Chapter 4

In Exercises \(1-16,\) find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation. $$\text { a. }2 x \quad \text { b. } x^{2} \quad \text { c. } x^{2}-2 x+1$$

Use Newton's method to estimate the solutions of the equation \(x^{2}+x-1=0 .\) Start with \(x_{0}=-1\) for the left-hand solution and with \(x_{0}=1\) for the solution on the right. Then, in each case, find \(x_{2}\) .

Minimizing perimeter for a rectangle whose area is \(16 \mathrm{in}^{2},\) and what are its dimensions?

Identify the inflection points and local maxima and minima of the functions graphed in Exercises \(1-8 .\) Identify the intervals on which the functions are concave up and concave down. \begin{equation} y=\frac{x^{3}}{3}-\frac{x^{2}}{2}-2 x+\frac{1}{3} \end{equation}

Answer the following questions about the functions whose derivatives are given: \begin{equation}\begin{array}{l}{\text { a. What are the critical points of } f ?} \\ {\text { b. On what open intervals is } f \text { increasing or decreasing? }} \\ {\text { c. At what points, if any, does } f \text { assume local maximum and }} \\ \quad {\text { minimum values? }}\end{array}\end{equation} \begin{equation}f^{\prime}(x)=x(x-1)\end{equation}

Find the value or values of \(c\) that satisfy the equation $$\frac{f(b)-f(a)}{b-a}=f^{\prime}(c)$$ in the conclusion of the Mean Value Theorem for the functions and intervals in Exercises \(1-6 .\) $$f(x)=x^{2}+2 x-1, \quad[0,1]$$

In Exercises \(1-16,\) find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation. $$\text { a. }6 x \quad \text { b. } x^{7} \quad \text { c. } x^{7}-6 x+8$$

Use Newton's method to estimate the one real solution of \(x^{3}+3 x+1=0 .\) Start with \(x_{0}=0\) and then find \(x_{2}\) .

Show that among all rectangles with an 8 -m perimeter, the one with largest area is a square.

Identify the inflection points and local maxima and minima of the functions graphed.Identify the intervals on which the functions are concave up and concave down. \begin{equation} y=\frac{x^{4}}{4}-2 x^{2}+4 \end{equation}

Answer the following questions about the functions whose derivatives are given: \begin{equation}\begin{array}{l}{\text { a. What are the critical points of } f ?} \\ {\text { b. On what open intervals is } f \text { increasing or decreasing? }} \\ {\text { c. At what points, if any, does } f \text { assume local maximum and }} \\ \quad {\text { minimum values? }}\end{array}\end{equation} \begin{equation}f^{\prime}(x)=(x-1)(x+2)\end{equation}

In Exercises \(1-16,\) find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation. $$\text { a. }-3 x^{-4} \quad \text { b. } x^{-4} \quad \text { c. } x^{-4}+2 x+3$$

Use Newton's method to estimate the two zeros of the function \(f(x)=x^{4}+x-3 .\) Start with \(x_{0}=-1\) for the left-hand zero and with \(x_{0}=1\) for the zero on the right. Then, in each case, find \(x_{2}\)

Identify the inflection points and local maxima and minima of the functions graphed.Identify the intervals on which the functions are concave up and concave down. \begin{equation} y=\frac{3}{4}\left(x^{2}-1\right)^{2 / 3} \end{equation}

Answer the following questions about the functions whose derivatives are given: \begin{equation}\begin{array}{l}{\text { a. What are the critical points of } f ?} \\ {\text { b. On what open intervals is } f \text { increasing or decreasing? }} \\ {\text { c. At what points, if any, does } f \text { assume local maximum and }} \\ \quad {\text { minimum values? }}\end{array}\end{equation} \begin{equation}f^{\prime}(x)=(x-1)^{2}(x+2)\end{equation}

Find the value or values of \(c\) that satisfy the equation $$\frac{f(b)-f(a)}{b-a}=f^{\prime}(c)$$ in the conclusion of the Mean Value Theorem for the functions and intervals in Exercises \(1-6 .\) $$f(x)=x+\frac{1}{x}, \quad\left[\frac{1}{2}, 2\right]$$

In Exercises \(1-16,\) find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation. $$\text { a. }2 x^{-3} \quad \text { b. } \frac{x^{-3}}{2}+x^{2} \quad \text { c. }-x^{-3}+x-1 $$

Use Newton's method to estimate the two zeros of the function \(f(x)=2 x-x^{2}+1 .\) Start with \(x_{0}=0\) for the left-hand zero and with \(x_{0}=2\) for the zero on the right. Then, in each case, find \(x_{2}\) .

