-3 x}}{e^{3 x}+e^{-3 x}} d x$$

A power line is attached at the same height to two utility poles that are separated by a distance of \(100 \mathrm{ft}\); the power line follows the curve \(f(x)=a \cosh (x / a) .\) Use the following steps to find the value of \(a\) that produces a sag of \(10 \mathrm{ft}\) midway between the poles. Use a coordinate system that places the poles at \(x=\pm 50\). a. Show that \(a\) satisfies the equation \(\cosh (50 / a)-1=10 / a\) b. Let \(t=10 / a,\) confirm that the equation in part (a) reduces to \(\cosh 5 t-1=t,\) and solve for \(t\) using a graphing utility. Report your answer accurate to two decimal places. c. Use your answer in part (b) to find \(a,\) and then compute the length of the power line.

A hemispherical bowl of radius 8 inches is filled to a depth of \(h\) inches, where \(0 \leq h \leq 8\) ( \(h=0\) corresponds to an empty bowl). Use the shell method to find the volume of water in the bowl as a function of \(h\). (Check the special cases \(h=0\) and \(h=8 .)\)

Consider the cubic polynomial \(f(x)=x(x-a)(x-b),\) where \(0 \leq a \leq b\) a. For a fixed value of \(b,\) find the function \(F(a)=\int_{0}^{b} f(x) d x\) For what value of \(a\) (which depends on \(b\) ) is \(F(a)=0 ?\) b. For a fixed value of \(b,\) find the function \(A(a)\) that gives the area of the region bounded by the graph of \(f\) and the \(x\) -axis between \(x=0\) and \(x=b .\) Graph this function and show that it has a minimum at \(a=b / 2 .\) What is the maximum value of \(A(a),\) and where does it occur (in terms of \(b\) )?

Evaluate the following integrals. $$\int_{0}^{1} \frac{16^{x}}{4^{2 x}} d x$$

Assume \(f\) and \(g\) are even, integrable functions on \([-a, a],\) where \(a>1 .\) Suppose \(f(x)>g(x)>0\) on \([-a, a]\) and that the area bounded by the graphs of \(f\) and \(g\) on \([-a, a]\) is \(10 .\) What is the value of \(\int_{0}^{\sqrt{a}} x\left[f\left(x^{2}\right)-g\left(x^{2}\right)\right] d x ?\)

Find the volume of the torus formed when a circle of radius 2 centered at (3,0) is revolved about the \(y\) -axis. Use the shell method. You may need a computer algebra system or table of integrals to evaluate the integral.

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 \(n ?\)

The velocity of a surface wave on the ocean is given by \(v=\sqrt{\frac{g \lambda}{2 \pi} \tanh \left(\frac{2 \pi d}{\lambda}\right)}(\text { Example } 8) .\) Use a graphing utility or root finder to approximate the wavelength \(\lambda\) of an ocean wave traveling at \(v=7 \mathrm{m} / \mathrm{s}\) in water that is \(d=10 \mathrm{m}\) deep.

Use Exercise 69 to do the following calculations. a. Find the velocity of a wave where \(\lambda=50 \mathrm{m}\) and \(d=20 \mathrm{m}\). b. Determine the depth of the water if a wave with \(\lambda=15 \mathrm{m}\) is traveling at \(v=4.5 \mathrm{m} / \mathrm{s}\).

Derivative of In \(|x|\) Differentiate \(\ln x\) for \(x>0\) and differentiate \(\ln (-x)\) for \(x<0\) to conclude that \(\frac{d}{d x}(\ln |x|)=\frac{1}{x}\).

Suppose \(R\) is the region bounded by \(y=f(x)\) and \(y=g(x)\) on the interval \([a, b],\) where \(f(x) \geq g(x)\). a. Show that if \(R\) is revolved about the vertical line \(x=x_{0}\), where \(x_{0}b ?\)

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 \(a \geq 1 / \sqrt{2}\) 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.

a. Confirm that the linear approximation to \(f(x)=\tanh x\) at \(a=0\) is \(L(x)=x\) b. Recall that the velocity of a surface wave on the ocean is \(v=\sqrt{\frac{g \lambda}{2 \pi} \tanh \left(\frac{2 \pi d}{\lambda}\right)} .\) In fluid dynamics, shallow water refers to water where the depth-to-wavelength ratio \(d / \lambda<0.05 .\) Use your answer to part (a) to explain why the shallow water velocity equation is \(v=\sqrt{g d}\) c. Use the shallow-water velocity equation to explain why waves tend to slow down as they approach the shore.

