Chapter 4

Use Newton's Method to approximate the indicated root of the given equation accurate to five decimal places. Begin by sketching a graph. $$ \text { The smallest positive root of } 2 \cot x=x $$

Find the general antiderivative \(F(x)+C\) for each of the following. $$ f(x)=x^{2}\left(x^{3}+5 x^{2}-3 x+\sqrt{3}\right) $$

, a function is defined and a closed interval is given. Decide whether the Mean Value Theorem applies to the given function on the given interval. If it does, find all possible values of \(c ;\) if not, state the reason. In each problem, sketch the graph of the given function on the given interval. $$ g(x)=x^{5 / 3} ;[-1,1] $$

Find the critical points and use the test of your choice to decide which critical points give a local maximum value and which give a local minimum value. What are these local maximum and minimum values? $$ f(x)=(x-2)^{5} $$

A farmer wishes to fence off three identical adjoining rectangular pens (see Figure 23), each with 300 square feet of area. What should the width and length of each pen be so that the least amount of fence is required?

Use the Concavity Theorem to determine where the given function is concave up and where it is concave down. Also find all inflection points. $$ f(z)=z^{2}-\frac{1}{z^{2}} $$

Identify the critical points and find the maximum value and minimum value on the given interval. $$ f(x)=x^{5}-\frac{25}{2} x^{3}+20 x-1 ; I=[-3,2] $$

Cesium- 137 and strontium-90 are two radioactive chemicals that were released at the Chernobyl nuclear reactor in April \(1986 .\) The half-life of cesium- 137 is \(30.22\) years, and that of strontium-90 is \(28.8\) years. In what year will the amount of cesium- 137 be equal to \(1 \%\) of what was released?

Find the general antiderivative \(F(x)+C\) for each of the following. $$ f(x)=\frac{3}{x^{2}}-\frac{2}{x^{3}} $$

, a function is defined and a closed interval is given. Decide whether the Mean Value Theorem applies to the given function on the given interval. If it does, find all possible values of \(c ;\) if not, state the reason. In each problem, sketch the graph of the given function on the given interval. $$ S(\theta)=\sin \theta ;[-\pi, \pi] $$

Find the critical points and use the test of your choice to decide which critical points give a local maximum value and which give a local minimum value. What are these local maximum and minimum values? $$ g(t)=\pi-(t-2)^{2 / 3} $$

Use the Concavity Theorem to determine where the given function is concave up and where it is concave down. Also find all inflection points. $$ q(x)=x^{4}-6 x^{3}-24 x^{2}+3 x+1 $$

Identify the critical points and find the maximum value and minimum value on the given interval. $$ g(x)=\frac{1}{1+x^{2}} ; I=(-\infty, \infty) $$

Find the \(x y\) -equation of the curve through \((1,2)\) whose slope at any point is three times the square of its \(y\) -coordinate.

Use Newton's Method to calculate \(\sqrt[4]{47}\) to five decimal places.

Find the general antiderivative \(F(x)+C\) for each of the following. $$ f(x)=\frac{\sqrt{2 x}}{x}+\frac{3}{x^{5}} $$

Find the critical points and use the test of your choice to decide which critical points give a local maximum value and which give a local minimum value. What are these local maximum and minimum values? $$ r(s)=3 s+s^{2 / 5} $$

Use the Concavity Theorem to determine where the given function is concave up and where it is concave down. Also find all inflection points. $$ f(x)=x^{4}+8 x^{3}-2 $$

Identify the critical points and find the maximum value and minimum value on the given interval. $$ f(x)=\frac{x}{1+x^{2}} ; I=[-1,4] $$

(Carbon Dating) All living things contain carbon 12 , which is stable, and carbon 14, which is radioactive. While a plant or animal is alive, the ratio of these two isotopes of carbon remains unchanged, since the carbon 14 is constantly renewed: after death, no more carbon 14 is absorbed. The half-life of carbon 14 is 5730 years. If charred logs of an old fort show only \(70 \%\) of the carbon 14 expected in living matter, when did the fort burn down? Assume that the fort burned soon after it was built of freshly cut logs.

Approximate the values of \(x\) that give maximum and minimum values of the function on the indicated intervals. $$ f(x)=x^{4}+x^{3}+x^{2}+x ;[-1,1] $$

Find the general antiderivative \(F(x)+C\) for each of the following. $$ f(x)=\frac{4 x^{6}+3 x^{4}}{x^{3}} $$

Find the points \(P\) and \(Q\) on the curve \(y=x^{2} / 4\), \(0 \leq x \leq 2 \sqrt{3}\), that are closest to and farthest from the point \((0,4) .\) Hint: The algebra is simpler if you consider the square of the required distance rather than the distance itself.

