Chapter 4

Find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation. $$\int\left(\sin 2 x-\csc ^{2} x\right) d x$$

The geometric mean of \(a\) and \(b\) The geometric mean of two positive numbers \(a\) and \(b\) is the number \(\sqrt{a b}\). Show that the value of \(c\) in the conclusion of the Mean Value Theorem for \(\quad f(x)=1 / x\) on an interval of positive numbers \([a, b]\) is \(c=\sqrt{a b}\).

You are to construct an open rectangular box with a square base and a volume of \(48 \mathrm{ft}^{3} .\) If material for the bottom costs \(\$ 6 / \mathrm{ft}^{2}\) and material for the sides costs \(\$ 4 / \mathrm{ft}^{2},\) what dimensions will result in the least expensive box? What is the minimum cost?

a. Find the local extrema of each function on the given interval, and say where they occur. b. Graph the function and its derivative together. Comment on the behavior of \(f\) in relation to the signs and values of \(f^{\prime}\) $$f(x)=\sin 2 x, \quad 0 \leq x \leq \pi$$

Find the limits. $$\lim _{x \rightarrow \infty}(1+2 x)^{1 /(2 \ln x)}$$

Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function. $$y=\frac{e^{x}}{1+e^{x}}$$

Find the extreme values (absolute and local) of the function over its natural domain, and where they occur. $$y=x-4 \sqrt{x}$$

Find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation. $$\int(2 \cos 2 x-3 \sin 3 x) d x$$

The geometric mean of \(a\) and \(b\) The geometric mean of two positive numbers \(a\) and \(b\) is the number \(\sqrt{a b}\). Show that the value of \(c\) in the conclusion of the Mean Value Theorem for \(\quad f(x)=1 / x\) on an interval of positive numbers \([a, b]\) is \(c=\sqrt{a b}\).

The 800 -room Mega Motel chain is filled to capacity when the room charge is \(\$ 50\) per night. For each \(\$ 10\) increase in room charge, 40 fewer rooms are filled each night. What charge per room will result in the maximum revenue per night?

a. Find the local extrema of each function on the given interval, and say where they occur. b. Graph the function and its derivative together. Comment on the behavior of \(f\) in relation to the signs and values of \(f^{\prime}\) $$f(x)=\sin x-\cos x, \quad 0 \leq x \leq 2 \pi$$

Find the limits. $$\lim _{x \rightarrow 0}\left(e^{x}+x\right)^{1 / x}$$

Find the extreme values (absolute and local) of the function over its natural domain, and where they occur. $$y=\frac{1}{\sqrt[3]{1-x^{2}}}$$

Find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation. $$\int \frac{1+\cos 4 t}{2} d t$$

Graph the function $$f(x)=\sin x \sin (x+2)-\sin ^{2}(x+1)$$ What does the graph do? Why does the function behave this way? Give reasons for your answers.

a. Find the local extrema of each function on the given interval, and say where they occur. b. Graph the function and its derivative together. Comment on the behavior of \(f\) in relation to the signs and values of \(f^{\prime}\) $$f(x)=\sqrt{3} \cos x+\sin x, \quad 0 \leq x \leq 2 \pi$$

Find the extreme values (absolute and local) of the function over its natural domain, and where they occur. $$y=\sqrt{3+2 x-x^{2}}$$

Find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation. $$\int \frac{1-\cos 6 t}{2} d t$$

Rolle's Theorem a. Construct a polynomial \(f(x)\) that has zeros at \(x=-2,-1,0\), \(1,\) and 2. b. Graph \(f\) and its derivative \(f^{\prime}\) together. How is what you see related to Rolle's Theorem? c. Do \(g(x)=\sin x\) and its derivative \(g^{\prime}\) illustrate the same phenomenon as \(f\) and \(f^{\prime} ?\)

a. When we cough, the trachea (windpipe) contracts to increase the velocity of the air going out. This raises the questions of how much it should contract to maximize the velocity and whether it really contracts that much when we cough. Under reasonable assumptions about the elasticity of the tracheal wall and about how the air near the wall is slowed by friction, the average flow velocity \(v\) can be modeled by the equation $$v=c\left(r_{0}-r\right) r^{2} \mathrm{cm} / \mathrm{sec}, \quad \frac{r_{0}}{2} \leq r \leq r_{0},$$ where \(r_{0}\) is the rest radius of the trachea in centimeters and \(c\) is a positive constant whose value depends in part on the length of the trachea. Show that \(v\) is greatest when \(r=(2 / 3) r_{0} ;\) that is, when the trachea is about \(33 \%\) contracted. The remarkable fact is that X-ray photographs confirm that the trachea contracts about this much during a cough. b. Take \(r_{0}\) to be 0.5 and \(c\) to be 1 and graph \(v\) over the interval \(0 \leq r \leq 0.5 .\) Compare what you see with the claim that \(v\) is at a maximum when \(r=(2 / 3) r_{0}\).

a. Find the local extrema of each function on the given interval, and say where they occur. b. Graph the function and its derivative together. Comment on the behavior of \(f\) in relation to the signs and values of \(f^{\prime}\) $$f(x)=-2 x+\tan x, \quad \frac{-\pi}{2}

