Chapter 3

Use Newton's method to find the zero(s) of \(\bar{f}\) to four decimal places by solving the equation \(f(x)=0 .\) Use the initial estimate \((s) x_{0}\). \(f(x)=-x^{3}-2 x+2, \quad x_{0}=1\)

evaluate the limit using l'Hôpital's Rule if appropriate. $$ \lim _{x \rightarrow 1} \frac{x-1}{x^{2}-1} $$

Find two positive numbers whose sum is 100 and whose product is a maximum.

In Exercises \(1-4\), use the information summarized in the table to sketch the graph of \(\bar{f}\). $$ f(x)=x^{3}-3 x^{2}+1 $$ $$ \begin{array}{|l|l|} \hline \text { Domain } & (-\infty, \infty) \\ \text { Intercepts } & y \text { -intercept: } 1 \\ \text { Symmetry } & \text { None } \\ \text { Asymptotes } & \text { None } \\ \text { Intervals where } f \text { is / or } \backslash & \ \text { on }(-\infty, 0) \text { and on }(2, \infty) ; \\ & \searrow \text { on }(0,2) \\ \text { Relative extrema } & \text { Rel. max. at }(0,1) ; \\ \text { Concavity } & \text { rel. min. at }(2,-3) \\ \text { Point of inflection } & \begin{array}{l} \text { Downward on }(-\infty, 1) ; \\ \text { upward on }(1, \infty) \\\ (1,-1) \end{array} \\ \hline \end{array} $$

In Exencises \(1-8\), verify that the function satisfies the hypotheses of Rolle's Theorem on the given interval, and find all values of \(c\) that satisfy the conclusion of the theorem. $$ f(x)=x^{2}-4 x+3 ; \quad[1,3] $$

Use Newton's method to find the zero(s) of \(\bar{f}\) to four decimal places by solving the equation \(f(x)=2 x^{3}-15 x^{2}+36 x-20, \quad x_{0}=1\)

evaluate the limit using l'Hôpital's Rule if appropriate. $$ \lim _{x \rightarrow-1} \frac{x^{2}-2 x-3}{x+1} $$

Find two numbers whose difference is 50 and whose product is a minimum.

In Exercises \(1-4\), use the information summarized in the table to sketch the graph of \(\bar{f}\). $$ f(x)=\frac{1}{9}\left(x^{4}-4 x^{3}\right) $$ $$ \begin{array}{|l|l|} \hline \text { Domain } & (-\infty, \infty) \\ \text { Intercepts } & x \text { -intercepts: } 0,4 \\ & y \text { -intercept: } 0 \\ \text { Symmetry } & \text { None } \\ \text { Asymptotes } & \text { None } \\ \text { Intervals where } f \text { is / or } \backslash & \text { / on }(3, \infty) ; \\ & \searrow \text { on }(-\infty, 3) \\ \text { Relative extrema } & \text { Rel. min. at }(3,-3) \\ \text { Concavity } & \text { Downward on }(0,2) ; \\ & \text { upward on }(-\infty, 0) \text { and on } \\ & (2, \infty) \\ \text { Points of inflection } & (0,0) \text { and }\left(2,-\frac{16}{9}\right) \\ \hline \end{array} $$

Use Newton's method to find the zero(s) of \(f\) to four decimal places by solving the equation \(f(x)=0 .\) Use the initial estimate \((s) x_{0}\). \(f(x)=\frac{3}{2} x^{4}-2 x^{3}-6 x^{2}+8, \quad x_{0}=1\) and \(x_{0}=3\)

evaluate the limit using l'Hôpital's Rule if appropriate. $$ \lim _{x \rightarrow 2} \frac{x^{3}-8}{x-2} $$

In Exercises \(1-4\), use the information summarized in the table to sketch the graph of \(\bar{f}\). $$ f(x)=\frac{4 x-4}{x^{2}} $$ $$ \begin{array}{|l|l|} \hline \text { Domain } & (-\infty, 0) \cup(0, \infty) \\ \text { Intercepts } & x \text { -intercept: } 1 \\ \text { Symmetry } & \text { None } \\ \text { Asymptotes } & x \text { -axis; } y \text { -axis } \\ \text { Intervals where } f \text { is } / \text { or } \backslash & \begin{array}{l} / \text { on }(0,2) ; \\ \text { and on }(2, \infty) \end{array} \\ \text { Relative extrema } & \text { Rel. max. at }(2,1) \\ \text { Concavity } & \text { Downward on }(-\infty, 0) \text { and } \\ & \text { on }(0,3) ; \text { upward on }(3, \infty) \\ \text { Point of inflection } & \left(3, \frac{8}{9}\right) \\ \hline \end{array} $$

