Chapter 4

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 \(x_{2}\) .

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

In Exercises \(1-6,\) use 1'Hopital's Rule to evaluate the limit. Then evaluate the limit using a method studied in Chapter 2 . $$ \lim _{x \rightarrow 2} \frac{x-2}{x^{2}-4} $$

Minimizing perimeter What is the smallest perimeter possible for a rectangle whose area is 16 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. $$ y=\frac{x^{3}}{3}-\frac{x^{2}}{2}-2 x+\frac{1}{3} $$

Answer the following questions about the functions whose derivatives are given in Exercises \(1-8 :\) a. What are the critical points of \(f ?\) b. On what intervals is \(f\) increasing or decreasing? c. At what points, if any, does \(f\) assume local maximum and minimum values? $$ f^{\prime}(x)=x(x-1) $$

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. \(f(x)=x^{2}+2 x-1, \quad[0,1]\)

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 \(x_{2}\) .

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

In Exercises \(1-6,\) use 1'Hopital's Rule to evaluate the limit. Then evaluate the limit using a method studied in Chapter 2 . $$ \lim _{x \rightarrow 0} \frac{\sin 5 x}{x} $$

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 in Exercises \(1-8 .\) Identify the intervals on which the functions are concave up and concave down. $$ y=\frac{x^{4}}{4}-2 x^{2}+4 $$

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. \(f(x)=x^{2 / 3}, \quad[0,1]\)

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 \(x_{2}\) .

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

In Exercises \(1-6,\) use 1'Hopital's Rule to evaluate the limit. Then evaluate the limit using a method studied in Chapter 2 . $$ \lim _{x \rightarrow \infty} \frac{5 x^{2}-3 x}{7 x^{2}+1} $$

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. $$ y=\frac{3}{4}\left(x^{2}-1\right)^{2 / 3} $$

Answer the following questions about the functions whose derivatives are given in Exercises \(1-8 :\) a. What are the critical points of \(f ?\) b. On what intervals is \(f\) increasing or decreasing? c. At what points, if any, does \(f\) assume local maximum and minimum values? $$ f^{\prime}(x)=(x-1)^{2}(x+2) $$

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. \(f(x)=x+\frac{1}{x}, \quad\left[\frac{1}{2}, 2\right]\)

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 \(x_{2}\) .

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

In Exercises \(1-6,\) use 1'Hopital's Rule to evaluate the limit. Then evaluate the limit using a method studied in Chapter 2 . $$ \lim _{x \rightarrow 1} \frac{x^{3}-1}{4 x^{3}-x-3} $$

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 in Exercises \(1-8 .\) Identify the intervals on which the functions are concave up and concave down. $$ y=\frac{9}{14} x^{1 / 3}\left(x^{2}-7\right) $$

Answer the following questions about the functions whose derivatives are given in Exercises \(1-8 :\) a. What are the critical points of \(f ?\) b. On what intervals is \(f\) increasing or decreasing? c. At what points, if any, does \(f\) assume local maximum and minimum values? $$ f^{\prime}(x)=(x-1)^{2}(x+2)^{2} $$

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. \(f(x)=\sqrt{x-1}, \quad[1,3]\)

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 \(x_{2}\) .

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

In Exercises \(1-6,\) use 1'Hopital's Rule to evaluate the limit. Then evaluate the limit using a method studied in Chapter 2 . $$ \lim _{x \rightarrow 0} \frac{1-\cos x}{x^{2}} $$

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?

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. $$ y=x+\sin 2 x,-\frac{2 \pi}{3} \leq x \leq \frac{2 \pi}{3} $$

Answer the following questions about the functions whose derivatives are given in Exercises \(1-8 :\) a. What are the critical points of \(f ?\) b. On what intervals is \(f\) increasing or decreasing? c. At what points, if any, does \(f\) assume local maximum and minimum values? $$ f^{\prime}(x)=(x-1)(x+2)(x-3) $$

Satisfy the hypotheses of the Mean Value Theorem on the given interval, and which do not? Give reasons for your answers. \(f(x)=x^{2 / 3}, \quad[-1,8]\)

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 \(x_{2}\) .

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

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 \(a=b\) .

In Exercises \(1-6,\) use 1'Hopital's Rule to evaluate the limit. Then evaluate the limit using a method studied in Chapter 2 . $$ \lim _{x \rightarrow \infty} \frac{2 x^{2}+3 x}{x^{3}+x+1} $$

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. $$ y=\tan x-4 x,-\frac{\pi}{2}

Answer the following questions about the functions whose derivatives are given in Exercises \(1-8 :\) a. What are the critical points of \(f ?\) b. On what intervals is \(f\) increasing or decreasing? c. At what points, if any, does \(f\) assume local maximum and minimum values? $$ f^{\prime}(x)=(x-7)(x+1)(x+5) $$

Satisfy the hypotheses of the Mean Value Theorem on the given interval, and which do not? Give reasons for your answers. \(f(x)=x^{4 / 5}, \quad[0,1]\)

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

Use 1'Hopital's Rule to find the limits in Exercises \(7-26\). $$ \lim _{t \rightarrow 0} \frac{\sin t^{2}}{t} $$

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 in Exercises \(1-8 .\) Identify the intervals on which the functions are concave up and concave down. $$ y=\sin |x|,-2 \pi \leq x \leq 2 \pi $$

Answer the following questions about the functions whose derivatives are given in Exercises \(1-8 :\) a. What are the critical points of \(f ?\) b. On what intervals is \(f\) increasing or decreasing? c. At what points, if any, does \(f\) assume local maximum and minimum values? $$ f^{\prime}(x)=x^{-1 / 3}(x+2) $$

Satisfy the hypotheses of the Mean Value Theorem on the given interval, and which do not? Give reasons for your answers. \(f(x)=\sqrt{x(1-x)}, \quad[0,1]\)

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.

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

Use 1'Hopital's Rule to find the limits in Exercises \(7-26\). $$ \lim _{x \rightarrow \pi / 2} \frac{2 x-\pi}{\cos x} $$