Chapter 1

Find all the values of \(x\) that satisfy at least one of the two inequalities. (a) \(2 x-7>1\) or \(2 x+1<3\) (b) \(2 x-7 \leq 1\) or \(2 x+1<3\) (c) \(2 x-7 \leq 1\) or \(2 x+1>3\)

Change each rational number to a decimal by performing long division. \(\frac{2}{7}\)

Use the addition identity for the tangent to show that \(\tan (t+\pi)=\tan t\) for all \(t\) in the domain of \(\tan t\).

In Problems \(29-34\), find an equation for each line. Then write your answer in the form \(A x+B y+C=0\). Through \((2,3)\) and \((4,8)\)

The magnitude \(M\) of an earthquake on the Richter scale is $$ M=0.67 \log _{10}(0.37 E)+1.46 $$ where E is the energy of the earthquake in kilowatt-hours. Find the energy of an earthquake of magnitude 7. Of magnitude 8.

In Problems 31-38, plot the graphs of both equations on the same coordinate plane. Find and label the points of intersection of the two graphs (see Example 4). $$ \begin{aligned} &y=-2 x+3 \\ &y=-2(x-4)^{2} \end{aligned} $$

Find the formula for the amount \(E(x)\) by which a number \(x\) exceeds its square. Plot a graph of \(E(x)\) for \(0 \leq x \leq 1\). Use the graph to estimate the positive number less than or equal to 1 that exceeds its square by the maximum amount.

By repeated use of the addition formula $$ \tan (x+y)=(\tan x+\tan y) /(1-\tan x \tan y) $$ show that $$ \frac{\pi}{4}=3 \tan ^{-1}\left(\frac{1}{4}\right)+\tan ^{-1}\left(\frac{5}{99}\right) $$

Solve for \(x\), expressing your answer in interval notation. (a) \((x+1)\left(x^{2}+2 x-7\right) \geq x^{2}-1\) (b) \(x^{4}-2 x^{2} \geq 8\) (c) \(\left(x^{2}+1\right)^{2}-7\left(x^{2}+1\right)+10<0\)

Change each rational number to a decimal by performing long division. \(\frac{3}{21}\)

Show that \(\cos (x-\pi)=-\cos x\) for all \(x\).

In Problems \(29-34\), find an equation for each line. Then write your answer in the form \(A x+B y+C=0\). Through \((4,1)\) and \((8,2)\)

The loudness of sound is measured in decibels in honor of Alexander Graham Bell (1847-1922), inventor of the telephone. If the variation in pressure is \(P\) pounds per square inch, then the loudness \(L\) in decibels is $$ L=20 \log _{10}(121.3 P) $$ Find the variation in pressure caused by music at 115 decibels.

In Problems 31-38, plot the graphs of both equations on the same coordinate plane. Find and label the points of intersection of the two graphs (see Example 4). $$ \begin{aligned} &y=-2 x+3 \\ &y=3 x^{2}-3 x+12 \end{aligned} $$

Let \(p\) denote the perimeter of an equilateral triangle. Find a formula for \(A(p)\), the area of such a triangle.

Verify that $$ \frac{\pi}{4}=4 \tan ^{-1}\left(\frac{1}{5}\right)-\tan ^{-1}\left(\frac{1}{239}\right) $$ a result discovered by John Machin in 1706 and used by him to calculate the first 100 decimal places of \(\pi\).

Solve each inequality. Express your solution in interval notation. (a) \(1.99<\frac{1}{x}<2.01\) (b) \(2.99<\frac{1}{x+2}<3.01\)

Change each rational number to a decimal by performing long division. \(\frac{5}{17}\)

Suppose that a tire on a truck has an outer radius of \(2.5\) feet. How many revolutions per minute does the tire make when the truck is traveling 60 miles per hour?

In Problems 35-38, find the slope and \(y\)-intercept of each line. \(3 y=-2 x+1\)

In the equally tempered scale to which keyed instruments have been tuned since the days of J.S. Bach \((1685-1750)\), the frequencies of successive notes \(\mathrm{C}, \mathrm{C} \\#, \mathrm{D}, \mathrm{D} \\#, \mathrm{E}, \mathrm{F}, \mathrm{F} \\#, \mathrm{G}, \mathrm{G} \\#, \mathrm{~A}, \mathrm{~A} \\#, \mathrm{~B}\), \(\overline{\mathrm{C}}\) form a geometric sequence (progression), with \(\overline{\mathrm{C}}\) having twice the frequency of \(C\) (C# is read \(C\) sharp and \(\bar{C}\) indicates one octave above C). What is the ratio \(r\) between the frequencies of successive notes? If the frequency of \(A\) is 440 , find the frequency of \(\overline{\mathrm{C}}\).

