Chapter 4

What number exceeds its square by the maximum amount? Begin by convincing yourself that this number is on the interval \([0,1]\).

Identify the critical points. Then use (a) the First Derivative Test and (if possible) (b) the Second Derivative Test to decide which of the critical points give a local maximum and which give a local minimum. $$ f(x)=\frac{x}{x^{2}+4} $$

Use the Monotonicity Theorem to find where the given function is increasing and where it is decreasing. $$ h(z)=\frac{z^{4}}{4}-\frac{4 z^{3}}{6} $$

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

The population of the United States was \(3.9\) million in 1790 and 178 million in 1960 . If the rate of growth is assumed proportional to the number present, what estimate would you give for the population in \(2000 ?\) (Compare your answer with the actual 2000 population, which was 275 million.)

First find the general solution (involving a constant \(\mathrm{C}\) ) for the given differential equation. Then find the particular solution that satisfies the indicated condition. $$ \frac{d y}{d x}=\sqrt{\frac{x}{y}} ; y=4 \text { at } x=1 $$

Find the general antiderivative \(F(x)+C\) for each of the following. $$ f(x)=7 x^{-3 / 4} $$

Show that for a rectangle of given perimeter \(K\) the one with maximum area is a square.

Identify the critical points. Then use (a) the First Derivative Test and (if possible) (b) the Second Derivative Test to decide which of the critical points give a local maximum and which give a local minimum. $$ g(z)=\frac{z^{2}}{1+z^{2}} $$

Use the Monotonicity Theorem to find where the given function is increasing and where it is decreasing. $$ H(t)=\sin t, 0 \leq t \leq 2 \pi $$

Identify the critical points and find the maximum value and minimum value on the given interval. $$ G(x)=\frac{1}{5}\left(2 x^{3}+3 x^{2}-12 x\right) ; I=[-3,3] $$

The population of a certain country is growing at \(3.2 \%\) per year; that is, if it is \(A\) at the beginning of a year, it is \(1.032 \mathrm{~A}\) at the end of that year. Assuming that it is \(4.5\) million now, what will it be at the end of 1 year? 2 years? 10 years? 100 years?

First find the general solution (involving a constant \(\mathrm{C}\) ) for the given differential equation. Then find the particular solution that satisfies the indicated condition. $$ \frac{d z}{d t}=t^{2} z^{2} ; z=1 / 3 \text { at } t=1 $$

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

Find the volume of the largest open box that can be made from a piece of cardboard 24 inches square by cutting equal squares from the corners and turning up the sides (see Example 1 ).

Identify the critical points. Then use (a) the First Derivative Test and (if possible) (b) the Second Derivative Test to decide which of the critical points give a local maximum and which give a local minimum. $$ h(y)=\tan ^{-1} y^{2} $$

Use the Monotonicity Theorem to find where the given function is increasing and where it is decreasing. $$ H(t)=\sin t, 0 \leq t \leq 2 \pi $$

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

First find the general solution (involving a constant \(\mathrm{C}\) ) for the given differential equation. Then find the particular solution that satisfies the indicated condition. $$ \frac{d y}{d t}=y^{4} ; y=1 \text { at } t=0 $$

Use Newton's Method to approximate the indicated root of the given equation accurate to five decimal places. Begin by sketching a graph. The root of \(x \ln x=2\)

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

, 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. $$ f(x)=\frac{x-4}{x-3} ;[0,4] $$

Identify the critical points. Then use (a) the First Derivative Test and (if possible) (b) the Second Derivative Test to decide which of the critical points give a local maximum and which give a local minimum. $$ f(x)=\frac{\ln (x+2)}{x+2} $$

A farmer has 80 feet of fence with which he plans to enclose a rectangular pen along one side of his 100 -foot barn, as shown in Figure 19 (the side along the barn needs no fence). What are the dimensions of the pen that has maximum area?

Use the Monotonicity Theorem to find where the given function is increasing and where it is decreasing. $$ f(x)=\frac{e^{-x}}{x^{2}} $$

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

A population is growing at a rate proportional to its size. After 5 years, the population size was 164,000 . After 12 years, the population size was 235,000 . What was the original population size?

First find the general solution (involving a constant \(\mathrm{C}\) ) for the given differential equation. Then find the particular solution that satisfies the indicated condition. $$ \frac{d s}{d t}=16 t^{2}+4 t-1 ; s=100 \text { at } t=0 $$

Use Newton's Method to approximate the indicated root of the given equation accurate to five decimal places. Begin by sketching a graph. All real roots of \(x^{4}-8 x^{3}+22 x^{2}-24 x+8=0\)

Find the general antiderivative \(F(x)+C\) for each of the following. $$ f(x)=4 x^{5}-x^{3} $$

, 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. $$ h(t)=t^{2 / 3} ;[0,2] $$

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? $$ f(x)=x^{3}-3 x $$

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. $$ f(x)=(x-1)^{2} $$

Identify the critical points and find the maximum value and minimum value on the given interval. $$ h(x)=e^{-x^{2}} ; I=[-1,3] $$

The mass of a tumor grows at a rate proportional to its size. The first measurement of its mass was \(4.0\) grams. Four months later its mass was \(6.76\) grams. How large was the tumor six months before the first measurement? If the instrument can detect tumors of mass 1 gram or greater, would the tumor have been detected at that time?

First find the general solution (involving a constant \(\mathrm{C}\) ) for the given differential equation. Then find the particular solution that satisfies the indicated condition. $$ \frac{d u}{d t}=u^{3}\left(t^{3}-t\right) ; u=4 \text { at } t=0 $$

Find the general antiderivative \(F(x)+C\) for each of the following. $$ f(x)=x^{100}+x^{99} $$

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? $$ g(x)=x^{4}+x^{2}+3 $$

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

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

A radioactive substance has a half-life of 700 years. If there were 10 grams initially, how much would be left after 300 years?

First find the general solution (involving a constant \(\mathrm{C}\) ) for the given differential equation. Then find the particular solution that satisfies the indicated condition. $$ \frac{d y}{d x}=(2 x+1)^{4} ; y=6 \text { at } x=0 $$

Use Newton's Method to approximate the indicated root of the given equation accurate to five decimal places. Begin by sketching a graph. $$ \text { The positive root of } 2 x^{2}-\sin x=0 $$

Find the general antiderivative \(F(x)+C\) for each of the following. $$ f(x)=27 x^{7}+3 x^{5}-45 x^{3}+\sqrt{2} x $$

, 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. $$ g(x)=x^{5 / 3} ;[0,1] $$

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? $$ H(x)=x^{4}-2 x^{3} $$

Use the Concavity Theorem to determine where the given function is concave up and where it is concave down. Also find all inflection points. $$ T(t)=3 t^{3}-18 t $$

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

If a radioactive substance loses \(15 \%\) of its radioactivity in 2 days, what is its half-life?

First find the general solution (involving a constant \(\mathrm{C}\) ) for the given differential equation. Then find the particular solution that satisfies the indicated condition. $$ \frac{d y}{d x}=-y^{2} x\left(x^{2}+2\right)^{4} ; y=1 \text { at } x=0 $$