Find the limits. $$\lim _{x \rightarrow 0^{+}}\left(1+\frac{1}{x}\right)^{x}$$

Find the extreme values (absolute and local) of the function over its natural domain, and where they occur. $$y=\frac{x}{x^{2}+1}$$

Find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation. $$\int\left(\frac{1}{x}-\frac{5}{x^{2}+1}\right) d x$$

Unique solution Assume that \(f\) is continuous on \([a, b]\) and differentiable on \((a, b) .\) Also assume that \(f(a)\) and \(f(b)\) have opposite signs and that \(f^{\prime} \neq 0\) between \(a\) and \(b .\) Show that \(f(x)=0\) exactly once between \(a\) and \(b\).

An inequality for positive integers Show that if \(a, b, c,\) and \(d\) are positive integers, then $$\frac{\left(a^{2}+1\right)\left(b^{2}+1\right)\left(c^{2}+1\right)\left(d^{2}+1\right)}{a b c d} \geq 16$$

a. Find the local extrema of each function on the given interval, and say where they occur. b. Graph the function and its derivative together. Comment on the behavior of \(f\) in relation to the signs and values of \(f^{\prime}\) $$f(x)=\frac{x}{2}-2 \sin \frac{x}{2}, \quad 0 \leq x \leq 2 \pi$$

Find the limits. $$\lim _{x \rightarrow \infty}\left(\frac{x+2}{x-1}\right)^{x}$$

Find the extreme values (absolute and local) of the function over its natural domain, and where they occur. $$y=\frac{x+1}{x^{2}+2 x+2}$$

Find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation. $$\int\left(\frac{2}{\sqrt{1-y^{2}}}-\frac{1}{y^{1 / 4}}\right) d y$$

Parallel tangents Assume that \(f\) and \(g\) are differentiable on \([a, b]\) and that \(f(a)=g(a)\) and \(f(b)=g(b) .\) Show that there is at least one point between \(a\) and \(b\) where the tangents to the graphs of \(\bar{f}\) and \(g\) are parallel or the same line. Illustrate with a sketch.

a. Find the local extrema of each function on the given interval, and say where they occur. b. Graph the function and its derivative together. Comment on the behavior of \(f\) in relation to the signs and values of \(f^{\prime}\) $$f(x)=-2 \cos x-\cos ^{2} x,-\pi \leq x \leq \pi$$

Find the limits. $$\lim _{x \rightarrow \infty}\left(\frac{x^{2}+1}{x+2}\right)^{1 / x}$$

Find the extreme values (absolute and local) of the function over its natural domain, and where they occur. $$y=e^{x}+e^{-x}$$

Suppose that \(f^{\prime}(x) \leq 1\) for \(1 \leq x \leq 4 .\) Show that \(f(4)-\) \(f(1) \leq 3\).

Find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation. $$\int 3 x^{\sqrt{3}} d x$$

a. Find the local extrema of each function on the given interval, and say where they occur. b. Graph the function and its derivative together. Comment on the behavior of \(f\) in relation to the signs and values of \(f^{\prime}\) $$f(x)=\csc ^{2} x-2 \cot x, \quad 0

Find the limits. $$\lim _{x \rightarrow 0^{+}} x^{2} \ln x$$

Find the extreme values (absolute and local) of the function over its natural domain, and where they occur. $$y=e^{x}-e^{-x}$$

Find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation. $$\int x^{\sqrt{2}-1} d x$$

Suppose that \(0

You have been asked to determine whether the function \(f(x)=\) \(3+4 \cos x+\cos 2 x\) is ever negative. a. Explain why you need to consider values of \(x\) only in the interval \([0,2 \pi]\) b. Is \(f\) ever negative? Explain.

a. Find the local extrema of each function on the given interval, and say where they occur. b. Graph the function and its derivative together. Comment on the behavior of \(f\) in relation to the signs and values of \(f^{\prime}\) $$f(x)=\sec ^{2} x-2 \tan x, \quad \frac{-\pi}{2}

Show that \(|\cos x-1| \leq|x|\) for all \(x\) -values. (Hint: Consider \(f(t)=\cos t \text { on }[0, x] .)\).

Find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation. $$\begin{aligned} &\int\left(1+\tan ^{2} \theta\right) d \theta\\\ &\text { (Hint: }\left.1+\tan ^{2} \theta=\sec ^{2} \theta\right) \end{aligned}$$

a. The function \(y=\cot x-\sqrt{2} \csc x\) has an absolute maximum value on the interval \(0

Find the limits. $$\lim _{x \rightarrow 0^{-}} x \tan \left(\frac{\pi}{2}-x\right)$$

Find the extreme values (absolute and local) of the function over its natural domain, and where they occur. $$y=x^{2} \ln x$$

Show that for any numbers \(a\) and \(b\), the sine inequality \(| \sin b-\) \(\sin a|\leq| b-a |\) is true.

Find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation. $$\int\left(2+\tan ^{2} \theta\right) d \theta$$