Chapter 4

$$ f(x)=\frac{x^{6}-x}{x^{3}} $$

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.

17\. 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 \((0,4)\).

In Problems 5-26, identify the critical points and find the maximum value and minimum value on the given interval. $$ r(\theta)=\sin \theta ; I=\left[-\frac{\pi}{4}, \frac{\pi}{6}\right] $$

In Problems 11-18, use the Concavity Theorem to determine where the given function is concave up and where it is concave down. Also find all inflection points. \(F(x)=2 x^{2}+\cos ^{2} x\)

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 \(v\) and directed distance s after 2 seconds (see Example 4). \(a=t ; v_{0}=3, s_{0}=0\)

In Problems 17-20, approximate the values of \(x\) that give maximum and minimum values of the function on the indicated intervals. $$ f(x)=\frac{x^{3}+1}{x^{4}+1} ;[-4,4] $$

$$ f(x)=\frac{x^{6}-x}{x^{3}} $$

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?

In Problems 5-26, identify the critical points and find the maximum value and minimum value on the given interval. $$ s(t)=\sin t-\cos t ; I=[0, \pi] $$

In Problems 11-18, use the Concavity Theorem to determine where the given function is concave up and where it is concave down. Also find all inflection points. \(G(x)=\arcsin 2 x\)

In each of the Problems 1-21, 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. $$ f(x)=x+\frac{1}{x} ;\left[-1, \frac{1}{2}\right] $$

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 \(v\) and directed distance s after 2 seconds (see Example 4). \(a=(1+t)^{-4}, v_{0}=0, s_{0}=10\)

In Problems 1-27, make an analysis as suggested in the summary above and then sketch the graph. $$ R(z)=z|z| $$

In Problems 17-20, approximate the values of \(x\) that give maximum and minimum values of the function on the indicated intervals. $$ f(x)=\frac{\sin x}{x} ;[\pi, 3 \pi] $$

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?

In Problems 5-26, identify the critical points and find the maximum value and minimum value on the given interval. $$ a(x)=|x-1| ; I=[0,3] $$

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). \(f(x)=x^{3}-12 x+1\)

In each of the Problems 1-21, 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. $$ f(x)=x+\frac{1}{x} ;[1,2] $$

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 \(v\) and directed distance s after 2 seconds (see Example 4). \(a=\sqrt[3]{2 t+1} ; v_{0}=0, s_{0}=10\)

In Problems 17-20, approximate the values of \(x\) that give maximum and minimum values of the function on the indicated intervals. $$ f(x)=x^{2} \sin \frac{x}{2} ;[0,4 \pi] $$

\(g(\theta)=|\sin \theta|, 0<\theta<2 \pi\)

In Problems 5-26, identify the critical points and find the maximum value and minimum value on the given interval. $$ f(s)=|3 s-2| ; I=[-1,4] $$

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). \(g(x)=4 x^{3}-3 x^{2}-6 x+12\)

In each of the Problems 1-21, 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. $$ f(x)=[x] ;[1,2] $$

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 \(v\) and directed distance s after 2 seconds (see Example 4). \(a=(3 t+1)^{-3} ; v_{0}=4, s_{0}=0\)

$$ \int\left(x^{2}+x\right) d x $$

In Problems 21-30, find, if possible, the (global) maximum and minimum values of the given function on the indicated interval. 21\. \(f(x)=\sin ^{2} 2 x\) on \([0,2]\)

In Problems 5-26, identify the critical points and find the maximum value and minimum value on the given interval. $$ g(x)=\sqrt[3]{x} ; I=[-1,27] $$

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). \(g(x)=3 x^{4}-4 x^{3}+2 \quad\)

Sketch the graph of \(y=x^{1 / 3}\). Obviously, its only \(x\) intercept is zero. Convince yourself that Newton's Method fails to converge to the root of \(x^{1 / 3}=0\). Explain this failure.

$$ \int\left(x^{3}+\sqrt{x}\right) d x $$

A powerhouse is located on one bank of a straight river that is \(w\) feet wide. A factory is situated on the opposite bank of the river, \(L\) feet downstream from the point \(A\) directly opposite the powerhouse. What is the most economical path for a cable connecting the powerhouse to the factory if it costs \(a\) dollars per foot to lay the cable under water and \(b\) dollars per foot on land \((a>b)\) ?

