Chapter 4

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. $$\int x^{-5 / 4} d x$$

Find all possible functions with the given derivative. a. \(y^{\prime}=2 x\) b. \(y^{\prime}=2 x-1 \quad\) c. \(y^{\prime}=3 x^{2}+2 x-1\)

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

Use I'Hópital's rule to find the limits. $$\lim _{x \rightarrow 0^{-}} \frac{\ln \left(e^{x}-1\right)}{\ln x}$$

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

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. $$\int(\sqrt{x}+\sqrt[3]{x}) d x$$

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

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 \(x=1 ?\)

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

Use I'Hópital's rule to find the limits. $$\lim _{y \rightarrow 0} \frac{\sqrt{5 y+25}-5}{y}$$

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. $$\int\left(\frac{\sqrt{x}}{2}+\frac{2}{\sqrt{x}}\right) d x$$

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

Use I'Hópital's rule to find the limits. $$\lim _{y \rightarrow 0} \frac{\sqrt{a y+a^{2}}-a}{y}, a>0$$

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

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. $$\int\left(8 y-\frac{2}{y^{1 / 4}}\right) d y$$

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

Vertical motion The height above ground of an object moving vertically is given by $$s=-16 t^{2}+96 t+112$$ with \(s\) in feet 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 \(s=0\)

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

Use I'Hópital's rule to find the limits. $4\lim _{x \rightarrow \infty}(\ln 2 x-\ln (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. $$h(x)=\ln (x+1), \quad 0 \leq x \leq 3$$

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

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

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

Quickest route Jane is 2 mi offshore in a boat and wishes to reach a coastal village 6 mi down a straight shoreline from the point nearest the boat. She can row 2 mph and can walk 5 mph. Where should she land her boat to reach the village in the least amount of time?

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. $$g(x)=x^{2 / 3}(x+5)$$

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

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. $$f(x)=\frac{1}{x}+\ln x, \quad 0.5 \leq x \leq 4$$

Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function. $$y=\sqrt{16-x^{2}}$$

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

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

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. $$h(x)=x^{1 / 3}\left(x^{2}-4\right)$$

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

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

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

Motion on a line The positions of two particles on the \(s\) -axis are \(s_{1}=\sin t\) and \(s_{2}=\sin (t+\pi / 3),\) with \(s_{1}\) and \(s_{2}\) in meters and \(t\) in seconds. a. At what time(s) in the interval \(0 \leq t \leq 2 \pi\) do the particles meet? b. What is the farthest apart that the particles ever get? c. When in the interval \(0 \leq t \leq 2 \pi\) is the distance between the particles changing the fastest?

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. $$k(x)=x^{2 / 3}\left(x^{2}-4\right)$$

Use I'Hópital's rule to find the limits. $$\lim _{x \rightarrow 0^{+}}\left(\frac{3 x+1}{x}-\frac{1}{\sin x}\right)$$

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

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

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

Find the function with the given derivative whose graph passes through the point \(P\). $$f^{\prime}(x)=e^{2 x}, \quad P\left(0, \frac{3}{2}\right)$$

The intensity of illumination at any point from a light source is proportional to the square of the reciprocal of the distance between the point and the light source. Two lights, one having an intensity eight times that of the other, are 6 m apart. How far from the stronger light is the total illumination least?

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. $$f(x)=e^{2 x}+e^{-x}$$