Chapter 4

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 \(-v_{0}^{2} / 2 k\).

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.

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.)

Evaluate the indicated indefinite integrals. $$ \int\left(x^{3}+\sqrt{x}\right) d x $$

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 \((a>b)\) ?

Find, if possible, the (global) maximum and minimum values of the given function on the indicated interval. $$ f(x)=\frac{2 x}{x^{2}+4} \text { on }[0, \infty) $$

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

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

A dead body is found at \(10 \mathrm{PM}\). to have temperature \(82^{\circ} \mathrm{F}\). One hour later the temperature was \(76^{\circ} \mathrm{F}\). The temperature of the room was a constant \(70^{\circ} \mathrm{F}\). Assuming that the temperature of the body was \(98.6^{\circ} \mathrm{F}\) when it was alive, estimate the time of death.

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.

23\. 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] (c) Find \(i\) accurate to five decimal places starting with \(i=0.012\), and then give the annual rate \(r\) as a percent \((r=1200 i) .\)

Evaluate the indicated indefinite integrals. $$ \int(x+1)^{2} d x $$

At \(7: 00 \mathrm{~A} . \mathrm{M}\). 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?

Find, if possible, the (global) maximum and minimum values of the given function on the indicated interval. $$ g(x)=\frac{x^{2}}{x^{3}+32} \text { on }[0, \infty) $$

Determine where the graph of the given function is increasing, decreasing, concave up, and concave down. Then sketch the graph (see Example 4). $$ G(x)=3 x^{5}-5 x^{3}+1 $$

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

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 \(\lim _{t \rightarrow \infty} T(t)=T_{1}\).

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 \(f^{\prime}(\bar{x}) \neq 0\).

Evaluate the indicated indefinite integrals. $$ \int(z+\sqrt{2} z)^{2} d z $$

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 \([a, b]\).

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).

Find, if possible, the (global) maximum and minimum values of the given function on the indicated interval. $$ h(x)=\frac{1}{x^{2}+4} \text { on }[0, \infty) $$

Determine where the graph of the given function is increasing, decreasing, concave up, and concave down. Then sketch the graph (see Example 4). $$ H(x)=\frac{x^{2}}{x^{2}+1} $$

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

25\. If \(\$ 375\) is put in the bank today, what will it be worth at the end of 2 years if interest is \(3.5 \%\) and is compounded as specified? (a) Annually (b) Monthly (c) Daily (d) Continuously

The rate of change of volume \(V\) of a melting snowball is proportional to the surface area \(S\) of the ball; that is, \(d V / d t=-k S\), where \(k\) is a positive constant. If the radius of the ball at \(t=0\) is \(r=2\) and at \(t=10\) is \(r=0.5\), show that \(r=-\frac{3}{20} t+2 .\)

Use the Fixed-Point Algorithm with \(x_{1}\) as indicated to solve the equations to five decimal places. $$ x=\frac{3}{2} \cos x ; x_{1}=1 $$

Evaluate the indicated indefinite integrals. $$ \int \frac{\left(z^{2}+1\right)^{2}}{\sqrt{z}} d z $$

Prove: If \(f\) is continuous on \((a, b)\) and if \(f^{\prime}(x)\) exists and satisfies \(f^{\prime}(x)>0\) except at one point \(x_{0}\) in \((a, b)\), then \(f\) is increasing on \((a, b) .\) Hint: Consider \(f\) on each of the intervals \(\left(a, x_{0}\right]\) and \(\left[x_{0}, b\right)\) separately.

Find the greatest volume that a right circular cylinder can have if it is inscribed in a sphere of radius \(r\).

Find, if possible, the (global) maximum and minimum values of the given function on the indicated interval. $$ F(x)=6 \sqrt{x}-4 x \text { on }[0,4] $$

Determine where the graph of the given function is increasing, decreasing, concave up, and concave down. Then sketch the graph (see Example 4). $$ f(x)=\sqrt{\sin x} \text { on }[0, \pi] $$

Identify the critical points and find the maximum value and minimum value on the given interval. $$ g(\theta)=\theta^{2} \sec \theta ; I=\left[-\frac{\pi}{4}, \frac{\pi}{4}\right] $$

From what height must a ball be dropped in order to strike the ground with a velocity of \(-136\) feet per second?

Use the Fixed-Point Algorithm with \(x_{1}\) as indicated to solve the equations to five decimal places. $$ x=2-\sin x ; x_{1}=2 $$

Evaluate the indicated indefinite integrals. $$ \int \frac{s(s+1)^{2}}{\sqrt{s}} d s $$

Show that the rectangle with maximum perimeter that can be inscribed in a circle is a square.

Find, if possible, the (global) maximum and minimum values of the given function on the indicated interval. $$ F(x)=6 \sqrt{x}-4 x \text { on }[0, \infty) $$

Identify the critical points and find the maximum value and minimum value on the given interval. $$ h(t)=\frac{t^{5 / 3}}{2+t} ; I=[-1,8] $$

How long does it take money to double in value for the specified interest rate? (a) \(6 \%\) compounded monthly (b) \(6 \%\) compounded continuously

Use the Fixed-Point Algorithm with \(x_{1}\) as indicated to solve the equations to five decimal places. $$ x=\sqrt{2.7+x} ; x_{1}=1 $$

Evaluate the indicated indefinite integrals. $$ \int(\sin \theta-\cos \theta) d \theta $$

What are the dimensions of the right circular cylinder with greatest curved surface area that can be inscribed in a sphere of radius \(r ?\)

Find, if possible, the (global) maximum and minimum values of the given function on the indicated interval. $$ f(x)=\frac{64}{\sin x}+\frac{27}{\cos x} \text { on }(0, \pi / 2) $$

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

Inflation between 1999 and 2004 ran at about \(2.5 \%\) per year. On this basis, what would you expect a car that would have cost \(\$ 20,000\) in 1999 to cost in \(2004 ?\)

If the brakes of a car, when fully applied, produce a constant deceleration of 11 feet per second per second, what is the shortest distance in which the car can be braked to a halt from a speed of 60 miles per hour?

Use the Fixed-Point Algorithm with \(x_{1}\) as indicated to solve the equations to five decimal places. $$ x=\sqrt{3.2+x} ; x_{1}=47 $$

Use the Mean Value Theorem to show that \(s=1 / t^{2}\) decreases on any interval to the right of the origin.

Evaluate the indicated indefinite integrals. $$ \int\left(t^{2}-2 \cos t\right) d t $$