In Problems 5-26, identify the critical points and find the maximum value and minimum value on the given interval. $$ s(t)=t^{2 / 5} ; I=[-1,32] $$

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). \(F(x)=x^{6}-3 x^{4}\)

If \(f\) is continuous on \([a, b]\) and differentiable on \((a, b)\), and if \(f(a)=f(b)\), then there is at least one number \(c\) in \((a, b)\) such that \(f^{\prime}(c)=0\). Show that Rolle's Theorem is just a special case of the Mean Value Theorem. (Michel Rolle (1652-1719) was a French mathematician.)

A ball is thrown upward from the surface of a planet where the acceleration of gravity is \(k\) (a negative constant) feet per second per second. If the initial velocity is \(v_{0}\), show that the maximum height is \(-v_{0}^{2} / 2 k\).

In installment buying, one would like to figure out the real interest rate (effective rate), but unfortunately this involves solving a complicated equation. If one buys an item worth $$\$ P$$ today and agrees to pay for it with payments of $$\$ R$$ at the end of each month for \(k\) months, then $$ P=\frac{R}{i}\left[1-\frac{1}{(1+i)^{k}}\right] $$ where \(i\) is the interest rate per month. Tom bought a used car for $$\$ 2000$$ and agreed to pay for it with $$\$ 100$$ payments at the end of each of the next 24 months. (a) Show that \(i\) satisfies the equation \(20 i(1+i)^{24}-(1+i)^{24}+1=0\) (b) Show that Newton's Method for this equation reduces to $$ i_{n+1}=i_{n}-\left[\frac{20 i_{n}^{2}+19 i_{n}-1+\left(1+i_{n}\right)^{-23}}{500 i_{n}-4}\right] $$ (c) Find \(i\) accurate to five decimal places starting with \(i=0.012\), and then give the annual rate \(r\) as a percent \((r=1200 i)\).

$$ \int(x+1)^{2} d x $$

\(g(x)=\frac{x^{2}}{x^{3}+32}\) on \([0, \infty)\)

At 7:00 A.M. one ship was 60 miles due east from a second ship. If the first ship sailed west at 20 miles per hour and the second ship sailed southeast at 30 miles per hour, when were they closest together?

In Problems 5-26, identify the critical points and find the maximum value and minimum value on the given interval. $$ f(x)=x e^{-x^{2}} ; I=[-1,2] $$

On the surface of the moon, the acceleration of gravity is \(-5.28\) feet per second per second. If an object is thrown upward from an initial height of 1000 feet with a velocity of 56 feet per second, find its velocity and height \(4.5\) seconds later.

In applying Newton's Method to solve \(f(x)=0\), one can usually tell by simply looking at the numbers \(x_{1}, x_{2}, x_{3}, \ldots\) whether the sequence is converging. But even if it converges, say to \(\bar{x}\), can we be sure that \(\bar{x}\) is a solution? Show that the answer is yes provided \(f\) and \(f^{\prime}\) are continuous at \(\bar{x}\) and \(f^{\prime}(\bar{x}) \neq 0\).

$$ \int(z+\sqrt{2} z)^{2} d z $$

Solve the differential equation for Newton's Law of Cooling for an arbitrary \(T_{0}, T_{1}\), and \(k\), assuming that \(T_{0}>T_{1}\). Show that \(\lim _{t \rightarrow \infty} T(t)=T_{1}\).

Find the equation of the line that is tangent to the ellipse \(b^{2} x^{2}+a^{2} y^{2}=a^{2} b^{2}\) in the first quadrant and forms with the coordinate axes the triangle with smallest possible area ( \(a\) and \(b\) are positive constants).

In Problems 5-26, identify the critical points and find the maximum value and minimum value on the given interval. $$ g(x)=\frac{\ln (x+1)}{x+1} ; I=[0,3] $$

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). \(H(x)=\frac{x^{2}}{x^{2}+1}\)

Show that if \(f\) is the quadratic function defined by \(f(x)=\alpha x^{2}+\beta x+\gamma, \alpha \neq 0\), then the number \(c\) of the Mean Value Theorem is always the midpoint of the given interval \([a, b]\).