Chapter 3

In Exercises \(1-8\), determine the open intervals on which the graph is concave upward or concave downward. \(y=x^{2}-x-2\)

Find two positive numbers whose sum is 110 and whose product is a maximum. (a) Analytically complete six rows of a table such as the one below. (The first two rows are shown.) $$ \begin{array}{|c|c|c|} \hline \begin{array}{c} \text { First } \\ \text { Number } \boldsymbol{x} \end{array} & \begin{array}{c} \text { Second } \\ \text { Number } \end{array} & \ {\text { Product } \boldsymbol{P}} \\ \hline 10 & 110-10 & 10(110-10)=1000 \\ \hline 20 & 110-20 & 20(110-20)=1800 \\ \hline \end{array} $$ (b) Use a graphing utility to generate additional rows of the table. Use the table to estimate the solution. (c) Write the product \(P\) as a function of \(x\). (d) Use a graphing utility to graph the function in part (c) and estimate the solution from the graph. (e) Use calculus to find the critical number of the function in part (c). Then find the two numbers.

In Exercises 1 and \(2,\) describe in your own words what the statement means. $$ \lim _{x \rightarrow \infty} f(x)=4 $$

In Exercises \(1-4,\) explain why Rolle's Theorem does not apply to the function even though there exist \(a\) and \(b\) such that \(f(a)=f(b)\). $$ f(x)=1-|x-1|, \quad[0,2] $$

Determine the open intervals on which the graph is concave upward or concave downward. \(y=-x^{3}+3 x^{2}-2\)

An open box of maximum volume is to be made from a square piece of material, 24 inches on a side, by cutting equal squares from the corners and turning up the sides (see figure). (a) Analytically complete six rows of a table such as the one below. (The first two rows are shown.) Use the table to guess the maximum volume. $$ \begin{array}{|c|c|c|} \hline \text { Height } & \begin{array}{c} \text { Length and } \\ \text { Width } \end{array} & \text { Volume } \\ \hline 1 & 24-2(1) & 1[24-2(1)]^{2}=484 \\ \hline 2 & 24-2(2) & 2[24-2(2)]^{2}=800 \\ \hline \end{array} $$ (b) Write the volume \(V\) as a function of \(x\). (c) Use calculus to find the critical number of the function in part (b) and find the maximum value. (d) Use a graphing utility to graph the function in part (b) and verify the maximum volume from the graph.

In Exercises 1 and \(2,\) describe in your own words what the statement means. $$ \lim _{x \rightarrow-\infty} f(x)=2 $$

Explain why Rolle's Theorem does not apply to the function even though there exist \(a\) and \(b\) such that \(f(a)=f(b)\). $$ f(x)=\cot \frac{x}{2}, \quad[\pi, 3 \pi] $$

Numerical and Graphical Analysis In Exercises 3-8, 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. $$ \begin{array}{|l|l|l|l|l|l|l|l|} \hline x & 10^{0} & 10^{1} & 10^{2} & 10^{3} & 10^{4} & 10^{5} & 10^{6} \\ \hline f(x) & & & & & & & \\ \hline \end{array} $$ $$ f(x)=\frac{4 x+3}{2 x-1} $$

Identify the open intervals on which the function is increasing or decreasing. $$ f(x)=x^{2}-6 x+8 $$

Determine the open intervals on which the graph is concave upward or concave downward. \(y=\frac{-3 x^{5}+40 x^{3}+135 x}{270}\)

Find two positive numbers that satisfy the given requirements. The product is 192 and the sum is a minimum.

Numerical and Graphical Analysis In Exercises 3-8, 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. $$ \begin{array}{|l|l|l|l|l|l|l|l|} \hline x & 10^{0} & 10^{1} & 10^{2} & 10^{3} & 10^{4} & 10^{5} & 10^{6} \\ \hline f(x) & & & & & & & \\ \hline \end{array} $$ $$ f(x)=\frac{2 x^{2}}{x+1} $$

Identify the open intervals on which the function is increasing or decreasing. $$ y=-(x+1)^{2} $$

Explain why Rolle's Theorem does not apply to the function even though there exist \(a\) and \(b\) such that \(f(a)=f(b)\). $$ \begin{array}{l} f(x)=\sqrt{\left(2-x^{2 / 3}\right)^{3}} \\ {[-1,1]} \end{array} $$

Determine the open intervals on which the graph is concave upward or concave downward. \(g(x)=3 x^{2}-x^{3}\)

Find two positive numbers that satisfy the given requirements. The product is 192 and the sum of the first plus three times the second is a minimum.

In Exercises 5 and \(6,\) use the information to evaluate and compare \(\Delta y\) and \(d y\). $$ y=\frac{1}{2} x^{3} \quad x=2 \quad \Delta x=d x=0.1 $$

Numerical and Graphical Analysis In Exercises 3-8, 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. $$ \begin{array}{|l|l|l|l|l|l|l|l|} \hline x & 10^{0} & 10^{1} & 10^{2} & 10^{3} & 10^{4} & 10^{5} & 10^{6} \\ \hline f(x) & & & & & & & \\ \hline \end{array} $$ $$ f(x)=\frac{-6 x}{\sqrt{4 x^{2}+5}} $$

In Exercises \(5-8,\) find the two \(x\) -intercepts of the function \(f\) and show that \(f^{\prime}(x)=0\) at some point between the two \(x\) -intercepts. $$ f(x)=x^{2}-x-2 $$

Identify the open intervals on which the function is increasing or decreasing. $$ y=\frac{x^{3}}{4}-3 x $$

Determine the open intervals on which the graph is concave upward or concave downward. \(h(x)=x^{5}-5 x+2\)

Sse the information to evaluate and compare \(\Delta y\) and \(d y\). $$ y=x^{4}+1 \quad x=-1 \quad \Delta x=d x=0.01 $$

Find two positive numbers that satisfy the given requirements. The second number is the reciprocal of the first and the sum is a minimum.