In Problems 31-38, plot the graphs of both equations on the same coordinate plane. Find and label the points of intersection of the two graphs (see Example 4). $$ \begin{aligned} &y=x \\ &x^{2}+y^{2}=4 \end{aligned} $$

A right triangle has a fixed hypotenuse of length \(h\) and one leg that has length \(x\). Find a formula for the length \(L(x)\) of the other leg.

Find formulas for \(f^{-1}(x)\) for each of the following functions \(f\), first indicating how you would restrict the domain so that \(f\) has an inverse. For example, if \(f(x)=3 \sin 2 x\) and we restrict the domain to \(-\pi / 4 \leq x \leq \pi / 4\), then \(f^{-1}(x)=\frac{1}{2} \sin ^{-1}(x / 3)\). (a) \(f(x)=3 \cos 2 x\) (b) \(f(x)=2 \sin 3 x\) (c) \(f(x)=\frac{1}{2} \tan x\) (d) \(f(x)=\sin \frac{1}{x}\)

Find the solution sets of the given inequalities. $$ |x-2| \geq 5 $$

Change each rational number to a decimal by performing long division. \(\frac{11}{3}\)

How far does a wheel of radius 2 feet roll along level ground in making 150 revolutions?

In Problems 35-38, find the slope and \(y\)-intercept of each line. \(-4 y=5 x-6\)

In Problems 31-38, plot the graphs of both equations on the same coordinate plane. Find and label the points of intersection of the two graphs (see Example 4). $$ \begin{aligned} &y=x-1 \\ &2 x^{2}+3 y^{2}=12 \end{aligned} $$

A right triangle has a fixed hypotenuse of length \(h\) and one leg that has length \(x\). Find a formula for the area \(A(x)\) of the triangle.

Draw the graphs of $$ y=\arcsin x \quad \text { and } \quad y=\arctan \left(x / \sqrt{1-x^{2}}\right) $$ using the same axes. Make a conjecture. Prove it.

Find the solution sets of the given inequalities. $$ |x+2|<1 $$

Change each rational number to a decimal by performing long division. \(\frac{11}{13}\)

In Problems 35-38, find the slope and \(y\)-intercept of each line. \(6-2 y=10 x-2\)

In Problems 31-38, plot the graphs of both equations on the same coordinate plane. Find and label the points of intersection of the two graphs (see Example 4). $$ \begin{aligned} &y-3 x=1 \\ &x^{2}+2 x+y^{2}=15 \end{aligned} $$

The Acme Car Rental Agency charges \(\$ 24\) a day for the rental of a car plus \(\$ 0.40\) per mile. (a) Write a formula for the total rental expense \(E(x)\) for one day, where \(x\) is the number of miles driven. (b) If you rent a car for one day, how many miles can you drive for \(\$ 120\) ?

Draw the graph of \(y=\pi / 2-\arcsin x\). Make a conjecture. Prove it.

Find the solution sets of the given inequalities. $$ |4 x+5| \leq 10 $$

Change each repeating decimal to a ratio of two integers \(0.123123123 \ldots\)

The angle of inclination \(\alpha\) of a line is the smallest positive angle from the positive \(x\)-axis to the line ( \(\alpha=0\) for a horizontal line). Show that the slope \(m\) of the line is equal to \(\tan \alpha\).

In Problems 35-38, find the slope and \(y\)-intercept of each line. \(4 x+5 y=-20\)

\(f(x)=(x-3)^{2}, x \geq 3\)

In Problems 31-38, plot the graphs of both equations on the same coordinate plane. Find and label the points of intersection of the two graphs (see Example 4). $$ \begin{aligned} &y=4 x+3 \\ &x^{2}+y^{2}=81 \end{aligned} $$

A right circular cylinder of radius \(r\) is inscribed in a sphere of radius \(2 r\). Find a formula for \(V(r)\), the volume of the cylinder, in terms of \(r\).

Draw the graph of \(y=\sin (\arcsin x)\) on \([-1,1]\). Then draw the graph of \(y=\arcsin (\sin x)\) on \([-2 \pi, 2 \pi]\). Explain the differences that you observe.

Find the solution sets of the given inequalities. $$ |2 x-1|>2 $$

Change each repeating decimal to a ratio of two integers \(0.217171717 \ldots\)

Find the angle of inclination of the following lines (see Problem 38). (a) \(y=\sqrt{3} x-7\) (b) \(\sqrt{3} x+3 y=6\)

Write an equation for the line through \((3,-3)\) that is (a) parallel to the line \(y=2 x+5\); (b) perpendicular to the line \(y=2 x+5\); (c) parallel to the line \(2 x+3 y=6\); (d) perpendicular to the line \(2 x+3 y=6\); (e) parallel to the line through \((-1,2)\) and \((3,-1)\); (f) parallel to the line \(x=8\); (g) perpendicular to the line \(x=8\).

A 1-mile track has parallel sides and equal semicircular ends. Find a formula for the area enclosed by the track, \(A(d)\), in terms of the diameter \(d\) of the semicircles. What is the natural domain for this function?