A rectangle has its base on the \(x\) -axis and its upper two vertices on the parabola \(y=12-x^{2}.\) What is the largest area the rectangle can have, and what are its dimensions?

Identify the inflection points and local maxima and minima of the functions graphed.Identify the intervals on which the functions are concave up and concave down. \begin{equation} y=\frac{9}{14} x^{1 / 3}\left(x^{2}-7\right) \end{equation}

Answer the following questions about the functions whose derivatives are given: \begin{equation}\begin{array}{l}{\text { a. What are the critical points of } f ?} \\ {\text { b. On what open intervals is } f \text { increasing or decreasing? }} \\ {\text { c. At what points, if any, does } f \text { assume local maximum and }} \\ \quad {\text { minimum values? }}\end{array}\end{equation} \begin{equation}f^{\prime}(x)=(x-1)^{2}(x+2)^{2}\end{equation}

Find the value or values of \(c\) that satisfy the equation $$\frac{f(b)-f(a)}{b-a}=f^{\prime}(c)$$ in the conclusion of the Mean Value Theorem for the functions and intervals in Exercises \(1-6 .\) $$f(x)=\sqrt{x-1}, \quad[1,3]$$

In Exercises \(1-16,\) find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation. $$\text { a. }\frac{1}{x^{2}} \quad \text { b. } \frac{5}{x^{2}} \quad \text { c. } 2-\frac{5}{x^{2}} $$

Use Newton's method to find the positive fourth root of 2 by solving the equation \(x^{4}-2=0 .\) Start with \(x_{0}=1\) and find \(x_{2}\) .

You are planning to make an open rectangular box from an 8-in.-by-15-in. piece of cardboard by cutting congruent squares from the corners and folding up the sides. What are the dimensions of the box of largest volume you can make this way, and what is its volume?

Answer the following questions about the functions whose derivatives are given: \begin{equation}\begin{array}{l}{\text { a. What are the critical points of } f ?} \\ {\text { b. On what open intervals is } f \text { increasing or decreasing? }} \\ {\text { c. At what points, if any, does } f \text { assume local maximum and }} \\ \quad {\text { minimum values? }}\end{array}\end{equation} \begin{equation}f^{\prime}(x)=(x-1)(x+2)(x-3)\end{equation}

Find the value or values of \(c\) that satisfy the equation $$\frac{f(b)-f(a)}{b-a}=f^{\prime}(c)$$ in the conclusion of the Mean Value Theorem for the functions and intervals in Exercises \(1-6 .\) $$f(x)=x^{3}-x^{2}, \quad[-1,2]$$

In Exercises \(1-16,\) find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation. $$\text { a. }-\frac{2}{x^{3}} \quad \text { b. } \frac{1}{2 x^{3}} \quad \text { c. } x^{3}-\frac{1}{x^{3}} $$

Use Newton's method to find the negative fourth root of 2 by solving the equation \(x^{4}-2=0 .\) Start with \(x_{0}=-1\) and find \(x_{2}\) .

You are planning to close off a corner of the first quadrant with a line segment 20 units long running from \((a, 0)\) to \((0, b) .\) Show that the area of the triangle enclosed by the segment is largest when \(a=b.\)

Identify the inflection points and local maxima and minima of the functions graphed.Identify the intervals on which the functions are concave up and concave down. \(\begin{equation} y=\tan x-4 x,-\frac{\pi}{2} < x < \frac{\pi}{2} \end{equation}\)

Answer the following questions about the functions whose derivatives are given: \begin{equation}\begin{array}{l}{\text { a. What are the critical points of } f ?} \\ {\text { b. On what open intervals is } f \text { increasing or decreasing? }} \\ {\text { c. At what points, if any, does } f \text { assume local maximum and }} \\ \quad {\text { minimum values? }}\end{array}\end{equation} \begin{equation}f^{\prime}(x)=(x-7)(x+1)(x+5)\end{equation}

In Exercises \(1-16,\) find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation. $$\text { a. }\frac{3}{2} \sqrt{x} \quad \text { b. } \frac{1}{2 \sqrt{x}} \quad \text { c. } \sqrt{x}+\frac{1}{\sqrt{x}}$$

Use Newton's method to find an approximate solution of \(3-x=x^{3} .\) Start with \(x_{0}=1\) and find \(x_{2}\) .

The best fencing plan A rectangular plot of farmland will be bounded on one side by a river and on the other three sides by a single-strand electric fence. With 800 \(\mathrm{m}\) of wire at your disposal, what is the largest area you can enclose, and what are its dimensions?

