Chapter 1

Prove that the sine of an angle inscribed in a circle of unit diameter is the length of the chord of the subtended arc. (Hint: First assume that one side of the angle is a diameter and use the fact that the resulting triangle is a right triangle (Figure 1.76). Then use the fact that all inscribed angles with the same subtended arc are equal.)

Solve the inequality. $$ \frac{2-5 x}{3-4 x} \geq-2 $$

Find an equation of the line that is parallel to the given line \(l\) and passes through the given point \(P\). \(l: 5 x-2 y-1=0 ; P=(3,3)\)

Determine the range of the function. $$ f(x)=\frac{x^{2}-1}{x^{2}+1} $$

Sketch the graph of the equation. In each case determine whether the graph is that of a function. $$ y^{2}=x^{3} $$

a. Let \(f(x)=a x+b\). Show that \(f(x+1)-f(x)=a\). b. Let \(g(x)=b a^{x}\), where \(a\) is positive and \(b \neq 0 .\) Show that \(g(x+1) / g(x)=a\).

Evaluate the expression. $$ -|-3| $$

Find an equation of the line that is perpendicular to the given line \(l\) and passes through the given point \(P\). \(l: y=2 x+1 ; P=(-1,-3)\)

Determine which of the following define a function. Explain your reason for any that do not define a function. a. The domain consists of the number \(-2\), which is assigned the number \(\pi\). b. The domain consists of the number \(-2\), which is assigned the numbers \(-2\) and \(\pi\). c. \(f(x)=\pm \sqrt{x}\) d. \(f(x)=\pm \sqrt{x^{2}+1}\) e. \(g(x)=\left\\{\begin{array}{l}x-1 \text { for } x<0 \\ 12 x-6 \text { for } x>0\end{array}\right.\) f. \(g(x)=\left\\{\begin{array}{l}2-4 x \text { for } x<0 \\ x^{2} \text { for } x>1\end{array}\right.\) g. \(g(x)=\left\\{\begin{array}{l}4 x+1 \text { for } x \leq 2 \\ 2 x^{3}-7 \text { for } x \geq 2\end{array}\right.\) h. \(g(x)=\left\\{\begin{array}{l}2-3 x^{3} \text { for } x \leq 1 \\ 3 x^{4}-3 \text { for } x \geq 1\end{array}\right.\) i. \(f(t)=\left\\{\begin{array}{l}t^{2} \text { for } t \text { rational } \\\ t \text { for } t \text { irrational }\end{array}\right.\) j. \(f(t)=\left\\{\begin{array}{l}t^{2} \text { for } t^{2} \text { rational } \\\ t \text { for } t \text { irrational }\end{array}\right.\)

Sketch the graph of the equation. In each case determine whether the graph is that of a function. $$ x y=0 $$

Let \(f(x)=1+\sqrt{x+1}, g(x)=1+\sqrt{2-x}\), and \(h(x)\) \(=f(x)+g(x)\). Plot the graphs of \(f, g\), and \(h\) on the same calculator screen with viewing windows \([-10,10]\). Explain why the graph of \(h\) is so short.

Evaluate the expression. $$ |-\sqrt{2}|^{2} $$

Find an equation of the line that is perpendicular to the given line \(l\) and passes through the given point \(P\). \(l: y=-\frac{1}{3} x-2 ; P=(0,0)\)

In each of the following, determine whether \(f\) and \(g\) are the same. a. \(f(x)=1-x^{2} ; g(x)=1-x^{2}\) for \(-1

Sketch the graph of the equation. In each case determine whether the graph is that of a function. $$ |x|=|y| $$

Approximate all zeros of the function to the nearest hundredth. $$ f(x)=-4.9 x^{2}+5.1 x+1.2 $$

a. Show that the perimeter \(p_{n}(r)\) of a regular polygon of \(n\) sides inscribed in a circle of radius \(r\) is given by $$ p_{n}(r)=2 n r \sin \frac{\pi}{n} $$ b. Using the result of part (a), find the radius of the smallest circle that can circumscribe the Pentagon building, cach of whose outer walls is 921 feet long.

Evaluate the expression. $$ |-5|+|5| $$

Find an equation of the line that is perpendicular to the given line \(l\) and passes through the given point \(P\). \(l: 2 x+3 y-6=0 ; P=(2,3)\)

Which of the following functions are the same? a. \(f_{1}(x)=\sqrt{1-6 x+9 x^{2}}\) b. \(f_{2}(x)=1-3 x\) c. \(f_{3}(t)=1-3 t\) d. \(f_{4}(w)=1-3 w\) for \(w \geq 0\) e. \(f_{5}(t)=|1-3 t|\) f. \(f_{6}(x)=\frac{(1-3 x)^{2}}{1-3 x}\)

Sketch the graph of the equation. In each case determine whether the graph is that of a function. $$ |x|+|y|=1 $$

Approximate all zeros of the function to the nearest hundredth. $$ f(x)=\sqrt{2} x^{2}+\pi x+1 $$

Evaluate the expression. $$ |-5|-|5| $$

Find an equation of the line that is perpendicular to the given line \(l\) and passes through the given point \(P\). \(l: 3 x-y=0 ; P=(1,3)\)

Let \(f(x)=\sqrt{x^{2}+1}-1\) and \(g(x)=\frac{x^{2}}{1+\sqrt{x^{2}+1}}\) a. Find the domains of \(f\) and \(g\). b. Show that \(f=g\).