Find the critical points and use the test of your choice to decide which critical points give a local maximum value and which give a local minimum value. What are these local maximum and minimum values? $$ f(t)=t-\frac{1}{t}, t \neq 0 $$

Use the Concavity Theorem to determine where the given function is concave up and where it is concave down. Also find all inflection points. $$ F(x)=2 x^{2}+\cos ^{2} x $$

Identify the critical points and find the maximum value and minimum value on the given interval. $$ r(\theta)=\sin \theta ; I=\left[-\frac{\pi}{4}, \frac{\pi}{6}\right] $$

An object is moving along a coordinate line subject to the indicated acceleration a (in centimeters per second per second) with the initial velocity \(v_{0}\) (in centimeters per second) and directed distance \(s_{0}\) (in centimeters). Find both the velocity \(\underline{v}\) and directed distance \(s\) after 2 seconds (see Example 4). $$ a=(1+t)^{-4} ; v_{0}=0, s_{0}=10 $$

Approximate the values of \(x\) that give maximum and minimum values of the function on the indicated intervals. $$ f(x)=\frac{x^{3}+1}{x^{4}+1} ;[-4,4] $$

Find the general antiderivative \(F(x)+C\) for each of the following. $$ f(x)=\frac{x^{6}-x}{x^{3}} $$

A right circular cone is to be inscribed in another right circular cone of given volume, with the same axis and with the vertex of the inner cone touching the base of the outer cone. What must be the ratio of their altitudes for the inscribed cone to have maximum volume?

Find the critical points and use the test of your choice to decide which critical points give a local maximum value and which give a local minimum value. What are these local maximum and minimum values? $$ f(x)=\frac{x^{2}}{\sqrt{x^{2}+4}} $$

Use the Concavity Theorem to determine where the given function is concave up and where it is concave down. Also find all inflection points. $$ G(x)=\arcsin 2 x $$

Identify the critical points and find the maximum value and minimum value on the given interval. $$ s(t)=\sin t-\cos t ; I=[0, \pi] $$

An object is taken from an oven at \(300^{\circ} \mathrm{F}\) and left to cool in a room at \(75^{\circ} \mathrm{F}\). If the temperature fell to \(200^{\circ} \mathrm{F}\) in \(\frac{1}{2}\) hour, what will it be after 3 hours?

An object is moving along a coordinate line subject to the indicated acceleration a (in centimeters per second per second) with the initial velocity \(v_{0}\) (in centimeters per second) and directed distance \(s_{0}\) (in centimeters). Find both the velocity \(\underline{v}\) and directed distance \(s\) after 2 seconds (see Example 4). $$ a=\sqrt[3]{2 t+1} ; v_{0}=0, s_{0}=10 $$

, a function is defined and a closed interval is given. Decide whether the Mean Value Theorem applies to the given function on the given interval. If it does, find all possible values of \(c ;\) if not, state the reason. In each problem, sketch the graph of the given function on the given interval. $$ f(x)=x+\frac{1}{x} ;[1,2] $$

Find the general antiderivative \(F(x)+C\) for each of the following. $$ f(x)=x-\cosh x $$

A small island is 2 miles from the nearest point \(P\) on the straight shoreline of a large lake. If a woman on the island can row a boat 3 miles per hour and can walk 4 miles per hour, where should the boat be landed in order to arrive at a town 10 miles down the shore from \(P\) in the least time?

Find the critical points and use the test of your choice to decide which critical points give a local maximum value and which give a local minimum value. What are these local maximum and minimum values? $$ \Lambda(\theta)=\frac{\cos \theta}{1+\sin \theta}, 0<\theta<2 \pi $$

In Problems 19-28, determine where the graph of the given function is increasing, decreasing, concave up, and concave down. Then sketch the graph (see Example 4). $$ f(x)=x^{3}-12 x+1 $$

Identify the critical points and find the maximum value and minimum value on the given interval. $$ a(x)=|x-1| ; I=[0,3] $$

An object is moving along a coordinate line subject to the indicated acceleration a (in centimeters per second per second) with the initial velocity \(v_{0}\) (in centimeters per second) and directed distance \(s_{0}\) (in centimeters). Find both the velocity \(\underline{v}\) and directed distance \(s\) after 2 seconds (see Example 4). $$ a=(3 t+1)^{-3}, v_{0}=4, s_{0}=0 $$

Approximate the values of \(x\) that give maximum and minimum values of the function on the indicated intervals. $$ f(x)=x^{2} \sin \frac{x}{2} ;[0,4 \pi] $$

Find the general antiderivative \(F(x)+C\) for each of the following. $$ f(x)=x^{2}+e^{x} $$

Determine where the graph of the given function is increasing, decreasing, concave up, and concave down. Then sketch the graph (see Example 4). $$ g(x)=4 x^{3}-3 x^{2}-6 x+12 $$

Identify the critical points and find the maximum value and minimum value on the given interval. $$ f(s)=|3 s-2| ; I=[-1,4] $$

Evaluate the indicated indefinite integrals. $$ \int\left(x^{2}+x\right) d x $$

Find, if possible, the (global) maximum and minimum values of the given function on the indicated interval. $$ f(x)=\sin ^{2} 2 x \text { on }[0,2] $$

Determine where the graph of the given function is increasing, decreasing, concave up, and concave down. Then sketch the graph (see Example 4). $$ g(x)=3 x^{4}-4 x^{3}+2 $$

Identify the critical points and find the maximum value and minimum value on the given interval. $$ g(x)=\sqrt[3]{x} ; I=[-1,27] $$