Identify the inflection points and local maxima and minima of the functions graphed.Identify the intervals on which the functions are concave up and concave down. \(\begin{equation} y=\sin |x|,-2 \pi \leq x \leq 2 \pi \end{equation}\)

Answer the following questions about the functions whose derivatives are given: \begin{equation}\begin{array}{l}{\text { a. What are the critical points of } f ?} \\ {\text { b. On what open intervals is } f \text { increasing or decreasing? }} \\ {\text { c. At what points, if any, does } f \text { assume local maximum and }} \\ \quad {\text { minimum values? }}\end{array}\end{equation} \begin{equation}f^{\prime}(x)=\frac{x^{2}(x-1)}{x+2}, \quad x \neq-2\end{equation}

In Exercises \(1-16,\) find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation. $$\text { a. }\frac{4}{3} \sqrt[3]{x} \quad \text { b. } \frac{1}{3 \sqrt[3]{x}} \quad \text { c. } \sqrt[3]{x}+\frac{1}{\sqrt[3]{x}}$$

The shortest fence \(A\) 216 \(\mathrm{m}^{2}\) rectangular pea patch is to be enclosed by a fence and divided into two equal parts by another fence parallel to one of the sides. What dimensions for the outer rectangle will require the smallest total length of fence? How much fence will be needed?

Identify the inflection points and local maxima and minima of the functions graphed.Identify the intervals on which the functions are concave up and concave down. \begin{equation} y=2 \cos x-\sqrt{2} x,-\pi \leq x \leq \frac{3 \pi}{2} \end{equation}

Answer the following questions about the functions whose derivatives are given: \begin{equation}\begin{array}{l}{\text { a. What are the critical points of } f ?} \\ {\text { b. On what open intervals is } f \text { increasing or decreasing? }} \\ {\text { c. At what points, if any, does } f \text { assume local maximum and }} \\ \quad {\text { minimum values? }}\end{array}\end{equation} \begin{equation}f^{\prime}(x)=\frac{(x-2)(x+4)}{(x+1)(x-3)}, \quad x \neq-1,3\end{equation}

In Exercises \(1-16,\) find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation. $$\text { a. }\frac{2}{3} x^{-1 / 3} \quad \text { b. } \frac{1}{3} x^{-2 / 3} \quad \text { c. }-\frac{1}{3} x^{-4 / 3}$$

Guessing a root Suppose that your first guess is lucky, in the sense that \(x_{0}\) is a root of \(f(x)=0 .\) Assuming that \(f^{\prime}\left(x_{0}\right)\) is defined and not \(0,\) what happens to \(x_{1}\) and later approximations?

In Exercises \(9-50\) , identify the coordinates of any local and absolute extreme points and inflection points. Graph the function. \begin{equation} y=x^{2}-4 x+3 \end{equation}

Answer the following questions about the functions whose derivatives are given: \begin{equation}\begin{array}{l}{\text { a. What are the critical points of } f ?} \\ {\text { b. On what open intervals is } f \text { increasing or decreasing? }} \\ {\text { c. At what points, if any, does } f \text { assume local maximum and }} \\ \quad {\text { minimum values? }}\end{array}\end{equation} \begin{equation}f^{\prime}(x)=1-\frac{4}{x^{2}}, \quad x \neq 0\end{equation}

In Exercises \(1-16,\) find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation. $$\text { a. }\frac{1}{2} x^{-1 / 2} \quad \text { b. }-\frac{1}{2} x^{-3 / 2} \quad \text { c. }-\frac{3}{2} x^{-5 / 2}$$

Estimating pi You plan to estimate \(\pi / 2\) to five decimal places by using Newton's method to solve the equation cos \(x=0\) . Does it matter what your starting value is? Give reasons for your answer.

Catching rainwater A 1125 \(\mathrm{ft}^{3}\) open-top rectangular tank with a square base \(x \mathrm{ft}\) on a side and \(y\) ft deep is to be built with its top flush with the ground to catch runoff water. The costs associated with the tank involve not only the material from which the tank is made but also an excavation charge proportional to the product \(x y .\) \begin{equation} \begin{aligned} \text { a. If the total cost is } \\ c &=5\left(x^{2}+4 x y\right)+10 x y, \end{aligned} \end{equation} \begin{equation} \begin{array}{l} \quad\quad\quad\quad\quad\quad {\text { what values of } x \text { and } y \text { will minimize it? }} \\ \quad\quad\quad\quad\quad {\text { b. Give a possible scenario for the cost function in part (a). }}\end{array} \end{equation}

Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function. \begin{equation} y=6-2 x-x^{2} \end{equation}

In Exercises \(1-16,\) find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation. $$\text { a. }-\pi \sin \pi x \quad \text { b. } 3 \sin x \quad \text { c. } \sin \pi x-3 \sin 3 x$$