Properties of \(e^{x}\) Use the inverse relations between \(\ln x\) and \(e^{x}\) and the properties of \(\ln x\) to prove the following properties. a. \(e^{x-y}=\frac{e^{x}}{e^{y}}\) b. \(\left(e^{x}\right)^{y}=e^{x y}\)

Suppose \(R\) is the region bounded by \(y=f(x)\) and \(y=g(x)\) on the interval \([a, b],\) where \(f(x) \geq g(x) \geq 0\). a. Show that if \(R\) is revolved about the horizontal line \(y=y_{0}\) that lies below \(R,\) then by the washer method, the volume of the resulting solid is $$ V=\int_{a}^{b} \pi\left[\left(f(x)-y_{0}\right)^{2}-\left(g(x)-y_{0}\right)^{2}\right] d x.$$ b. How is this formula changed if the line \(y=y_{0}\) lies above \(R ?\)

Use the following argument to show that \(\lim _{x \rightarrow \infty} \ln x\) \(=\infty\) and \(\lim _{x \rightarrow 0^{+}}\) \(\ln x=-\infty\). a. Make a sketch of the function \(f(x)=1 / x\) on the interval \([1,2] .\) Explain why the area of the region bounded by \(y=f(x)\) and the \(x\) -axis on [1,2] is \(\ln 2\) b. Construct a rectangle over the interval [1,2] with height \(\frac{1}{2}\) Explain why \(\ln 2>\frac{1}{2}\) c. Show that \(\ln 2^{n}>n / 2\) and \(\ln 2^{-n}<-n / 2\) d. Conclude that \(\lim _{x \rightarrow \infty} \ln x=\infty\) and \(\lim _{x \rightarrow 0^{+}} \ln x=-\infty\)

A tsunami is an ocean wave often caused by earthquakes on the ocean floor; these waves typically have long wavelengths, ranging between 150 to \(1000 \mathrm{km}\). Imagine a tsunami traveling across the Pacific Ocean, which is the deepest ocean in the world, with an average depth of about 4000 m. Explain why the shallow-water velocity equation (Exercise 71 ) applies to tsunamis even though the actual depth of the water is large. What does the shallow- water equation say about the speed of a tsunami in the Pacific Ocean (use \(d=4000 \mathrm{m}) ?\)

An ellipse centered at the origin is described by the equation \(x^{2} / a^{2}+y^{2} / b^{2}=1 .\) If an ellipse \(R\) is revolved about either axis, the resulting solid is an ellipsoid. a. Find the volume of the ellipsoid generated when \(R\) is revolved about the \(x\) -axis (in terms of \(a\) and \(b\) ). b. Find the volume of the ellipsoid generated when \(R\) is revolved about the \(y\) -axis (in terms of \(a\) and \(b\) ). c. Should the results of parts (a) and (b) agree? Explain.

Bounds on \(e\) Use a left Riemann sum with at least \(n=2\) subintervals 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 \(n=7\) subintervals of equal length to approximate \(\ln 3=\int_{1}^{3} \frac{d t}{t}\) and show that \(\ln 3>1 .\)

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. \(\int_{0}^{1} \frac{d x}{4-x^{2}}=\frac{1}{2}\left(\operatorname{coth}^{-1} \frac{1}{2}-\operatorname{coth}^{-1} 0\right)\)

Use a calculator to evaluate each expression, or state that the value does not exist. Report answers accurate to four decimal places. a. \(\operatorname{coth} 4\) b. \(\tanh ^{-1} 2\) c. \(\operatorname{csch}^{-1} 5\) d. \(\left.\operatorname{csch} x\right|_{1 / 2} ^{2}\) e. \(\ln | \tanh (x / 2) \|_{1}^{10} \quad\) f. \(\left.\tan ^{-1}(\sinh x)\right|_{-3} ^{3} \quad\) g. \(\left.\frac{1}{4} \operatorname{coth}^{-1}\left(\frac{x}{4}\right)\right|_{20} ^{\frac{36}{6}}\)

Alternative proof of product property 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, 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 why the following equalities are true. (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. \(\mathrm{cosh 0}\) b. \(\mathrm{tanh 0}\) c. \(\mathrm{csch 0}\) d. \(\mathrm{sech}(sinh 0)\) e. \(\operatorname{coth}(\ln 5) \quad\) f. \(\sinh (2 \ln 3)\) g. \(\cosh ^{2} 1 \quad\) h. \(\operatorname{sech}^{-1}(\ln 3)\) i. \(\cosh ^{-1}(17 / 8)\) j. \(\sinh ^{-1}\left(\frac{e^{2}-1}{2 e}\right)\)

The harmonic sum is \(1+\frac{1}{2}+\frac{1}{3}+\dots+\frac{1}{n} .\) Use a right Riemann sum to approximate \(\int_{1}^{n} \frac{d x}{x}(\) 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? (There is enough information 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 \(f(x)=\tanh ^{2} x\).

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 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. \(\lim _{x \rightarrow 0} \frac{\tanh ^{-1} x}{\tan (\pi x / 2)}\)