Chapter 3

Determining Concavity In Exercises \(3-14\) , determine the open intervals on which the graph is concave upward or concave downward. $$ f(x)=\frac{x^{2}+1}{x^{2}-1} $$

Intervals on Which \(f\) Is Increasing or Decreasing In Exercises \(9-16\) , identify the open intervals on which the function is increasing or decreasing. $$ g(x)=x^{2}-2 x-8 $$

Determine whether Rolle's Theorem can be applied to \(f\) on the closed interval \([a, b] .\) If Rolle's Theorem can be applied, find all values of \(c\) in the open interval \((a, b)\) such that \(f^{\prime}(c)=0 .\) If Rolle's Theorem cannot be applied, explain why not. \(f(x)=-x^{2}+3 x, \quad[0,3]\)

Comparing \(\Delta y\) and \(d y\) In Exercises \(7-10\) , use the information to evaluate and compare \(\Delta y\) and \(d y .\) $$ \begin{array}{ll}{\text { Function }} & {x \text { -Value }} \\ {y=2-x^{4}} & {x=2}\end{array} \quad \Delta x=d x=0.01 $$

Numerical and Graphical Analysis In Exercises \(7-12\) , use a graphing utility to complete the table and estimate the limit as \(x\) approaches infinity. Then use a graphing utility to graph the function and estimate the limit graphically. $$ \begin{array}{|c|c|c|c|c|c|c|}\hline x & {10^{0}} & {10^{1}} & {10^{2}} & {10^{3}} & {10^{4}} & {10^{5}} & {10^{6}} \\ \hline f(x) & {} & {} & {} & {} \\\ \hline\end{array} $$ $$ f(x)=\frac{10}{\sqrt{2 x^{2}-1}} $$

Approximate the zero(s) of the function. Use Newton’s Method and continue the process until two successive approximations differ by less than 0.001. Then find the zero(s) using a graphing utility and compare the results. \(f(x)=x-2 \sqrt{x+1}\)

In Exercises 5–24, analyze and sketch a graph of the function. Label any intercepts, relative extrema, points of inflection, and asymptotes. Use a graphing utility to verify your results. $$ f(x)=\frac{x-3}{x} $$

Determining Concavity In Exercises \(3-14\) , determine the open intervals on which the graph is concave upward or concave downward. $$ y=\frac{-3 x^{5}+40 x^{3}+135 x}{270} $$

Intervals on Which \(f\) Is Increasing or Decreasing In Exercises \(9-16\) , identify the open intervals on which the function is increasing or decreasing. $$ h(x)=12 x-x^{3} $$

Determine whether Rolle's Theorem can be applied to \(f\) on the closed interval \([a, b] .\) If Rolle's Theorem can be applied, find all values of \(c\) in the open interval \((a, b)\) such that \(f^{\prime}(c)=0 .\) If Rolle's Theorem cannot be applied, explain why not. \(f(x)=x^{2}-8 x+5, \quad[2,6]\)

Finding a Differential In Exercises \(11-20\) , find the differential \(d y\) of the given function. $$ y=3 x^{2}-4 $$

Minimum Perimeter In Exercises 11 and \(12,\) find the length and width of a rectangle that has the given area and a minimum perimeter. Area: 32 square feet

Numerical and Graphical Analysis In Exercises \(7-12\) , use a graphing utility to complete the table and estimate the limit as \(x\) approaches infinity. Then use a graphing utility to graph the function and estimate the limit graphically. $$ \begin{array}{|c|c|c|c|c|c|c|}\hline x & {10^{0}} & {10^{1}} & {10^{2}} & {10^{3}} & {10^{4}} & {10^{5}} & {10^{6}} \\ \hline f(x) & {} & {} & {} & {} \\\ \hline\end{array} $$ $$ f(x)=5-\frac{1}{x^{2}+1} $$

Approximate the zero(s) of the function. Use Newton’s Method and continue the process until two successive approximations differ by less than 0.001. Then find the zero(s) using a graphing utility and compare the results. \(f(x)=x^{3}-3.9 x^{2}+4.79 x-1.881\)

In Exercises 5–24, analyze and sketch a graph of the function. Label any intercepts, relative extrema, points of inflection, and asymptotes. Use a graphing utility to verify your results. $$ f(x)=x+\frac{32}{x^{2}} $$