Use the graph of the function \(f\) to find the given limits. a. \(\lim _{x \rightarrow 0} f(x)\) b. \(\lim _{x \rightarrow-x} f(x)\) c. \(\lim _{x \rightarrow \infty} f(x)\)

In Exencises \(1-8\), verify that the function satisfies the hypotheses of Rolle's Theorem on the given interval, and find all values of \(c\) that satisfy the conclusion of the theorem. $$ f(x)=x^{3}+x^{2}-2 x ; \quad[-2,0] $$

Use Newton's method to find the zero(s) of \(f\) to four decimal places by solving the equation \(f(x)=0 .\) Use the initial estimate \((s) x_{0}\). \(f(x)=x-\sqrt{1-x^{2}}, \quad x_{0}=0.5\)

evaluate the limit using l'Hôpital's Rule if appropriate. $$ \lim _{x \rightarrow 1} \frac{x^{7}-1}{x^{4}-1} $$

In Exercises \(1-4\), use the information summarized in the table to sketch the graph of \(\bar{f}\). $$ f(x)=x-3 x^{1 / 3} $$ $$ \begin{array}{|l|l|} \hline \text { Domain } & (-\infty, \infty) \\ \text { Intercepts } & x \text { -intercepts: } \pm 3 \sqrt{3}, 0 ; \\ & y \text { -intercept: } 0 \\ \text { Symmetry } & \text { With respect to the origin } \\ \text { Asymptotes } & \text { None } \\ \text { Intervals where } f \text { is / or } \backslash & \begin{array}{l} \prime \text { on }(-\infty,-1) \text { and on } \\ (1, \infty) ; \backslash \text { on }(-1,1) \end{array} \\ \text { Relative extrema } & \begin{array}{l} \text { Rel. max. at }(-1,2) ; \\ \text { rel. min. at }(1,-2) \end{array} \\ \text { Concavity } & \begin{array}{l} \text { Downward on }(-\infty, 0) \\ \text { upward on }(0, \infty) \\ \text { Point of inflection } & (0,0) \end{array} \\ \hline \end{array} $$

In Exencises \(1-8\), verify that the function satisfies the hypotheses of Rolle's Theorem on the given interval, and find all values of \(c\) that satisfy the conclusion of the theorem. $$ h(x)=x^{3}(x-7)^{4} ; \quad[0,7] $$

Use Newton's method to find the point of intersection of the graphs to four decimal places of accuracy by solving the equation \(f(x)-g(x)=0 .\) Use the initial estimate \(x_{0}\) for the \(x\) -coordinate. f(x)=x^{2}, g(x)=\sin x, \quad x_{0}=1

evaluate the limit using l'Hôpital's Rule if appropriate. $$ \lim _{x \rightarrow 0} \frac{e^{x}-1}{x^{2}+x} $$

In Exercises \(5-38\), sketch the graph of the function using the curve- sketching guidelines on page \(348 .\) $$ f(x)=4-3 x-2 x^{3} $$

In Exencises \(1-8\), verify that the function satisfies the hypotheses of Rolle's Theorem on the given interval, and find all values of \(c\) that satisfy the conclusion of the theorem. $$ f(x)=x \sqrt{1-x^{2}} ; \quad[-1,1] $$

Use Newton's method to find the point of intersection of the graphs to four decimal places of accuracy by solving the equation \(f(x)-g(x)=0 .\) Use the initial estimate \(x_{0}\) for the \(x\) -coordinate. f(x)=\tan x, g(x)=1-x, \quad x_{0}=1

evaluate the limit using l'Hôpital's Rule if appropriate. $$ \lim _{x \rightarrow 1} \frac{\ln x}{x-1} $$

Find the dimensions of a rectangle of area \(144 \mathrm{ft}^{2}\) that has the smallest possible perimeter.

In Exercises \(5-38\), sketch the graph of the function using the curve- sketching guidelines on page \(348 .\) $$ f(x)=x^{3}-3 x^{2}+2 $$

In Exencises \(1-8\), verify that the function satisfies the hypotheses of Rolle's Theorem on the given interval, and find all values of \(c\) that satisfy the conclusion of the theorem. $$ f(t)=t^{2 / 3}(6-t)^{1 / 3} ; \quad[0,6] $$

Use Newton's method to find the point of intersection of the graphs to four decimal places of accuracy by solving the equation \(f(x)-g(x)=0 .\) Use the initial estimate \(x_{0}\) for the \(x\) -coordinate. f(x)=\frac{1}{2} \cos x, g(x)=x, \quad x_{0}=0.5

evaluate the limit using l'Hôpital's Rule if appropriate. $$ \lim _{t \rightarrow \pi} \frac{\sin t}{\pi-t} $$

A Fencing Problem A rancher has \(400 \mathrm{ft}\) of fencing with which to enclose two adjacent rectangular parts of a corral. What are the dimensions of the parts if the area enclosed is to be as large as possible and she uses all of the fencing available?

