Chapter 4

Find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation. $$\int(5-6 x) d x$$

Show that the functions have exactly one zero in the given interval. $$r(\theta)=2 \theta-\cos ^{2} \theta+\sqrt{2}, \quad(-\infty, \infty)$$

a. Find the open intervals on which the function is increasing and decreasing. b. Identify the function's local and absolute extreme values, if any, saying where they occur. $$h(r)=(r+7)^{3}$$

Use I'Hópital's rule to find the limits. $$\lim _{x \rightarrow(\pi / 2)}\left(\frac{\pi}{2}-x\right) \tan x$$

Find the approximate values of \(r_{1}\) through \(r_{4}\) in the factorization $$8 x^{4}-14 x^{3}-9 x^{2}+11 x-1=8\left(x-r_{1}\right)\left(x-r_{2}\right)\left(x-r_{3}\right)\left(x-r_{4}\right)$$ (GRAPH CAN'T COPY)

Find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates. $$h(x)=\sqrt[3]{x}, \quad-1 \leq x \leq 8$$

Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function. $$y=\sin x \cos x, \quad 0 \leq x \leq \pi$$

Find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation. $$\int\left(3 t^{2}+\frac{t}{2}\right) d t$$

a. Find the open intervals on which the function is increasing and decreasing. b. Identify the function's local and absolute extreme values, if any, saying where they occur.. $$f(x)=x^{4}-8 x^{2}+16$$

Use I'Hópital's rule to find the limits. $$\lim _{\theta \rightarrow 0} \frac{3^{\sin \theta}-1}{\theta}$$

Use Newton's method to find the zeros of \(f(x)=4 x^{4}-4 x^{2}\) using the given starting values. a. \(x_{0}=-2\) and \(x_{0}=-0.8,\) lying in \((-\infty,-\sqrt{2} / 2)\) b. \(x_{0}=-0.5\) and \(x_{0}=0.25,\) lying in \((-\sqrt{21} / 7, \sqrt{21} / 7)\) c. \(x_{0}=0.8\) and \(x_{0}=2,\) lying in \((\sqrt{2} / 2, \infty)\) d. \(x_{0}=-\sqrt{21} / 7\) and \(x_{0}=\sqrt{21} / 7\)

Find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates. $$h(x)=-3 x^{2 / 3}, \quad-1 \leq x \leq 1$$

Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function. $$y=\cos x+\sqrt{3} \sin x, \quad 0 \leq x \leq 2 \pi$$

Find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation. $$\int\left(\frac{t^{2}}{2}+4 t^{3}\right) d t$$

Show that the functions have exactly one zero in the given interval. $$r(\theta)=\tan \theta-\cot \theta-\theta, \quad(0, \pi / 2)$$

a. Find the open intervals on which the function is increasing and decreasing. b. Identify the function's local and absolute extreme values, if any, saying where they occur. $$g(x)=x^{4}-4 x^{3}+4 x^{2}$$

Find the point on the line \(\frac{x}{a}+\frac{y}{b}=1\) that is closest to the origin.

In submarine location problems, it is often necessary to find a submarine's closest point of approach (CPA) to a sonobuoy (sound detector) in the water. Suppose that the submarine travels on the parabolic path \(y=x^{2}\) and that the buoy is located at the point \((2,-1 / 2)\) a. Show that the value of \(x\) that minimizes the distance between the submarine and the buoy is a solution of the equation \(x=1 /\left(x^{2}+1\right)\) b. Solve the equation \(x=1 /\left(x^{2}+1\right)\) with Newton's method. (GRAPH CAN'T COPY)

Use I'Hópital's rule to find the limits. $$\lim _{\theta \rightarrow 0} \frac{(1 / 2)^{6}-1}{\theta}$$

Find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates. $$g(x)=\sqrt{4-x^{2}},-2 \leq x \leq 1$$

Find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation. $$\int\left(3 t^{2}+\frac{t}{2}\right) d t$$

Suppose that \(f(-1)=3\) and that \(f^{\prime}(x)=0\) for all \(x .\) Must \(f(x)=3\) for all \(x ?\) Give reasons for your answer.

a. Find the open intervals on which the function is increasing and decreasing. b. Identify the function's local and absolute extreme values, if any, saying where they occur. $$H(t)=\frac{3}{2} t^{4}-t^{6}$$

Find a positive number for which the sum of it and its reciprocal is the smallest (least) possible.