Sketch the graph of the equation. In each case determine whether the graph is that of a function. $$ |x|-|2 y|=1 $$

For which functions \(f\) is there a function \(g\) such that \(f=1 / g\) ?

Suppose a ball of mass \(m\) is attached to a string of length \(L\) and is rotated in a vertical plane with enough velocity \(v\) so that the string remains taut (Figure \(1.78)\). Then the tension \(T\) in the string, which depends on the angle \(\theta\) that the string makes with the downward vertical, is given by $$ T=m\left(\frac{v^{2}}{L}+g \cos \theta\right) \text { for } 0 \leq \theta<2 \pi $$ where \(g\) is the (negative) acceleration due to gravity. a. From your intuition, at which point in the path of the ball would the tension be greatest, and at which would it be least? b. From (14), find the value of \(\theta\) at which \(T\) is greatest and the value at which \(T\) is least. Do these values agree with your intuition?

Solve the equation. $$ |x|=1 $$

Find an equation of the line that is perpendicular to the given line \(l\) and passes through the given point \(P\). \(l: y-1=2(x-3) ; P=(4,-5)\)

Let \(f(x)=x^{3}\). a. On the same screen, plot the graphs of \(y=f(x), y=\) \(f(x-1)\), and \(y=f(x-2)\). b. On the same screen, plot the graphs of \(y=f(x), y=\) \(f(x+1)\), and \(y=f(x+2)\). c. On the basis of parts (a) and (b), how do you think one obtains the graph of \(y=f(x+c)\) from the graph of \(f ?\) Consider the two cases \(c>0\) and \(c<0\) separately.

For which functions \(f\) is there a function \(g\) such that $$ f=\sqrt{1+g} $$

Approximate all zeros of the function to the nearest hundredth. Let \(f(x)=a x^{2}+b x+c\) with \(a \neq 0\). Suppose \(f\) has two zeros, \(z_{1}\) and \(z_{2}\). Express \(z_{1}+z_{2}\) in terms of \(a, b\), and \(c\).

In a certain industrial area the amount of sulfur dioxide pollutant released into the atmosphere due to burning fossil fuels varies according to the season. Suppose that we wish to model the amount \(A\) of pollutant (in tons) released into the atmosphere at time \(t\) (in wecks) by means of the formula $$ A=1+b \cos \frac{\pi}{26} t $$

Solve the equation. $$ |x|=\pi $$

Find an equation of the line that is perpendicular to the given line \(l\) and passes through the given point \(P\). \(l: y+4=-\frac{3}{5}\left(x-\frac{1}{2}\right) ; P=\left(-1, \frac{1}{2}\right)\)

Find a formula for the function \(f\) that assigns to each \(x\) greater than \(-1\) the number obtained by squaring \(x\), then subtracting \(2 x\), and finally adding \(\sqrt{2}\).

Let \(f\) and \(g\) be even functions. a. Show that \(f+g\) is an even function. b. Show that \(f g\) is an even function.

Use the zoom feature of a graphics calculator to approximate the coordinates of the points of intersection of \(f\) and \(g\). Zoom until successive values of the \(x\) coordinate have identical first three digits. $$ f(x)=x^{4}-1, g(x)=x^{3}+1 $$

A single respiratory cycle includes one inhalation and one exhalation. During one respiratory cycle of a certain person at rest, the rate of flow \(R\) (in liters per second) of air into a person's lungs at time \(t\) (in seconds) is given by $$ R=0.5 \sin \frac{2 \pi}{5} t $$ a. How long does it take to complete one respiratory cycle? b. How many respiratory cycles are completed in one minute? c. Graph one complete cycle, starting at time \(t=0\). d. Interpret the meaning of positive and negative values of \(R\) e. To the nearest hundredth, find \(R\) when \(t=3\) seconds.

Solve the equation. $$ |x-1|=2 $$

Let \(l\) be the line that contains two given points, \(\left(x_{1}, y_{1}\right)\) and \(\left(x_{2}, y_{2}\right)\), with \(x_{1} \neq x_{2} .\) Show that an equation of \(l\) is $$ y-y_{1}=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\left(x-x_{1}\right) $$ (This equation is called a two-point equation of \(l\).)

Find a formula for the function \(f\) that assigns to each nonnegative \(x\) the number obtained by dividing \(x\) by 5 , then taking the cube root of the quotient, and finally multiplying the result by the product of \(\frac{1}{2}\) and \(x^{2}\).

Let \(f\) be even and \(g\) odd. Show that \(f g\) is an odd function.

Solve the equation. $$ \left|2 x-\frac{1}{2}\right|=\frac{1}{2} $$

Find a two-point equation of the given line. The line containing \((3,4)\) and \((1,3)\)

The volume \(V\) of a rectangular box with square base is 60 cubic centimeters. Express the length \(l\) (in centimeters) of a vertical side as a function of the length \(s\) of a side of the base.

Can a horizontal line pass through more than one point on the graph of a function? Explain.

Let \(f(x)=1 / x\). Show that \(f(f(x))=x\) for all \(x \neq 0\).

The strongest earthquakes ever recorded occurred oft the coast of Ecuador and Colombia in 1906, and in Japan in \(1933 .\) Each had a magnitude of \(8.9 .\) Find the ratio of the amplitude of the largest wave of such a quake to the corresponding amplitude of a zero-level quake.