Chapter 4

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 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. $$g(x)=\sqrt{4-x^{2}}, \quad-2 \leq x \leq 1$$

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. $$\int\left(\frac{1}{7}-\frac{1}{y^{5 / 4}}\right) d y$$

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

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. $$K(t)=15 t^{3}-t^{5}$$

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. $$g(x)=-\sqrt{5-x^{2}}, \quad-\sqrt{5} \leq x \leq 0$$

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. $$\int 2 x\left(1-x^{-3}\right) d x$$

A wire \(b \mathrm{m}\) 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?

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. $$f(x)=x-6 \sqrt{x-1}$$

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 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. $$f(\theta)=\sin \theta, \quad-\frac{\pi}{2} \leq \theta \leq \frac{5 \pi}{6}$$

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. $$\int x^{-3}(x+1) d x$$

Find all possible functions with the given derivative. $$\begin{aligned} &\text { a. } y^{\prime}=2 x\\\ &\text { b. } y^{\prime}=2 x-1 \quad \text { c. } y^{\prime}=3 x^{2}+2 x-1 \end{aligned}$$

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. $$g(x)=4 \sqrt{x}-x^{2}+3$$

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. $$f(\theta)=\tan \theta, \quad-\frac{\pi}{3} \leq \theta \leq \frac{\pi}{4}$$

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

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. $$\int \frac{t \sqrt{t}+\sqrt{t}}{t^{2}} d t$$

Find all possible functions with the given derivative. $$\text { a. } y^{\prime}=-\frac{1}{x^{2}} \quad \text { b. } y^{\prime}=1-\frac{1}{x^{2}} \quad \text { c. } y^{\prime}=5+\frac{1}{x^{2}}$$

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. $$g(x)=x \sqrt{8-x^{2}}$$

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. $$\int \frac{4+\sqrt{t}}{t^{3}} d t$$

Determine the dimensions of the rectangle of largest area that can be inscribed in a semicircle of radius 3 (See accompanying figure.)

Find all possible functions with the given derivative. a. \(y^{\prime}=\frac{1}{2 \sqrt{x}}\) b. \(y^{\prime}=\frac{1}{\sqrt{x}} \quad\) c. \(y^{\prime}=4 x-\frac{1}{\sqrt{x}}\)

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. $$g(x)=x^{2} \sqrt{5-x}$$

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. $$g(x)=\sec x, \quad-\frac{\pi}{3} \leq x \leq \frac{\pi}{6}$$

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

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. $$\int(-2 \cos t) d t$$

What value of \(a\) makes \(f(x)=x^{2}+(a / x)\) have a. a local minimum at \(x=2 ?\) b. a point of inflection at \(x=1 ?\)

Find all possible functions with the given derivative. a. \(y^{\prime}=\sin 2 t\) b. \(y^{\prime}=\cos \frac{t}{2} \quad\) c. \(y^{\prime}=\sin 2 t+\cos \frac{t}{2}\)

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. $$f(x)=\frac{x^{2}-3}{x-2}, \quad x \neq 2$$

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. $$f(t)=2-|t|, \quad-1 \leq t \leq 3$$

Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function. $$y=x^{2 / 3}\left(\frac{5}{2}-x\right)$$

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. $$\int(-5 \sin t) d t$$

Find all possible functions with the given derivative. a. \(y^{\prime}=\sec ^{2} \theta\) b. \(y^{\prime}=\sqrt{\theta} \quad\) c. \(y^{\prime}=\sqrt{\theta}-\sec ^{2} \theta\)

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. $$f(x)=\frac{x^{3}}{3 x^{2}+1}$$

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. $$f(t)=|t-5|, \quad 4 \leq t \leq 7$$

Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function. $$y=x^{2 / 3}(x-5)$$

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. $$\int 7 \sin \frac{\theta}{3} d \theta$$

The height above ground of an object moving vertically is given by $$s=-4.9 t^{2}+29.4 t+34.3$$ with \(s\) in meters and \(t\) in seconds. Find a. the object's velocity when \(t=0\) b. its maximum height and when it occurs; c. its velocity when \(s=0\)

Find the function with the given derivative whose graph passes through the point \(P\). $$f^{\prime}(x)=2 x-1, \quad P(0,0)$$

Find the function's absolute maximum and minimum values and say where they are assumed. $$f(x)=x^{4 / 3}, \quad-1 \leq x \leq 8$$

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. $$f(x)=x^{1 / 3}(x+8)$$

Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function. $$y=x \sqrt{8-x^{2}}$$

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. $$\int 3 \cos 5 \theta d \theta$$

Jane is \(2 \mathrm{km}\) offshore in a boat and wishes to reach a coastal village 6 km down a straight shoreline from the point nearest the boat. She can row \(2 \mathrm{km} / \mathrm{h}\) and can walk \(5 \mathrm{km} / \mathrm{h}\). Where should she land her boat to reach the village in the least amount of time?

Find the function with the given derivative whose graph passes through the point \(P\). $$g^{\prime}(x)=\frac{1}{x^{2}}+2 x, \quad P(-1,1)$$

Find the function's absolute maximum and minimum values and say where they are assumed. $$f(x)=x^{5 / 3}, \quad-1 \leq x \leq 8$$