Some curves are so flat that, in practice, Newton's method stops too far from the root to give a useful estimate. Try Newton's method on \(f(x)=(x-1)^{40}\) with a starting value of \(x_{0}=2\) to see how close your machine comes to the root \(x=1 .\) See the accompanying graph. (GRAPH CAN'T COPY)

Use I'Hópital's rule to find the limits. $$\lim _{x \rightarrow 0} \frac{x 2^{x}}{2^{x}-1}$$

Find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates. $$g(x)=-\sqrt{5-x^{2}},-\sqrt{5} \leq x \leq 0$$

Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function. $$y=x^{2 / 5}$$

Find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation. $$\int\left(1-x^{2}-3 x^{5}\right) d x$$

Suppose that \(f(0)=5\) and that \(f^{\prime}(x)=2\) for all \(x .\) Must \(f(x)=\) \(2 x+5\) for all \(x ?\) Give reasons for your answer.

a. Find the open intervals on which the function is increasing and decreasing. b. Identify the function's local and absolute extreme values, if any, saying where they occur. $$K(t)=15 t^{3}-t^{5}$$

Find a positive number for which the sum of its reciprocal and four times its square is the smallest possible.

Use I'Hópital's rule to find the limits. $$\lim _{x \rightarrow 0} \frac{3^{x}-1}{2^{x}-1}$$

Find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates. $$f(\theta)=\sin \theta, \quad-\frac{\pi}{2} \leq \theta \leq \frac{5 \pi}{6}$$

Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function. $$y=\frac{x}{\sqrt{x^{2}+1}}$$

Find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation. $$\int\left(\frac{1}{x^{2}}-x^{2}-\frac{1}{3}\right) d x$$

Suppose that \(f^{\prime}(x)=2 x\) for all \(x\). Find \(f(2)\) if a. \(f(0)=0\) b. \(f(1)=0 \quad\) c. \(f(-2)=3\)

a. Find the open intervals on which the function is increasing and decreasing. b. Identify the function's local and absolute extreme values, if any, saying where they occur. $$f(x)=x-6 \sqrt{x-1}$$

A wire \(b\) m long is cut into two pieces. One piece is bent into an equilateral triangle and the other is bent into a circle. If the sum of the areas enclosed by each part is a minimum, what is the length of each part?

Use I'Hópital's rule to find the limits. $$\lim _{x \rightarrow \infty} \frac{\ln (x+1)}{\log _{2} x}$$

Find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation. $$\int\left(\frac{1}{5}-\frac{2}{x^{3}}+2 x\right) d x$$

What can be said about functions whose derivatives are constant? Give reasons for your answer.

a. Find the open intervals on which the function is increasing and decreasing. b. Identify the function's local and absolute extreme values, if any, saying where they occur. $$g(x)=4 \sqrt{x}-x^{2}+3$$

Use I'Hópital's rule to find the limits. $$\lim _{x \rightarrow \infty} \frac{\log _{2} x}{\log _{3}(x+3)}$$

Find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates. $$g(x)=\csc x, \quad \frac{\pi}{3} \leq x \leq \frac{2 \pi}{3}$$

Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function. $$y=2 x-3 x^{2 / 3}$$

Find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation. $$\int x^{-1 / 3} d x$$

Find all possible functions with the given derivative. a. \(y^{\prime}=x\) b. \(y^{\prime}=x^{2}\) c. \(y^{\prime}=x^{3}\)

a. Find the open intervals on which the function is increasing and decreasing. b. Identify the function's local and absolute extreme values, if any, saying where they occur. $$g(x)=x \sqrt{8-x^{2}}$$

Use I'Hópital's rule to find the limits. $$\lim _{x \rightarrow 0^{-}} \frac{\ln \left(x^{2}+2 x\right)}{\ln x}$$