91Ó°ÊÓ

Rolle's Theorem In your own words, describe Rolle's Theorem.

Optimization Problems In your own words, describe the guidelines for solving applied minimum and maximum problems.

Horizontal Asymptote What does it mean for the graph of a function to have a horizontal asymptote?

Limits at Infinity In your own words, summarize the guidelines for finding limits at infinity of rational functions.

Polynomial What are the maximum numbers of relative extrema and points of inflection that a fifth-degree polynomial can have? Explain.

Using Newton's Method In Exercises \(7-16,\) use Newton's Method to approximate the zero(s) of the function. Continue the iterations until two successive approximations differ by less than 0.001. Then find the zero(s) using a graphing utility and compare the results. \(f(x)=2-x^{3}\)

Maximum Area , find the length and width of a rectangle that has the given perimeter and a maximum area. Perimeter: 80 meters

Comparing \(\Delta y\) and \(d y\) In Exercises \(13-18\) use the information to find and compare \(\Delta y\) and \(d y\) . $$\begin{array}{ll}{\text { Function }} & {x \text { -Value }} \\ {y=0.5 x^{3}} & {x=1}\end{array} \quad \begin{array}{ll}{\text { Differential of } x} \\\ {\Delta x=d x=0.1}\end{array}$$

In Exercises 13-16, find each limit, if it exists. (a) $$\lim _{x \rightarrow \infty} \frac{5-2 x^{3 / 2}}{3 x^{2}-4}$$ (b) $$\lim _{x \rightarrow \infty} \frac{5-2 x^{3 / 2}}{3 x^{3 / 2}-4}$$ (c) $$\lim _{x \rightarrow \infty} \frac{5-2 x^{3 / 2}}{3 x-4}$$

Determining Concavity In Exercises \(5-16,\) determine the open intervals on which the graph of the function is concave upward or concave downward. \(y=x+\frac{2}{\sin x}, \quad(-\pi, \pi)\)