Chapter 3

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

In Problems \(I-4,\) 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. $$ x^{3}+2 x-6=0 ;[1,2] $$

In Problems \(1-12,\) a function \(f\) and its domain are given. Determine the critical points, evaluate \(f\) at these points, and find the (global) maximum and minimum values. \(f(x)=x^{2}-2 x ;[0,4]\)

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

A function is defined and a closed in terval 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. $$ f(x)=|x| ;[1,2] $$

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

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

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. $$\frac{d y}{d x}+\frac{x}{y}=0 ; y=\sqrt{1-x^{2}}$$

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

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 \(f\) and its domain are given. Determine the critical points, evaluate \(f\) at these points, and find the (global) maximum and minimum values. \(f(t)=\frac{1}{t} ;[1,4]\)

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

A function is defined and a closed in terval 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. $$ g(x)=|x| ;[-2,2] $$

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

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

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. $$ 2 \cos x-\sin x=0 ;[1,2] $$

A function \(f\) and its domain are given. Determine the critical points, evaluate \(f\) at these points, and find the (global) maximum and minimum values. \(f(z)=\frac{1}{z^{2}} ;\left[-2,-\frac{1}{2}\right]\)

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

A function is defined and a closed in terval 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. $$ f(x)=x^{2}+x ;[-2,2] $$

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

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

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

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

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)=\frac{1}{2} x+\sin x, 0

A function is defined and a closed in terval 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. $$ g(x)=(x+1)^{3} ;[-1,1] $$

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

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$$

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

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

A function \(f\) and its domain are given. Determine the critical points, evaluate \(f\) at these points, and find the (global) maximum and minimum values. \(f(x)=|x| ;\left[-\frac{1}{2}, 1\right]\)

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

A function is defined and a closed in terval 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. $$ H(s)=s^{2}+3 s-1 ;[-3,1] $$

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).

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 \(C\) ) 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$$

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

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

A function \(f\) and its domain are given. Determine the critical points, evaluate \(f\) at these points, and find the (global) maximum and minimum values. \(f(s)=s+|s| ;[-1,1]\)

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

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

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 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 \(C\) ) 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$$

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

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 \(x-2+2 \cos x=0\) (see Problem 4 )

A function \(f\) and its domain are given. Determine the critical points, evaluate \(f\) at these points, and find the (global) maximum and minimum values. \(f(x)=3 x^{4}-4 x^{3} ;[-2,3]\)

A function is defined and a closed in terval 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. $$ f(z)=\frac{1}{3}\left(z^{3}+z-4\right) ;[-1,2] $$

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. $$ f(x)=\frac{x}{x^{2}+4} $$