Chapter 3

Using a Tangent Line Approximation In Exercises \(1-6,\) find the tangent line approximation \(T\) to the graph of \(f\) at the given point. Use this linear approximation to complete the table. $$ \begin{array}{|c|c|c|c|c|c|}\hline x & {1.9} & {1.99} & {2} & {2.01} & {2.1} \\\ \hline f(x) & {} & {} \\ \hline T(x) & {} & {} \\ \hline\end{array} $$ $$ f(x)=x^{2}, \quad(2,4) $$

Numerical, Graphical, and Analytic Analysis 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.) $$ \begin{array}{|c|c|c|}\hline \text { First } & {\text { Second }} \\ {\text { Number, } x} & {\text { Number }} & {\text { Product, } P} \\ \hline 10 & {110-10} & {10(110-10)=1000} \\ \hline 20 & {110-20} & {20(110-20)=1800} \\\ \hline\end{array} $$ (b) Use a graphing utility to generate additional rows of the table. Use the table to estimate the solution. (Hint: Use the table feature of the graphing utility.) (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.

Complete two iterations of Newton’s Method to approximate a zero of the function using the given initial guess. \(f(x)=x^{2}-5, \quad x_{1}=2.2\)

Finding the Value of the Derivative at Relative Extrema In Exercises \(1-6,\) find the value of the derivative (if it exists) at each indicated extremum. $$ f(x)=\frac{x^{2}}{x^{2}+4} $$

Using a Tangent Line Approximation In Exercises \(1-6,\) find the tangent line approximation \(T\) to the graph of \(f\) at the given point. Use this linear approximation to complete the table. $$ \begin{array}{|c|c|c|c|c|c|}\hline x & {1.9} & {1.99} & {2} & {2.01} & {2.1} \\\ \hline f(x) & {} & {} \\ \hline T(x) & {} & {} \\ \hline\end{array} $$ $$ f(x)=\frac{6}{x^{2}}, \quad\left(2, \frac{3}{2}\right) $$

Numerical, Graphical, and Analytic Analysis 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. \(\begin{array}{|c|c|c|}\hline \text { Height, } x & {\text { Length and Width }} & {\text { Volume, } V} \\ \hline 1 & {24-2(1)} & {1[24-2(1)]^{2}=484} \\\ \hline 2 & {24-2(2)} & {2[24-2(2)]^{2}=800} \\ \hline\end{array}\) (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.

Complete two iterations of Newton’s Method to approximate a zero of the function using the given initial guess. \(f(x)=x^{3}-3, \quad x_{1}=1.4\)

Finding the Value of the Derivative at Relative Extrema In Exercises \(1-6,\) find the value of the derivative (if it exists) at each indicated extremum. $$ f(x)=\cos \frac{\pi x}{2} $$

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

Using a Tangent Line Approximation In Exercises \(1-6,\) find the tangent line approximation \(T\) to the graph of \(f\) at the given point. Use this linear approximation to complete the table. $$ \begin{array}{|c|c|c|c|c|c|}\hline x & {1.9} & {1.99} & {2} & {2.01} & {2.1} \\\ \hline f(x) & {} & {} \\ \hline T(x) & {} & {} \\ \hline\end{array} $$ $$ f(x)=x^{5}, \quad(2,32) $$

Complete two iterations of Newton’s Method to approximate a zero of the function using the given initial guess. \(f(x)=\cos x, \quad x_{1}=1.6\)

Finding the Value of the Derivative at Relative Extrema In Exercises \(1-6,\) find the value of the derivative (if it exists) at each indicated extremum. $$ g(x)=x+\frac{4}{x^{2}} $$

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

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

Using a Tangent Line Approximation In Exercises \(1-6,\) find the tangent line approximation \(T\) to the graph of \(f\) at the given point. Use this linear approximation to complete the table. $$ \begin{array}{|c|c|c|c|c|c|}\hline x & {1.9} & {1.99} & {2} & {2.01} & {2.1} \\\ \hline f(x) & {} & {} \\ \hline T(x) & {} & {} \\ \hline\end{array} $$ $$ f(x)=\sqrt{x}, \quad(2, \sqrt{2}) $$

Finding Numbers In Exercises \(3-8,\) find two positive numbers that satisfy the given requirements. The product is 185 and the sum is a minimum.

Complete two iterations of Newton’s Method to approximate a zero of the function using the given initial guess. \(f(x)=\tan x, \quad x_{1}=0.1\)

Finding the Value of the Derivative at Relative Extrema In Exercises \(1-6,\) find the value of the derivative (if it exists) at each indicated extremum. $$ f(x)=-3 x \sqrt{x+1} $$

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

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

Using a Tangent Line Approximation In Exercises \(1-6,\) find the tangent line approximation \(T\) to the graph of \(f\) at the given point. Use this linear approximation to complete the table. $$ \begin{array}{|c|c|c|c|c|c|}\hline x & {1.9} & {1.99} & {2} & {2.01} & {2.1} \\\ \hline f(x) & {} & {} \\ \hline T(x) & {} & {} \\ \hline\end{array} $$ $$ f(x)=\sin x, \quad(2, \sin 2) $$

Finding Numbers In Exercises \(3-8,\) find two positive numbers that satisfy the given requirements. The product is 147 and the sum of the first number plus three times the second number is a minimum.

