Chapter 4

In Problems 1-4, use the Bisection Method to approximate the real root of the given equation on the given interval. Each answer should be accurate to two decimal places. $$ x^{3}+2 x-6=0 ;[1,2] $$

Solve the given differential equation subject to the given condition. Note that \(y(a)\) denotes the value of \(y a t=a\). \(\frac{d y}{d t}=-6 y, y(0)=4\)

In Problems 1-10, 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. 1\. \(f(x)=x^{3}-6 x^{2}+4\)

Find two numbers whose product is \(-16\) and the sum of whose squares is a minimum.

In Problems 1-10, use the Monotonicity Theorem to find where the given function is increasing and where it is decreasing. \(f(x)=3 x+3\)

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.$$ f(x)=|x| ;[1,2] $$

Show that the indicated function is a solution of the given differential equation, that is, substitute the indicated function for y to see that it produces an equality. $$\frac{d y}{d x}+\frac{x}{y}=0 ; y=\sqrt{1-x^{2}}$$

In Problems 1-4, use the Bisection Method to approximate the real root of the given equation on the given interval. Each answer should be accurate to two decimal places. $$ x^{4}+5 x^{3}+1=0 ;[-1,0] $$

Solve the given differential equation subject to the given condition. Note that \(y(a)\) denotes the value of \(y a t=a\). \(\frac{d y}{d t}=6 y, y(0)=1\)

For what number does the principal square root exceed eight times the number by the largest amount?

In Problems 1-10, use the Monotonicity Theorem to find where the given function is increasing and where it is decreasing. \(g(x)=(x+1)(x-2)\)

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. $$ g(x)=|x| ;[-2,2] $$

Show that the indicated function is a solution of the given differential equation, that is, substitute the indicated function for y to see that it produces an equality. $$-x \frac{d y}{d x}+y=0 ; y=C x$$

In Problems 1-4, use the Bisection Method to approximate the real root of the given equation on the given interval. Each answer should be accurate to two decimal places. $$ 2 \cos x-e^{-x}=0 ;[1,2] $$

Solve the given differential equation subject to the given condition. Note that \(y(a)\) denotes the value of \(y a t=a\). \(\frac{d y}{d t}=0.005 y, y(10)=2\)

\(f(\theta)=\sin 2 \theta, 0<\theta<\frac{\pi}{4}\)

For what number does the principal fourth root exceed twice the number by the largest amount?

In Problems 1-10, use the Monotonicity Theorem to find where the given function is increasing and where it is decreasing. \(h(t)=t^{2}+2 t-3\)

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. $$ f(x)=x^{2}+x ;[-2,2] $$

Show that the indicated function is a solution of the given differential equation, that is, substitute the indicated function for y to see that it produces an equality. $$\frac{d^{2} y}{d x^{2}}+y=0 ; y=C_{1} \sin x+C_{2} \cos x$$

In Problems 1-4, use the Bisection Method to approximate the real root of the given equation on the given interval. Each answer should be accurate to two decimal places. $$ x-2+2 \ln x=0 ;[1,2] $$

Solve the given differential equation subject to the given condition. Note that \(y(a)\) denotes the value of \(y a t=a\). \(\frac{d y}{d t}=-0.003 y, y(-2)=3\)

Find two numbers whose product is \(-12\) and the sum of whose squares is a minimum.

In Problems 1-10, use the Monotonicity Theorem to find where the given function is increasing and where it is decreasing. \(f(x)=x^{3}-1\)

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

Show that the indicated function is a solution of the given differential equation, that is, substitute the indicated function for y to see that it produces an equality. \(\left(\frac{d y}{d x}\right)^{2}+y^{2}=1 ; y=\sin (x+C)\) and \(y=\pm 1\)

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 largest root of \(x^{3}+6 x^{2}+9 x+1=0\)

5\. Find the points on the parabola \(y=x^{2}\) that are closest to the point \((0,5)\).

In Problems 5-26, identify the critical points and find the maximum value and minimum value on the given interval. $$ f(x)=x^{2}+4 x+4 ; I=[-4,0] $$

In Problems 1-10, use the Monotonicity Theorem to find where the given function is increasing and where it is decreasing. \(G(x)=2 x^{3}-9 x^{2}+12 x\)

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. $$ H(s)=s^{2}+3 s-1 ;[-3,1] $$

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.) \(\frac{d y}{d x}=x^{2}+1 ; y=1\) 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 real root of \(7 x^{3}+x-5=0\)

f(x)=4 x^{5}-x^{3}

6\. Find the points on the parabola \(x=2 y^{2}\) that are closest to the point \((10,0)\).

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

In Problems 1-10, use the Monotonicity Theorem to find where the given function is increasing and where it is decreasing. \(f(t)=t^{3}+3 t^{2}-12\)

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. $$ F(x)=\frac{x^{3}}{3} ;[-2,2] $$

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.) \(\frac{d y}{d x}=x^{-3}+2 ; y=3\) at \(x=1\)

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

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

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. $$ f(z)=\frac{1}{3}\left(z^{3}+z-4\right) ;[-1,2] $$

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.) \(\frac{d y}{d x}=\frac{x}{y} ; y=1\) at \(x=1\)

In Problems 1-10, use the Monotonicity Theorem to find where the given function is increasing and where it is decreasing. \(h(z)=\frac{z^{4}}{4}-\frac{4 z^{3}}{6}\)

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 \(2 \cos x-e^{-x}=0\) (see Prob\(\operatorname{lem} 3\) )

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.)

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

In Problems 5-26, identify the critical points and find the maximum value and minimum value on the given interval. $$ G(x)=\frac{1}{5}\left(2 x^{3}+3 x^{2}-12 x\right) ; I=[-3,3] $$

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

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. $$ F(t)=\frac{1}{t-1} ;[0,2] $$