Chapter 4

Answer the following questions about the functions whose derivatives are given in Exercises \(1-14:\) a. What are the critical points of \(f ?\) b. On what open intervals is \(f\) increasing or decreasing? c. At what points, if any, does \(f\) assume local maximum and minimum values? $$f^{\prime}(x)=3-\frac{6}{\sqrt{x}}, \quad x \neq 0$$

Find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation. a. \(-\pi \sin \pi x\) b. \(3 \sin x\) c. \(\sin \pi x-3 \sin 3 x\)

Explain why the following four statements ask for the same information: i) Find the roots of \(f(x)=x^{3}-3 x-1\). ii) Find the \(x\) -coordinates of the intersections of the curve \(y=x^{3}\) with the line \(y=3 x+1\). iii) Find the \(x\) -coordinates of the points where the curve \(y=x^{3}-3 x\) crosses the horizontal line \(y=1\). iv) Find the values of \(x\) where the derivative of \(g(x)=\) \((1 / 4) x^{4}-(3 / 2) x^{2}-x+5\) equals zero.

You are designing a rectangular poster to contain \(312.5 \mathrm{cm}^{2}\) of printing with a \(10 \mathrm{cm}\) margin at the top and bottom and a \(5 \mathrm{cm}\) margin at each side. What overall dimensions will minimize the amount of paper used?

Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function. $$y=x^{3}-3 x+3$$

Answer the following questions about the functions whose derivatives are given in Exercises \(1-14:\) a. What are the critical points of \(f ?\) b. On what open intervals is \(f\) increasing or decreasing? c. At what points, if any, does \(f\) assume local maximum and minimum values? $$f^{\prime}(x)=x^{-1 / 3}(x+2)$$

Find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation. a. \(\pi \cos \pi x\) a. \(\pi \cos \pi x\) c. \(\cos \frac{\pi x}{2}+\pi \cos x\)

To calculate a planet's space coordinates, we have to solve equations like \(x=1+0.5 \sin x .\) Graphing the function \(f(x)=x-1-0.5 \sin x\) suggests that the function has a root near \(x=1.5 .\) Use one application of Newton's method to improve this estimate. That is, start with \(x_{0}=1.5\) and find \(x_{1}\). (The value of the root is 1.49870 to five decimal places.) Remember to use radians.

Find the volume of the largest right circular cone that can be inscribed in a sphere of radius 3 .

Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function. $$y=x(6-2 x)^{2}$$

Which of the functions satisfy the hypotheses of the Mean Value Theorem on the given interval, and which do not? Give reasons for your answers. $$f(x)=\left\\{\begin{array}{ll}2 x-3, & 0 \leq x \leq 2 \\\6 x-x^{2}-7, & 2

Answer the following questions about the functions whose derivatives are given in Exercises \(1-14:\) a. What are the critical points of \(f ?\) b. On what open intervals is \(f\) increasing or decreasing? c. At what points, if any, does \(f\) assume local maximum and minimum values? $$f^{\prime}(x)=x^{-1 / 2}(x-3)$$

Find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation. a. \(\frac{1}{2} \sec ^{2} x\) b. \(\frac{2}{3} \sec ^{2} \frac{x}{3}\) c. \(-\sec ^{2} \frac{3 x}{2}\)

The curve \(y=\tan x\) crosses the line \(y=2 x\) between \(x=0\) and \(x=\pi / 2 .\) Use Newton's method to find where.

Two sides of a triangle have lengths \(a\) and \(b\), and the angle between them is \(\theta\). What value of \(\theta\) will maximize the triangle's area? (Hint: \(A=(1 / 2) a b \sin \theta\).)

The function $$f(x)=\left\\{\begin{array}{ll} x, & 0 \leq x<1 \\ 0, & x=1 \end{array}\right.$$ is zero at \(x=0\) and \(x=1\) and differentiable on \((0,1),\) but its derivative on (0,1) is never zero. How can this be? Doesn't Rolle's Theorem say the derivative has to be zero somewhere in (0, 1)? Give reasons for your answer.

Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function. $$y=-2 x^{3}+6 x^{2}-3$$

Answer the following questions about the functions whose derivatives are given in Exercises \(1-14:\) a. What are the critical points of \(f ?\) b. On what open intervals is \(f\) increasing or decreasing? c. At what points, if any, does \(f\) assume local maximum and minimum values? $$f^{\prime}(x)=(\sin x-1)(2 \cos x+1), 0 \leq x \leq 2 \pi$$

Find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation. a. \(\csc ^{2} x\) b. \(-\frac{3}{2} \csc ^{2} \frac{3 x}{2}\) b. \(-\frac{3}{2} \csc ^{2} \frac{3 x}{2}\)

Use Newton's method to find the two real solutions of the equation \(x^{4}-2 x^{3}-x^{2}-2 x+2=0\).

Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function. $$y=1-9 x-6 x^{2}-x^{3}$$

Answer the following questions about the functions whose derivatives are given in Exercises \(1-14:\) a. What are the critical points of \(f ?\) b. On what open intervals is \(f\) increasing or decreasing? c. At what points, if any, does \(f\) assume local maximum and minimum values? $$f^{\prime}(x)=(\sin x+\cos x)(\sin x-\cos x), 0 \leq x \leq 2 \pi$$

Find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation. a. \(\csc x \cot x\) b. \(-\csc 5 x \cot 5 x\) c. \(-\pi \csc \frac{\pi x}{2} \cot \frac{\pi x}{2}\)

You are designing a \(1000 \mathrm{cm}^{3}\) right circular cylindrical can whose manufacture will take waste into account. There is no waste in cutting the aluminum for the side, but the top and bottom of radius \(r\) will be cut from squares that measure \(2 r\) units on a side. The total amount of aluminum used up by the can will therefore be $$A=8 r^{2}+2 \pi r h$$ rather than the \(A=2 \pi r^{2}+2 \pi r h\) in Example \(2 .\) In Example 2 the ratio of \(h\) to \(r\) for the most economical can was 2 to \(1 .\) What is the ratio now?

a. Plot the zeros of each polynomial on a line together with the zeros of its first derivative. i) \(y=x^{2}-4\) ii) \(y=x^{2}+8 x+15\) iii) \(y=x^{3}-3 x^{2}+4=(x+1)(x-2)^{2}\) iv) \(y=x^{3}-33 x^{2}+216 x=x(x-9)(x-24)\) b. Use Rolle's Theorem to prove that between every two zeros of \(x^{n}+a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0}\) there lies a zero of $$ n x^{n-1}+(n-1) a_{n-1} x^{n-2}+\cdots+a_{1} $$

Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function. $$y=(x-2)^{3}+1$$

Sketch the graph of each function and determine whether the function has any absolute extreme values on its domain. Explain how your answer is consistent with Theorem 1. $$f(x)=|x|, \quad-1 < x < 2$$

Find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation. a. \(\sec x \tan x\) b. \(4 \sec 3 x \tan 3 x\) c. \(\sec \frac{\pi x}{2} \tan \frac{\pi x}{2}\)

Suppose that \(f^{\prime \prime}\) is continuous on \([a, b]\) and that \(f\) has three zeros in the interval. Show that \(f^{\prime \prime}\) has at least one zero in \((a, b)\) Generalize this result.

Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function. $$y=1-(x+1)^{3}$$

Sketch the graph of each function and determine whether the function has any absolute extreme values on its domain. Explain how your answer is consistent with Theorem 1. $$y=\frac{6}{x^{2}+2}, \quad-1 < x < 1$$

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. $$\int(x+1) d x$$

Find the four real zeros of the function \(f(x)=2 x^{4}-4 x^{2}+1\).

Show that if \(f^{\prime \prime}>0\) throughout an interval \([a, b],\) then \(f^{\prime}\) has at most one zero in \([a, b] .\) What if \(f^{\prime \prime}<0\) throughout \([a, b]\) instead?

Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function. $$y-x^{4}-2 x^{2}-x^{2}\left(x^{2}-2\right)$$

Sketch the graph of each function and determine whether the function has any absolute extreme values on its domain. Explain how your answer is consistent with Theorem 1. $$g(x)=\left\\{\begin{array}{ll} -x, & 0 \leq x < 1 \\ x-1, & 1 \leq x \leq 2 \end{array}\right.$$

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. $$\int(5-6 x) d x$$

Estimate \(\pi\) to as many decimal places as your calculator will display by using Newton's method to solve the equation \(\tan x=0\) with \(x_{0}=3\).

A rectangle is to be inscribed under the arch of the curve \(y=4 \cos (0.5 x)\) from \(x=-\pi\) to \(x=\pi .\) What are the dimensions of the rectangle with largest area, and what is the largest area?

Show that a cubic polynomial can have at most three real zeros.

Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function. $$y=-x^{4}+6 x^{2}-4=x^{2}\left(6-x^{2}\right)-4$$

Sketch the graph of each function and determine whether the function has any absolute extreme values on its domain. Explain how your answer is consistent with Theorem 1. $$h(x)=\left\\{\begin{array}{ll} \frac{1}{x}, & -1 \leq x < 0 \\ \sqrt{x}, & 0 \leq x \leq 4 \end{array}\right.$$

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. $$\int\left(3 t^{2}+\frac{t}{2}\right) d t$$

At what value(s) of \(x\) does \(\cos x=2 x ?\)

Find the dimensions of a right circular cylinder of maximum volume that can be inscribed in a sphere of radius \(10 \mathrm{cm} .\) What is the maximum volume?

Show that the functions that have exactly one zero in the given interval. $$f(x)=x^{4}+3 x+1, \quad[-2,-1]$$

Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function. $$y=4 x^{3}-x^{4}=x^{3}(4-x)$$

a. Find the open intervals on which the function is increasing and decreasing. b. Identify the function's local and absolute extreme values, if any, saying where they occur. $$g(t)=-t^{2}-3 t+3$$

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. $$\int\left(\frac{t^{2}}{2}+4 t^{3}\right) d t$$

Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function. $$y=x^{4}+2 x^{3}=x^{3}(x+2)$$