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

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

Find the function with the given derivative whose graph passes through the point \(P\). $$r^{\prime}(\theta)=8-\csc ^{2} \theta, \quad P\left(\frac{\pi}{4}, 0\right)$$

Find the function's absolute maximum and minimum values and say where they are assumed. $$g(\theta)=\theta^{3 / 5}, \quad-32 \leq \theta \leq 1$$

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

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

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?

Find the function with the given derivative whose graph passes through the point \(P\). $$r^{\prime}(t)=\sec t \tan t-1, \quad P(0,0)$$

Find the function's absolute maximum and minimum values and say where they are assumed. $$h(\theta)=3 \theta^{2 / 3}, \quad-27 \leq \theta \leq 8$$

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

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

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. $$\int \frac{\csc \theta \cot \theta}{2} d \theta$$

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 \(6 \mathrm{m}\) apart. How far from the stronger light is the total illumination least?

Give the velocity \(v=d s / d t\) and initial position of an object moving along a coordinate line. Find the object's position at time \(t\). $$\boldsymbol{v}=9.8 t+5, \quad s(0)=10$$

a. Identify the function's local extreme values in the given domain, and say where they occur. b. Which of the extreme values, if any, are absolute? c. Support your findings with a graphing calculator or computer grapher. $$f(x)=2 x-x^{2}, \quad-\infty

Determine all critical points for each function. $$y=x^{2}-6 x+7$$

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. $$\int \frac{2}{5} \sec \theta \tan \theta d \theta$$

The range \(R\) of a projectile fired from the origin over horizontal ground is the distance from the origin to the point of impact. If the projectile is fired with an initial velocity \(v_{0}\) at an angle \(\alpha\) with the horizontal, then in Chapter 13 we find that $$R=\frac{v_{0}^{2}}{g} \sin 2 \alpha$$ where \(g\) is the downward acceleration due to gravity. Find the angle \(\alpha\) for which the range \(R\) is the largest possible.

Give the velocity \(v=d s / d t\) and initial position of an object moving along a coordinate line. Find the object's position at time \(t\). $$v=32 t-2, \quad s(0.5)=4$$

Determine all critical points for each function. $$f(x)=6 x^{2}-x^{3}$$

a. Identify the function's local extreme values in the given domain, and say where they occur. b. Which of the extreme values, if any, are absolute? c. Support your findings with a graphing calculator or computer grapher. $$f(x)=(x+1)^{2}, \quad-\infty

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. $$\int\left(4 \sec x \tan x-2 \sec ^{2} x\right) d x$$

Give the velocity \(v=d s / d t\) and initial position of an object moving along a coordinate line. Find the object's position at time \(t\). $$\boldsymbol{v}=\sin \pi t, \quad s(0)=0$$

Determine all critical points for each function. $$f(x)=x(4-x)^{3}$$

a. Identify the function's local extreme values in the given domain, and say where they occur. b. Which of the extreme values, if any, are absolute? c. Support your findings with a graphing calculator or computer grapher. $$g(x)=x^{2}-4 x+4, \quad 1 \leq x<\infty$$

Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function. $$y=\frac{8 x}{x^{2}+4}$$

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

Give the velocity \(v=d s / d t\) and initial position of an object moving along a coordinate line. Find the object's position at time \(t\). $$v=\frac{2}{\pi} \cos \frac{2 t}{\pi}, \quad s\left(\pi^{2}\right)=1$$

Determine all critical points for each function. $$g(x)=(x-1)^{2}(x-3)^{2}$$

a. Identify the function's local extreme values in the given domain, and say where they occur. b. Which of the extreme values, if any, are absolute? c. Support your findings with a graphing calculator or computer grapher. $$g(x)=-x^{2}-6 x-9, \quad-4 \leq x<\infty$$

Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function. $$y=\frac{5}{x^{4}+5}$$

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. $$\int\left(\sin 2 x-\csc ^{2} x\right) d x$$

A small frictionless cart, attached to the wall by a spring, is pulled \(10 \mathrm{cm}\) from its rest position and released at time \(t=0\) to roll back and forth for 4 s. Its position at time \(t\) is \(s=10 \cos \pi t\) a. What is the cart's maximum speed? When is the cart moving that fast? Where is it then? What is the magnitude of the acceleration then? b. Where is the cart when the magnitude of the acceleration is greatest? What is the cart's speed then?

Give the acceleration \(a=d^{2} s / d t^{2},\) initial velocity, and initial position of an object moving on a coordinate line. Find the object's position at time \(t\). $$a=32, \quad v(0)=20, \quad s(0)=5$$

Determine all critical points for each function. $$y=x^{2}+\frac{2}{x}$$

a. Identify the function's local extreme values in the given domain, and say where they occur. b. Which of the extreme values, if any, are absolute? c. Support your findings with a graphing calculator or computer grapher. $$f(t)=12 t-t^{3}, \quad-3 \leq t<\infty$$

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

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. $$\int(2 \cos 2 x-3 \sin 3 x) d x$$

Two masses hanging side by side from springs have positions \(s_{1}=2 \sin t\) and \(s_{2}=\sin 2 t,\) respectively. a. At what times in the interval \(0

Give the acceleration \(a=d^{2} s / d t^{2},\) initial velocity, and initial position of an object moving on a coordinate line. Find the object's position at time \(t\). $$a=9.8, \quad v(0)=-3, \quad s(0)=0$$

Determine all critical points for each function. $$f(x)=\frac{x^{2}}{x-2}$$

a. Identify the function's local extreme values in the given domain, and say where they occur. b. Which of the extreme values, if any, are absolute? c. Support your findings with a graphing calculator or computer grapher. $$f(t)=t^{3}-3 t^{2}, \quad-\infty

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

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. $$\int \frac{1+\cos 4 t}{2} d t$$

At noon, ship A was 12 nautical miles due north of ship \(B\). Ship \(A\) was sailing south at 12 knots (nautical miles per hour; a nautical mile is \(1852 \mathrm{m}\) ) and continued to do so all day. Ship \(B\) was sailing east at 8 knots and continued to do so all day. a. Start counting time with \(t=0\) at noon and express the distance \(s\) between the ships as a function of \(t\) b. How rapidly was the distance between the ships changing at noon? One hour later? c. The visibility that day was 5 nautical miles. Did the ships ever sight each other? d. Graph \(s\) and \(d s / d t\) together as functions of \(t\) for \(-1 \leq t \leq 3\) using different colors if possible. Compare the graphs and reconcile what you see with your answers in parts (b) and (c). e. The graph of \(d s / d t\) looks as if it might have a horizontal asymptote in the first quadrant. This in turn suggests that \(d s / d t\) approaches a limiting value as \(t \rightarrow \infty .\) What is this value? What is its relation to the ships' individual speeds?

Give the acceleration \(a=d^{2} s / d t^{2},\) initial velocity, and initial position of an object moving on a coordinate line. Find the object's position at time \(t\). $$a=-4 \sin 2 t, \quad v(0)=2, \quad s(0)=-3$$

Determine all critical points for each function. $$y=x^{2}-32 \sqrt{x}$$

a. Identify the function's local extreme values in the given domain, and say where they occur. b. Which of the extreme values, if any, are absolute? c. Support your findings with a graphing calculator or computer grapher. $$h(x)=\frac{x^{3}}{3}-2 x^{2}+4 x, \quad 0 \leq x<\infty$$