91Ó°ÊÓ

In Exercises \(31-34,\) find the function's absolute maximum and minimum values and say where they are assumed. $$ f(x)=x^{5 / 3}, \quad-1 \leq x \leq 8 $$

\(\infty / \infty\) Form Give an example of two differentiable functions \(f\) and \(g\) with \(\lim _{x \rightarrow \infty} f(x)=\lim _{x \rightarrow \infty} g(x)=\infty\) that satisfy the following. $$ \begin{array}{ll}{\text { a. } \lim _{x \rightarrow \infty} \frac{f(x)}{g(x)}=3} & {\text { b. } \lim _{x \rightarrow \infty} \frac{f(x)}{g(x)}=0} \\ {\text { c. } \lim _{x \rightarrow \infty} \frac{f(x)}{g(x)}=\infty}\end{array} $$

Continuous extension Find a value of \(c\) that makes the function $$f(x)=\left\\{\begin{array}{ll}{\frac{9 x-3 \sin 3 x}{5 x^{3}},} & {x \neq 0} \\\ {c,} & {x=0}\end{array}\right.$$ continuous at \(x=0 .\) Explain why your value of \(c\) works.

Use the steps of the graphing procedure on page 272 to graph the equations in Exercises \(9-40 .\) Include the coordinates of any local extreme points and inflection points. $$ y=x \sqrt{8-x^{2}} $$

In Exercises \(31-34,\) find the function's absolute maximum and minimum values and say where they are assumed. $$ g(\theta)=\theta^{3 / 5}, \quad-32 \leq \theta \leq 1 $$

Find the function with the given derivative whose graph passes through the point \(P .\) \(f^{\prime}(x)=2 x-1, \quad P(0,0)\)

In Exercises \(17-54\) , find the most general antiderivative or indefinite integral. Check your answers by differentiation. $$ \int \frac{t \sqrt{t}+\sqrt{t}}{t^{2}} d t $$

In Exercises \(29-36 :\) a. Identify the function's local extreme values in the given domain, and say where they are assumed. b. Which of the extreme values, if any, are absolute? c. Support your findings with a graphing calculator or computer grapher. $$ f(t)=12 t-t^{3}, \quad-3 \leq t<\infty $$

In Exercises \(29-36 :\) a. Identify the function's local extreme values in the given domain, and say where they are assumed. b. Which of the extreme values, if any, are absolute? c. Support your findings with a graphing calculator or computer grapher. $$ f(t)=t^{3}-3 t^{2}, \quad-\infty

In Exercises \(31-34,\) find the function's absolute maximum and minimum values and say where they are assumed. $$ h(\theta)=3 \theta^{2 / 3}, \quad-27 \leq \theta \leq 8 $$