In Exercises \(5-38\), sketch the graph of the function using the curve- sketching guidelines on page \(348 .\) $$ f(x)=x^{3}-6 x^{2}+9 x+2 $$

In Exercises \(7-36\), find the limit. $$ \lim _{x \rightarrow-1^{-}} \frac{1}{x+1} $$

In Exencises \(1-8\), verify that the function satisfies the hypotheses of Rolle's Theorem on the given interval, and find all values of \(c\) that satisfy the conclusion of the theorem. $$ h(t)=\sin ^{2} t ; \quad[0, \pi] $$

In Exercises \(7-24\), sketch the graph of the function and find its absolute maximum and absolute minimum values, if any. $$ f(x)=2 x+3 \text { on }[-1, \infty) $$

Use Newton's method to find the point of intersection of the graphs to four decimal places of accuracy by solving the equation \(f(x)-g(x)=0 .\) Use the initial estimate \(x_{0}\) for the \(x\) -coordinate. f(x)=\sin x, g(x)=\frac{1}{5} x, \quad x_{0}=2

evaluate the limit using l'Hôpital's Rule if appropriate. $$ \lim _{x \rightarrow 0} \frac{e^{x}-1}{x+\sin x} $$

A Fencing Problem The owner of the Rancho Grande has 3000 yd of fencing with which to enclose a rectangular piece of grazing land situated along the straight portion of a river. If fencing is not required along the river, what are the dimensions of the largest area he can enclose? What is the area?

In Exercises \(5-38\), sketch the graph of the function using the curve- sketching guidelines on page \(348 .\) $$ y=2 t^{3}-15 t^{2}+36 t-20 $$

Find the limit. $$ \lim _{t \rightarrow-3^{+}} \frac{t}{t+3} $$

In Exencises \(1-8\), verify that the function satisfies the hypotheses of Rolle's Theorem on the given interval, and find all values of \(c\) that satisfy the conclusion of the theorem. $$ f(x)=\cos 2 x-1 ; \quad[0, \pi] $$

In Exercises \(7-24\), sketch the graph of the function and find its absolute maximum and absolute minimum values, if any. $$ g(x)=-3 x+2 \text { on }(-1,2] $$

Use Newton's method to approximate the indicated zero of the function. Continue with the iteration until two successive approximations differ by less than \(0.0001\). The zero of \(f(x)=x^{3}+x-4\) between \(x=0\) and \(x=2\). Take \(x_{0}=1\).

evaluate the limit using l'Hôpital's Rule if appropriate. $$ \lim _{\theta \rightarrow 0} \frac{\tan 2 \theta}{\theta} $$

In Exercises \(5-38\), sketch the graph of the function using the curve- sketching guidelines on page \(348 .\) $$ f(x)=2 x^{3}-9 x^{2}+12 x-3 $$

Packaging An open box is made from a rectangular piece of cardboard of dimensions \(16 \times 10\) in. by cutting out identical squares from each corner and bending up the resulting flaps. Find the dimensions of the box with the largest volume that can be made.

(a) find the intervals on which \(f\) is increasing or decreasing, and (b) find the relative maxima and relative minima of \(\vec{f}\). $$ f(x)=x^{2}-2 x $$

Find the limit. $$ \lim _{x \rightarrow 1^{-}} \frac{1+x}{1-x} $$

In Exercises 9-16, verify that the function satisfies the hypotheses of the Mean Value Theorem on the given interval, and find all values of \(c\) that satisfy the conclusion of the theorem. $$ f(x)=x^{2}+1 ; \quad[0,2] $$

In Exercises \(7-24\), sketch the graph of the function and find its absolute maximum and absolute minimum values, if any. $$ h(t)=t^{2}-1 \text { on }(-1,0) $$

Use Newton's method to approximate the indicated zero of the function. Continue with the iteration until two successive approximations differ by less than \(0.0001\). \begin{array}{l} \text { The zero of } f(x)=x^{3}+2 x^{2}+x-6 \text { between } x=1 \text { and }\\\ x=2 . \text { Take } x_{0}=1.5 \text { . } \end{array}