Numerical and Graphical Analysis In Exercises 3-8, 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. $$ \begin{array}{|l|l|l|l|l|l|l|l|} \hline x & 10^{0} & 10^{1} & 10^{2} & 10^{3} & 10^{4} & 10^{5} & 10^{6} \\ \hline f(x) & & & & & & & \\ \hline \end{array} $$ $$ f(x)=\frac{8 x}{\sqrt{x^{2}-3}} $$

Find the two \(x\) -intercepts of the function \(f\) and show that \(f^{\prime}(x)=0\) at some point between the two \(x\) -intercepts. $$ f(x)=x(x-3) $$

Identify the open intervals on which the function is increasing or decreasing. $$ f(x)=x^{4}-2 x^{2} $$

In Exercises \(7-14,\) find the differential \(d y\) of the given function. $$ y=3 x^{2}-4 $$

Determine the open intervals on which the graph is concave upward or concave downward. \(y=2 x-\tan x, \quad\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\)

Numerical and Graphical Analysis In Exercises 3-8, 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. $$ \begin{array}{|l|l|l|l|l|l|l|l|} \hline x & 10^{0} & 10^{1} & 10^{2} & 10^{3} & 10^{4} & 10^{5} & 10^{6} \\ \hline f(x) & & & & & & & \\ \hline \end{array} $$ $$ f(x)=5-\frac{1}{x^{2}+1} $$

Find the two \(x\) -intercepts of the function \(f\) and show that \(f^{\prime}(x)=0\) at some point between the two \(x\) -intercepts. $$ f(x)=x \sqrt{x+4} $$

Identify the open intervals on which the function is increasing or decreasing. $$ f(x)=\sin x+2,0

Find the differential \(d y\) of the given function. $$ y=\sqrt{9-x^{2}} $$

Determine the open intervals on which the graph is concave upward or concave downward. \(y=x+\frac{2}{\sin x}, \quad(-\pi, \pi)\)

Numerical and Graphical Analysis In Exercises 3-8, 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. $$ \begin{array}{|l|l|l|l|l|l|l|l|} \hline x & 10^{0} & 10^{1} & 10^{2} & 10^{3} & 10^{4} & 10^{5} & 10^{6} \\ \hline f(x) & & & & & & & \\ \hline \end{array} $$ $$ f(x)=4+\frac{3}{x^{2}+2} $$

Find the two \(x\) -intercepts of the function \(f\) and show that \(f^{\prime}(x)=0\) at some point between the two \(x\) -intercepts. $$ f(x)=-3 x \sqrt{x+1} $$

Identify the open intervals on which the function is increasing or decreasing. $$ h(x)=\cos \frac{x}{2}, 0

Find the length and width of a rectangle that has the given perimeter and a maximum area. Perimeter: 100 meters

Find the differential \(d y\) of the given function. $$ y=\ln \sqrt{4-x^{2}} $$

In Exercises \(9-20\), find the points of inflection and discuss the concavity of the graph of the function. \(f(x)=x^{3}-6 x^{2}+12 x\)

In Exercises 9 and \(10,\) find \(\lim _{x \rightarrow \infty} h(x),\) if possible. \(f(x)=5 x^{3}-3 x^{2}+10\) (a) \(h(x)=\frac{f(x)}{x^{2}}\) (b) \(h(x)=\frac{f(x)}{x^{3}}\) (c) \(h(x)=\frac{f(x)}{x^{4}}\)

Identify the open intervals on which the function is increasing or decreasing. $$ f(x)=\frac{1}{x^{2}} $$

In Exercises 9-16, find any critical numbers of the function. $$ f(x)=x^{2}(x-3) $$

Find the differential \(d y\) of the given function. $$ y=\sqrt{x}+1 / \sqrt{x} $$

Find the points of inflection and discuss the concavity of the graph of the function. \(f(x)=2 x^{4}-8 x+3\)

In Exercises 9 and \(10,\) find \(\lim _{x \rightarrow \infty} h(x),\) if possible. \(f(x)=5 x^{2}-3 x+7\) (a) \(h(x)=\frac{f(x)}{x}\) (b) \(h(x)=\frac{f(x)}{x^{2}}\) (c) \(h(x)=\frac{f(x)}{x^{3}}\)

Identify the open intervals on which the function is increasing or decreasing. $$ y=\frac{x^{2}}{x+1} $$

Find any critical numbers of the function. $$ g(x)=x^{2}\left(x^{2}-4\right) $$

Find the differential \(d y\) of the given function. $$ y=2 x-\cot ^{2} x $$

Find the points of inflection and discuss the concavity of the graph of the function. \(f(x)=x(x-4)^{3}\)