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. \(f(x)=x^{3}+4\)

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. $$ y=\frac{1}{x-2}-3 $$

Finding the Value of the Derivative at Relative Extrema In Exercises \(1-6,\) find the value of the derivative (if it exists) at each indicated extremum. $$ f(x)=(x+2)^{2 / 3} $$

Determining Concavity In Exercises \(3-14\) , determine the open intervals on which the graph is concave upward or concave downward. $$ f(x)=-x^{3}+6 x^{2}-9 x-1 $$

Find the two \(x\)-intercepts of the function \(f\) and show that \(f^{\prime}(x)=0\) at some point between the two \(x\)-intercepts. \(f(x)=x^{2}-x-2\)

Using a Tangent Line Approximation In Exercises \(1-6,\) find the tangent line approximation \(T\) to the graph of \(f\) at the given point. Use this linear approximation to complete the table. $$ \begin{array}{|c|c|c|c|c|c|}\hline x & {1.9} & {1.99} & {2} & {2.01} & {2.1} \\\ \hline f(x) & {} & {} \\ \hline T(x) & {} & {} \\ \hline\end{array} $$ $$ f(x)=\csc x, \quad(2, \csc 2) $$

Finding Numbers In Exercises \(3-8,\) find two positive numbers that satisfy the given requirements. The second number is the reciprocal of the first number and the sum is a minimum.

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. \(f(x)=2-x^{3}\)

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. $$ y=\frac{x}{x^{2}+1} $$

Finding the Value of the Derivative at Relative Extrema In Exercises \(1-6,\) find the value of the derivative (if it exists) at each indicated extremum. $$ f(x)=4-|x| $$

Determining Concavity In Exercises \(3-14\) , determine the open intervals on which the graph is concave upward or concave downward. $$ h(x)=x^{5}-5 x+2 $$

Comparing \(\Delta y\) and \(d y\) In Exercises \(7-10\) , use the information to evaluate and compare \(\Delta y\) and \(d y .\) Comparing \(\Delta y\) and \(d y\) In Exercises \(7-10\) , use the information to evaluate and compare \(\Delta y\) and \(d y .\) $$ \begin{array}{ll}{\text { Function }} & {x \text { -Value }} \\ {y=x^{3}} & {x=1}\end{array} \quad \Delta x=d x=0.1 $$

Finding Numbers In Exercises \(3-8,\) find two positive numbers that satisfy the given requirements. The sum of the first number and twice the second number is 108 and the product is a maximum.

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. $$ \begin{array}{|c|c|c|c|c|c|c|}\hline x & {10^{0}} & {10^{1}} & {10^{2}} & {10^{3}} & {10^{4}} & {10^{5}} & {10^{6}} \\ \hline f(x) & {} & {} & {} & {} \\\ \hline\end{array} $$ $$ f(x)=\frac{4 x+3}{2 x-1} $$

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. \(f(x)=x^{3}+x-1\)

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. $$ y=\frac{x^{2}}{x^{2}+3} $$

Determining Concavity In Exercises \(3-14\) , determine the open intervals on which the graph is concave upward or concave downward. $$ f(x)=\frac{24}{x^{2}+12} $$

Comparing \(\Delta y\) and \(d y\) In Exercises \(7-10\) , use the information to evaluate and compare \(\Delta y\) and \(d y .\) $$ \begin{array}{ll}{\text { Function }} & {x \text { -Value }} \\ {y=6-2 x^{2}} & {x=-2}\end{array} \quad \Delta x=d x=0.1 $$

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. $$ \begin{array}{|c|c|c|c|c|c|c|}\hline x & {10^{0}} & {10^{1}} & {10^{2}} & {10^{3}} & {10^{4}} & {10^{5}} & {10^{6}} \\ \hline f(x) & {} & {} & {} & {} \\\ \hline\end{array} $$ $$ f(x)=\frac{2 x^{2}}{x+1} $$

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. \(f(x)=x^{5}+x-1\)

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. $$ y=\frac{x^{2}+1}{x^{2}-4} $$

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

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

Comparing \(\Delta y\) and \(d y\) In Exercises \(7-10\) , use the information to evaluate and compare \(\Delta y\) and \(d y .\) $$ \begin{array}{ll}{\text { Function }} & {x \text { -Value }} \\ {y=x^{4}+1} & {x=-1}\end{array} \quad \Delta x=d x=0.01 $$

Maximum Area In Exercises 9 and \(10,\) find the length and width of a rectangle that has the given perimeter and a maximum area. Perimeter: 80 meters

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. $$ \begin{array}{|c|c|c|c|c|c|c|}\hline x & {10^{0}} & {10^{1}} & {10^{2}} & {10^{3}} & {10^{4}} & {10^{5}} & {10^{6}} \\ \hline f(x) & {} & {} & {} & {} \\\ \hline\end{array} $$ $$ f(x)=\frac{-6 x}{\sqrt{4 x^{2}+5}} $$

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. \(f(x)=5 \sqrt{x-1}-2 x\)

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. $$ y=\frac{3 x}{x^{2}-1} $$