Finding Critical Numbers In Exercises \(11-16,\) find the critical numbers of the function. $$ f(x)=x^{3}-3 x^{2} $$

Determining Concavity In Exercises \(3-14\) , determine the open intervals on which the graph is concave upward or concave downward. $$ g(x)=\frac{x^{2}+4}{4-x^{2}} $$

Intervals on Which \(f\) Is Increasing or Decreasing In Exercises \(9-16\) , identify the open intervals on which the function is increasing or decreasing. $$ y=x \sqrt{16-x^{2}} $$

Determine whether Rolle's Theorem can be applied to \(f\) on the closed interval \([a, b] .\) If Rolle's Theorem can be applied, find all values of \(c\) in the open interval \((a, b)\) such that \(f^{\prime}(c)=0 .\) If Rolle's Theorem cannot be applied, explain why not. \(f(x)=(x-1)(x-2)(x-3), \quad[1,3]\)

Finding a Differential In Exercises \(11-20\) , find the differential \(d y\) of the given function. $$ y=3 x^{2 / 3} $$

Numerical and Graphical Analysis In Exercises \(7-12\) , use a graphing utility to complete the table and estimate the limit as \(x\) approaches infinity. Then use a graphing utility to graph the function and estimate the limit graphically. $$ \begin{array}{|c|c|c|c|c|c|c|}\hline x & {10^{0}} & {10^{1}} & {10^{2}} & {10^{3}} & {10^{4}} & {10^{5}} & {10^{6}} \\ \hline f(x) & {} & {} & {} & {} \\\ \hline\end{array} $$ $$ f(x)=4+\frac{3}{x^{2}+2} $$

Approximate the zero(s) of the function. Use Newton’s Method and continue the process until two successive approximations differ by less than 0.001. Then find the zero(s) using a graphing utility and compare the results. \(f(x)=x^{4}+x^{3}-1\)

In Exercises 5–24, analyze and sketch a graph of the function. Label any intercepts, relative extrema, points of inflection, and asymptotes. Use a graphing utility to verify your results. $$ f(x)=\frac{x^{3}}{x^{2}-9} $$

Finding Critical Numbers In Exercises \(11-16,\) find the critical numbers of the function. $$ g(x)=x^{4}-8 x^{2} $$

Determining Concavity In Exercises \(3-14\) , determine the open intervals on which the graph is concave upward or concave downward. $$ h(x)=\frac{x^{2}-1}{2 x-1} $$

Intervals on Which \(f\) Is Increasing or Decreasing In Exercises \(9-16\) , identify the open intervals on which the function is increasing or decreasing. $$ y=x+\frac{9}{x} $$

Determine whether Rolle's Theorem can be applied to \(f\) on the closed interval \([a, b] .\) If Rolle's Theorem can be applied, find all values of \(c\) in the open interval \((a, b)\) such that \(f^{\prime}(c)=0 .\) If Rolle's Theorem cannot be applied, explain why not. \(f(x)=(x-4)(x+2)^{2}, \quad[-2,4]\)

Finding a Differential In Exercises \(11-20\) , find the differential \(d y\) of the given function. $$ y=x \tan x $$

Minimum Distance In Exercises \(13-16,\) find the point on the graph of the function that is closest to the given point. $$ f(x)=x^{2},\left(2, \frac{1}{2}\right) $$

Finding Limits at Infinity In Exercises 13 and 14, find \(\lim _{x \rightarrow \infty} h(x),\) if possible. $$ \begin{array}{l}{f(x)=5 x^{3}-3 x^{2}+10 x} \\ {\text { (a) } h(x)=\frac{f(x)}{x^{2}}} \\ {\text { (b) } h(x)=\frac{f(x)}{x^{3}}} \\\ {\text { (c) } h(x)=\frac{f(x)}{x^{4}}}\end{array} $$

Approximate the zero(s) of the function. Use Newton’s Method and continue the process until two successive approximations differ by less than 0.001. Then find the zero(s) using a graphing utility and compare the results. \(f(x)=1-x+\sin x\)

In Exercises 5–24, analyze and sketch a graph of the function. Label any intercepts, relative extrema, points of inflection, and asymptotes. Use a graphing utility to verify your results. $$ y=\frac{x^{2}-6 x+12}{x-4} $$

Finding Critical Numbers In Exercises \(11-16,\) find the critical numbers of the function. $$ g(t)=t \sqrt{4-t}, t<3 $$

