Chapter 4

Use I'Hópital's rule to find the limits. $$\lim _{\theta \rightarrow-\pi / 3} \frac{3 \theta+\pi}{\sin (\theta+(\pi / 3))}$$

A rectangle is to be inscribed under the arch of the curve \(y=4 \cos (0.5 x)\) from \(x=-\pi\) to \(x=\pi .\) What are the dimensions of the rectangle with largest area, and what is the largest area?

Sketch the graph of each function and determine whether the function has any absolute extreme values on its domain. Explain how your answer is consistent with Theorem 1. $$y=3 \sin x, \quad 0

Find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation. a. \(e^{3 x}\) b. \(e^{-x}\) c. \(e^{x / 2}\)

Show that if \(f^{\prime \prime} \geq 0\) throughout an interval \([a, b],\) then \(f^{\prime}\) has at most one zero in \([a, b] .\) What if \(f^{\prime \prime}<0\) throughout \([a, b]\) instead?

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. $$g(t)=-t^{2}-3 t+3$$

At what value(s) of \(x\) does \(\cos x=2 x ?\)

Find the dimensions of a right circular cylinder of maximum volume that can be inscribed in a sphere of radius \(10 \mathrm{cm} .\) What is the maximum volume?

Use I'Hópital's rule to find the limits. $$\lim _{\theta \rightarrow \pi / 2} \frac{1-\sin \theta}{1+\cos 2 \theta}$$

Sketch the graph of each function and determine whether the function has any absolute extreme values on its domain. Explain how your answer is consistent with Theorem 1. $$f(x)=\left\\{\begin{array}{lr} x+1, & -1 \leq x<0 \\ \cos x, & 0

Find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation. a. \(e^{-2 x}\) b. \(e^{4 x / 3}\) c. \(e^{-x / 5}\)

Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function. $$y=x^{4}+2 x^{3}=x^{3}(x+2)$$

Show that a cubic polynomial can have at most three real zeros.

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. $$g(t)=-3 t^{2}+9 t+5$$

a. The U.S. Postal Service will accept a box for domestic shipment only if the sum of its length and girth (distance around) does not exceed 108 in. What dimensions will give a box with a square end the largest possible volume? GRAPH CANT COPY b. Graph the volume of a 108 -in. box (length plus girth equals 108 in.) as a function of its length and compare what you see with your answer in part (a). (Continuation of Exercise \(20 .\) ) a. Suppose that instead of having a box with square ends you have a box with square sides so that its dimensions are \(h\) by \(h\) by \(w\) and the girth is \(2 h+2 w .\) What dimensions will give the box its largest volume now?b. Graph the volume as a function of \(h\) and compare what you see with your answer in part (a). GRAPH CANT COPY

Use I'Hópital's rule to find the limits. $$\lim _{x \rightarrow 1} \frac{x-1}{\ln x-\sin \pi x}$$

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. $$f(x)=\frac{2}{3} x-5, \quad-2 \leq x \leq 3$$

Find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation. a. \(3^{x}\) b. \(2^{-x}\) c. \(\left(\frac{5}{3}\right)^{x}\)

Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function. $$y=x^{5}-5 x^{4}=x^{4}(x-5)$$

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. $$h(x)=-x^{3}+2 x^{2}$$

a. Suppose that instead of having a box with square ends you have a box with square sides so that its dimensions are \(h\) by \(h\) by \(w\) and the girth is \(2 h+2 w .\) What dimensions will give the box its largest volume now? GRAPH CANT COPY b. Graph the volume as a function of \(h\) and compare what you see with your answer in part (a).

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

The graphs of \(y=x^{2}(x+1)\) and \(y=1 / x(x>0)\) intersect at one point \(x=r .\) Use Newton's method to estimate the value of \(r\) to four decimal places. (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. $$f(x)=-x-4, \quad-4 \leq x \leq 1$$

Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function. $$y=x\left(\frac{x}{2}-5\right)^{4}$$

Find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation. a. \(x^{\sqrt{3}}\) b. \(x^{\pi}\) c. \(x^{\sqrt{2}-1}\)

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. $$h(x)=2 x^{3}-18 x$$

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

The graphs of \(y=\sqrt{x}\) and \(y=3-x^{2}\) intersect at one point \(x=r .\) Use Newton's method to estimate the value of \(r\) to four decimal places.

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

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

Find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation. a. \(\frac{2}{\sqrt{1-x^{2}}}\) b. \(\frac{1}{2\left(x^{2}+1\right)}\) c. \(\frac{1}{1+4 x^{2}}\)

Show that the functions have exactly one zero in the given interval. $$g(t)=\sqrt{t}+\sqrt{1+t}-4, \quad(0, \infty)$$

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. $$f(\theta)=3 \theta^{2}-4 \theta^{3}$$

A silo (base not included) is to be constructed in the form of a cylinder surmounted by a hemisphere. The cost of construction per square unit of surface area is twice as great for the hemisphere as it is for the cylindrical sidewall. Determine the dimensions to be used if the volume is fixed and the cost of construction is to be kept to a minimum. Neglect the thickness of the silo and waste in construction.

Use I'Hópital's rule to find the limits. $$\lim _{t \rightarrow 0} \frac{t(1-\cos t)}{t-\sin t}$$

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. $$f(x)=4-x^{3},-2 \leq x \leq 1$$

Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function. $$y=x-\sin x, \quad 0 \leq x \leq 2 \pi$$

Find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation. a. \(x-\left(\frac{1}{2}\right)^{x}\) b. \(x^{2}+2^{x}\) c. \(\pi^{x}-x^{-1}\)

Show that the functions have exactly one zero in the given interval. $$g(t)=\frac{1}{1-t}+\sqrt{1+t}-3.1, \quad(-1,1)$$

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. $$f(\theta)=6 \theta-\theta^{3}$$

At what value(s) of \(x\) does \(\ln \left(1-x^{2}\right)=\) \(x-1 ?\)

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. $$F(x)=-\frac{1}{x^{2}}, \quad 0.5 \leq x \leq 2$$

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

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. $$\int(x+1) d x$$

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. $$f(r)=3 r^{3}+16 r$$

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

Use the Intermediate Value Theorem from Section 2.5 to show that \(f(x)=x^{3}+2 x-4\) has a root between \(x=1\) and \(x=2\) Then find the root to five decimal places.

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. $$F(x)=-\frac{1}{x},-2 \leq x \leq-1$$

Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function. $$y=\frac{4}{3} x-\tan x, \quad \frac{-\pi}{2}< x<\frac{\pi}{2}$$