Chapter 4

In Problems \(1-4\), solve the given differential equation subject to the given condition. Note that \(y(a)\) denotes the value of \(y\) at \(t=a\). $$ \frac{d y}{d t}=-6 y, y(0)=4 $$

In Problems \(1-4\), 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. $$ \frac{d y}{d x}+\frac{x}{y}=0 ; y=\sqrt{1-x^{2}} $$

Find the general antiderivative \(F(x)+C\) for each of the following. $$ f(x)=5 $$

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. $$ x^{3}+2 x-6=0 ;[1,2] $$

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

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. $$ f(x)=x^{3}-6 x^{2}+4 $$

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. $$ f(x)=3 x+3 $$

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

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. $$ -x \frac{d y}{d x}+y=0 ; y=C x $$

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

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. $$ x^{4}+5 x^{3}+1=0 ;[-1,0] $$

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

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

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

Use the Monotonicity Theorem to find where the given function is increasing and where it is decreasing. $$ g(x)=(x+1)(x-2) $$

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

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. $$ \frac{d^{2} y}{d x^{2}}+y=0 ; y=C_{1} \sin x+C_{2} \cos x $$

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

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. $$ 2 \cos x-e^{-x}=0 ;[1,2] $$

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

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

Use the Monotonicity Theorem to find where the given function is increasing and where it is decreasing. $$ h(t)=t^{2}+2 t-3 $$

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

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. $$ \left(\frac{d y}{d x}\right)^{2}+y^{2}=1 ; y=\sin (x+C) \text { and } y=\pm 1 $$

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. $$ x-2+2 \ln x=0 ;[1,2] $$

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

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

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{1}{2} x+\sin x, 0

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

Use the Monotonicity Theorem to find where the given function is increasing and where it is decreasing. $$ f(x)=x^{3}-1 $$

A bacterial population grows at a rate proportional to its size. Initially, it is 10,000 , and after 10 days it is 20,000 . What is the population after 25 days?

In Problems \(5-14\), 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. $$ \frac{d y}{d x}=x^{2}+1 ; y=1 \text { at } x=1 $$

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

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

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. $$ \Psi(\theta)=\sin ^{2} \theta,-\pi / 2<\theta<\pi / 2 $$

Find the points on the parabola \(y=x^{2}\) that are closest to the point \((0,5) .\) Hint: Minimize the square of the distance between \((x, y)\) and \((0,5)\)

Use the Monotonicity Theorem to find where the given function is increasing and where it is decreasing. $$ G(x)=2 x^{3}-9 x^{2}+12 x $$

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

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

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 \(7 x^{3}+x-5=0\)

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

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

Find the points on the parabola \(x=2 y^{2}\) that are closest to the point \((10,0) .\) Hint: Minimize the square of the distance between \((x, y)\) and \((10,0)\).

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. $$ r(z)=z^{4}+4 $$

Use the Monotonicity Theorem to find where the given function is increasing and where it is decreasing. $$ f(t)=t^{3}+3 t^{2}-12 $$

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

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. $$ \frac{d y}{d x}=\frac{x}{y} ; y=1 \text { at } x=1 $$

Find the general antiderivative \(F(x)+C\) for each of the following. $$ f(x)=1 / \sqrt[3]{x^{2}} $$

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