Chapter 4

First find the general solution (involving a constant C) for the given differential equation. Then find the particular solution that satisfies the indicated condition. (See Example 2.) \(\frac{d y}{d x}=\sqrt{\frac{x}{y}} ; y=4\) at \(x=1\)

In Problems 5-14, 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 \(\cos x=2 x\)

$$ f(x)=\frac{4 x^{6}+3 x^{4}}{x^{3}} $$

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?

\(h(y)=\tan ^{-1} y^{2}\)

In Problems 5-26, identify the critical points and find the maximum value and minimum value on the given interval. $$ f(x)=x^{3}-3 x+1 ; I=\left(-\frac{3}{2}, 3\right) \text { Hint: Sketch the graph. } $$

In Problems 1-10, use the Monotonicity Theorem to find where the given function is increasing and where it is decreasing. \(H(t)=\sin t, 0 \leq t \leq 2 \pi\)

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. $$ h(x)=e^{-x} ;[0,3] $$

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

In Problems 5-14, 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 \(x \ln x=2\)

In Problems 5-26, identify the critical points and find the maximum value and minimum value on the given interval. $$ f(x)=x^{3}-3 x+1 ; I=\left[-\frac{3}{2}, 3\right] $$

In Problems 1-10, use the Monotonicity Theorem to find where the given function is increasing and where it is decreasing. \(f(x)=\frac{e^{-x}}{x^{2}}\)

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

In Problems 5-14, 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 \(x^{4}-8 x^{3}+22 x^{2}-24 x+8=0\)

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?

In Problems 11-20, 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? 11\. \(f(x)=x^{3}-3 x\)

In Problems 5-26, identify the critical points and find the maximum value and minimum value on the given interval. $$ h(x)=e^{-x^{2}} ; I=[-1,3] $$

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

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

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

In Problems 5-14, 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 \(x^{4}+6 x^{3}+2 x^{2}+24 x-8=0\)

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?

In Problems 5-26, identify the critical points and find the maximum value and minimum value on the given interval. $$ g(x)=\frac{1}{1+x^{2}} ; I=[-3,1] $$

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. \(G(w)=w^{2}-1\)

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

First find the general solution (involving a constant C) for the given differential equation. Then find the particular solution that satisfies the indicated condition. (See Example 2.) \(\frac{d u}{d t}=u^{3}\left(t^{3}-t\right) ; u=4\) at \(t=0\)

In Problems 5-14, use Newton's Method to approximate the indicated root of the given equation accurate to five decimal places. Begin by sketching a graph. The positive root of \(2 x^{2}-\sin x=0\)

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

In Problems 5-26, identify the critical points and find the maximum value and minimum value on the given interval. $$ f(x)=x^{4}-2 x^{2}+2 ; I=[-2,2] $$

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. \(T(t)=3 t^{3}-18 t\)

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

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

In Problems 5-14, use Newton's Method to approximate the indicated root of the given equation accurate to five decimal places. Begin by sketching a graph. The smallest positive root of \(2 \cot x=x\)

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

In Problems 5-26, identify the critical points and find the maximum value and minimum value on the given interval. $$ f(x)=x^{5}-\frac{25}{3} x^{3}+20 x-1 ; I=[-3,2] $$

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. \(f(z)=z^{2}-\frac{1}{z^{2}}\)

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

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

Use Newton's Method to calculate \(\sqrt[3]{6}\) to five decimal places. Hint: Solve \(x^{3}-6=0\).

$$ f(x)=x^{100}+x^{99} $$

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? Answer this question for strontium- 90 .

\(g(t)=\pi-(t-2)^{2 / 3}\)

In Problems 5-26, identify the critical points and find the maximum value and minimum value on the given interval. $$ g(x)=\frac{1}{1+x^{2}} ; I=(-\infty, \infty) \text { Hint: Sketch the graph. } $$

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. \(q(x)=x^{4}-6 x^{3}-24 x^{2}+3 x+1\)

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. $$ S(\theta)=\sin \theta ;[-\pi, \pi] $$

An unknown amount of a radioactive substance is being studied. After two days, the mass is \(15.231\) grams. After eight days, the mass is \(9.086\) grams. How much was there initially? What is the half-life of this substance?

In Problems 5-26, identify the critical points and find the maximum value and minimum value on the given interval. $$ f(x)=\frac{x}{1+x^{2}} ; I=[-1,4] $$

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. \(f(x)=x^{4}+8 x^{3}-2\)

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.

In Problems 17-20, approximate the values of \(x\) that give maximum and minimum values of the function on the indicated intervals. $$ f(x)=x^{4}+x^{3}+x^{2}+x ;[-1,1] $$