Chapter 4

Use I'Hópital's rule to find the limits. $$\lim _{r \rightarrow-3} \frac{t^{3}-4 t+15}{t^{2}-t-12}$$

Show that if \(h>0,\) applying Newton's method to $$f(x)=\left\\{\begin{array}{ll}\sqrt{x}, & x \geq 0 \\\\\sqrt{-x}, & x<0\end{array}\right.$$ leads to \(x_{1}=-h\) if \(x_{0}=h\) and to \(x_{1}=h\) if \(x_{0}=-h .\) Draw a picture that shows what is going on.

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. $$f(x)=x^{2 / 3}, \quad[-1,8]$$

Find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation. a. \(\frac{1}{2} x^{-1 / 2}\) b. \(-\frac{1}{2} x^{-3 / 2}\) c. \(-\frac{3}{2} x^{-5 / 2}\)

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

Answer the following questions about the functions whose derivatives are given. 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? $$f^{\prime}(x)=3-\frac{6}{\sqrt{x}}, \quad x \neq 0$$

Catching rainwater \(\quad\) A \(1125 \mathrm{ft}^{3}\) open-top rectangular tank with a square base \(x\) ft on a side and \(y\) ft deep is to be built with its tope 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 \(x y .\) a. If the total cost is $$c=5\left(x^{2}+4 x y\right)+10 x y$$ what values of \(x\) and \(y\) will minimize it? b. Give a possible scenario for the cost function in part (a).

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. $$f(x)=x^{4 / 5}, \quad[0,1]$$

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. 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? $$f^{\prime}(x)=x^{-1 / 3}(x+2)$$

Use I'Hópital's rule to find the limits. $$\lim _{x \rightarrow \infty} \frac{5 x^{3}-2 x}{7 x^{3}+3}$$

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.

Designing a poster You are designing a rectangular poster to contain 50 in \(^{2}\) of printing with a 4 -in. margin at the top and bottom and a 2 -in. margin at each side. What overall dimensions will minimize the amount of paper used?

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. $$f(x)=\sqrt{x(1-x)}, \quad[0,1]$$

Find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation. 2\. a. \(\frac{1}{3 x}\) b. \(\frac{2}{5 x}\) c. \(1+\frac{4}{3 x}-\frac{1}{x^{2}}\)

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

Use I'Hópital's rule to find the limits. $$\lim _{x \rightarrow \infty} \frac{x-8 x^{2}}{12 x^{2}+5 x}$$

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

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. \(\sin \pi x-3 \sin 3 x\)

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. 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? $$f^{\prime}(x)=(\sin x-1)(2 \cos x+1), 0 \leq x \leq 2 \pi$$

Use I'Hópital's rule to find the limits. $$\lim _{t \rightarrow 0} \frac{\sin t^{2}}{t}$$

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: \(A=(1 / 2) a b \sin \theta .)\)

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. $$f(x)=\left\\{\begin{array}{ll}x^{2}-x, & -2 \leq x \leq-1 \\ 2 x^{2}-3 x-3, & -1 < x \leq 0\end{array}\right.$$

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

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. 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? $$f^{\prime}(x)=(\sin x+\cos x)(\sin x-\cos x), 0 \leq x \leq 2 \pi$$

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

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. $$f(x)=\left\\{\begin{array}{ll}2 x-3, & 0 \leq x \leq 2 \\ 6 x-x^{2}-7, & 2 < x \leq 3\end{array}\right.$$

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. $$f(x)=|x|, \quad-1

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

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

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.

Use I'Hópital's rule to find the limits. $$\lim _{x \rightarrow 0} \frac{8 x^{2}}{\cos x-1}$$

Designing a can 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?

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

Find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation. a. \(\csc ^{2} x\) b. \(-\frac{3}{2} \csc ^{2} \frac{3 x}{2}\) c. \(1-8 \csc ^{2} 2 x\)

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

Use I'Hópital's rule to find the limits. $$\lim _{x \rightarrow 0} \frac{\sin x-x}{x^{3}}$$

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. $$g(x)=\left\\{\begin{array}{ll} -x, & 0 \leq x<1 \\ x-1, & 1 \leq x \leq 2 \end{array}\right.$$

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

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

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 $$n x^{n-1}+(n-1) a_{n-1} x^{n-2}+\cdots+a_{1}$$

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

Use I'Hópital's rule to find the limits. $$\lim _{\theta \rightarrow \pi / 2} \frac{2 \theta-\pi}{\cos (2 \pi-\theta)}$$

Designing a suitcase \(\quad\) A 24 -in.-by- 36 -in. sheet of cardboard is folded in half to form a 24 -in.-by-18-in. rectangle as shown in the accompanying figure. Then four congruent squares of side length \(x\) are cut from the corners of the folded rectangle. The sheet is unfolded, and the six tabs are folded up to form a box with sides and a lid. a. Write a formula \(V(x)\) for the volume of the box. b. Find the domain of \(V\) for the problem situation and graph \(V\) over this domain. c. Use a graphical method to find the maximum volume and the value of \(x\) that gives it. d. Confirm your result in part (c) analytically. e. Find a value of \(x\) that yields a volume of 1120 in \(^{3}\) I. Write a paragraph describing the issues that arise in part (b). GRAPHS CANT COPY

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. $$h(x)=\left\\{\begin{array}{l}\frac{1}{x},-1 \leq x<0 \\\\\sqrt{x}, 0 \leq x \leq 4\end{array}\right.$$

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. \(\sec \frac{\pi x}{2} \tan \frac{\pi x}{2}\)

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

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.