Determining Concavity In Exercises \(3-14\) , determine the open intervals on which the graph is concave upward or concave downward. $$ y=2 x-\tan x, \quad\left(-\frac{\pi}{2}, \frac{\pi}{2}\right) $$

Intervals on Which \(f\) Is Increasing or Decreasing In Exercises \(9-16\) , identify the open intervals on which the function is increasing or decreasing. $$ f(x)=\sin x-1, \quad 0 < x < 2 \pi $$

Determine whether Rolle's Theorem can be applied to \(f\) on the closed interval \([a, b] .\) If Rolle's Theorem can be applied, find all values of \(c\) in the open interval \((a, b)\) such that \(f^{\prime}(c)=0 .\) If Rolle's Theorem cannot be applied, explain why not. \(f(x)=x^{2 / 3}-1, \quad[-8,8]\)

Finding a Differential In Exercises \(11-20\) , find the differential \(d y\) of the given function. $$ y=\csc 2 x $$

Minimum Distance In Exercises \(13-16,\) find the point on the graph of the function that is closest to the given point. $$ f(x)=(x-1)^{2},(-5,3) $$

Finding Limits at Infinity In Exercises 13 and 14, find \(\lim _{x \rightarrow \infty} h(x),\) if possible. $$ \begin{array}{l}{f(x)=-4 x^{2}+2 x-5} \\ {\text { (a) } h(x)=\frac{f(x)}{x}} \\\ {\text { (b) } h(x)=\frac{f(x)}{x^{2}}} \\ {\text { (c) } h(x)=\frac{f(x)}{x^{3}}}\end{array} $$

Approximate the zero(s) of the function. Use Newton’s Method and continue the process until two successive approximations differ by less than 0.001. Then find the zero(s) using a graphing utility and compare the results. \(f(x)=x^{3}-\cos x\)

In Exercises 5–24, analyze and sketch a graph of the function. Label any intercepts, relative extrema, points of inflection, and asymptotes. Use a graphing utility to verify your results. $$ y=\frac{-x^{2}-4 x-7}{x+3} $$

Finding Critical Numbers In Exercises \(11-16,\) find the critical numbers of the function. $$ f(x)=\frac{4 x}{x^{2}+1} $$

Determining Concavity In Exercises \(3-14\) , determine the open intervals on which the graph is concave upward or concave downward. $$ y=x+\frac{2}{\sin x}, \quad(-\pi, \pi) $$

Intervals on Which \(f\) Is Increasing or Decreasing In Exercises \(9-16\) , identify the open intervals on which the function is increasing or decreasing. $$ h(x)=\cos \frac{x}{2}, \quad 0 < x < 2 \pi $$

Determine whether Rolle's Theorem can be applied to \(f\) on the closed interval \([a, b] .\) If Rolle's Theorem can be applied, find all values of \(c\) in the open interval \((a, b)\) such that \(f^{\prime}(c)=0 .\) If Rolle's Theorem cannot be applied, explain why not. \(f(x)=3-|x-3|, \quad[0,6]\)

Finding a Differential In Exercises \(11-20\) , find the differential \(d y\) of the given function. $$ y=\frac{x+1}{2 x-1} $$

Finding Limits at Infinity In Exercises \(15-18\) , find each limit, if possible. $$ \begin{array}{l}{\text { (a) } \lim _{x \rightarrow \infty} \frac{x^{2}+2}{x^{3}-1}} \\ {\text { (b) } \lim _{x \rightarrow \infty} \frac{x^{2}+2}{x^{2}-1}} \\ {\text { (c) } \lim _{x \rightarrow \infty} \frac{x^{2}+2}{x-1}}\end{array} $$

Apply Newton's Method to approximate the \(x\) -value(s) of the indicated point(s) of intersection of the two graphs. Continue the process until two successive approximations differ by less than 0.001 .[Hint: Let \(h(x)=f(x)-g(x).\)] \(f(x)=2 x+1\) \(g(x)=\sqrt{x+4}\)

In Exercises 5–24, analyze and sketch a graph of the function. Label any intercepts, relative extrema, points of inflection, and asymptotes. Use a graphing utility to verify your results. $$ y=x \sqrt{4-x} $$

Finding Critical Numbers In Exercises \(11-16,\) find the critical numbers of the function. $$ \begin{array}{l}{h(x)=\sin ^{2} x+\cos x} \\ {0