Chapter 1

Trigonometric functions that have high frequencies pose special problems for graphing. We now explore how to plot such functions. Graph the function \(f(x)=\sin 50 x\) using the window given by a \(y\) range of \(-1.5 \leq y \leq 1.5\) and the \(x\) range given by (a) \([-15,15]\) (b) \([-10,10]\) (c) \([-8,8]\) (d) \([-1,1]\) (e) \([-0.25,0.25]\) Indicate briefly which \(x\)-window shows the true behavior of the function, and discuss reasons why the other \(x\)-windows give results that look different.

Find the equation of the circle circumscribed about the right triangle whose vertices are \((0,0),(8,0)\), and \((0,6)\).

Show that the indicated implication is true. $$ |x+4|<\frac{\varepsilon}{2} \Rightarrow|2 x+8|<\varepsilon $$

Find the best decimal approximation that your calculator allows. Begin by making a mental estimate. \((\sqrt{2}-\sqrt{3})^{4}\)

Graph the function \(f(x)=\cos x+\frac{1}{50} \sin 50 x\) using the windows given by the following ranges of \(x\) and \(y\). (a) \(-5 \leq x \leq 5,-1 \leq y \leq 1\) (b) \(-1 \leq x \leq 1,0.5 \leq y \leq 1.5\) (c) \(-0.1 \leq x \leq 0.1,0.9 \leq y \leq 1.1\) Indicate briefly which \((x, y)\)-window shows the true behavior of the function, and discuss reasons why the other \((x, y)\)-windows give results that look different. In this case, is it true that only one window gives the important behavior, or do we need more than one window to graphically communicate the behavior of this function?

Show that the two circles \(x^{2}+y^{2}-4 x-2 y-11=0\) and \(x^{2}+y^{2}+20 x-12 y+72=0\) do not intersect. Hint: Find the distance between their centers.

Suppose that both \(f\) and \(g\) have inverses and that \(h(x)=(f \circ g)(x)=f(g(x))\). Show that \(h\) has an inverse given by \(h^{-1}=g^{-1} \circ f^{-1}\).

Find \(\delta\) (depending on \(\mathrm{s}\) ) so that the given implication is true. $$ |x-5|<\delta \Rightarrow|3 x-15|<\varepsilon $$

Find the best decimal approximation that your calculator allows. Begin by making a mental estimate. \(\sqrt[4]{1.123}-\sqrt[3]{1.09}\)

Let \(f(x)=\frac{3 x+2}{x^{2}+1}\) and \(g(x)=\frac{1}{100} \cos (100 x)\). (a) Use functional composition to form \(h(x)=(f \circ g)(x)\), as well as \(j(x)=(g \circ f)(x)\). (b) Find the appropriate window or windows that give a clear picture of \(h(x)\). (c) Find the appropriate window or windows that give a clear picture of \(j(x)\).

What relationship between \(a, b\), and \(c\) must hold if \(x^{2}+a x+y^{2}+b y+c=0\) is the equation of a circle?

Find the best decimal approximation that your calculator allows. Begin by making a mental estimate. \((3.1415)^{-1 / 2}\)

Let \(f(x)=\frac{a x+b}{c x+d}\) and assume \(b c-a d \neq 0\). (a) Find the formula for \(f^{-1}(x)\). (b) Why is the condition \(b c-a d \neq 0\) needed? (c) What condition on \(a, b, c\), and \(d\) will make \(f=f^{-1}\) ?

Find the best decimal approximation that your calculator allows. Begin by making a mental estimate. \(\sqrt{8.9 \pi^{2}+1}-3 \pi\)

Suppose that a continuous function is periodic with period 2 and is quadratic between \(-0.25\) and \(0.25\) and linear between \(-1.75\) and \(-0.25\). In addition, it has the value 0 at 0 and \(0.0625\) at \(\pm 0.25\). Sketch the function over the domain \([-2,2]\), and give a piecewise definition of the function.

Let \(f(x)=\frac{x-3}{x+1}\). Show that \(f(f(f(x)))=x\), provided \(x \neq \pm 1\).

Find the best decimal approximation that your calculator allows. Begin by making a mental estimate. \(\sqrt[4]{\left(6 \pi^{2}-2\right) \pi}\)

Let \(f(x)=\frac{x}{x-1}\). Find and simplify each value. (a) \(f(1 / x)\) (b) \(f(f(x))\) (c) \(f(1 / f(x))\)

On a lathe, you are to turn out a disk (thin right circular cylinder) of circumference 10 inches. This is done by continually measuring the diameter as you make the disk smaller. How closely must you measure the diameter if you can tolerate an error of at most \(0.02\) inch in the circumference?

Show that between any two different real numbers there is a rational number. (Hint: If \(a0\), so there is a natural number \(n\) such that \(1 / nb\\}\) and use the fact that a set of integers that is bounded from below contains a least element.) Show that between any two different real numbers there are infinitely many rational numbers.

Show that the set of points that are twice as far from \((3,4)\) as from \((1,1)\) form a circle. Find its center and radius.

Let \(f(x)=\frac{x}{\sqrt{x}-1}\). Find and simplify. (a) \(f\left(\frac{1}{x}\right)\) (b) \(f(f(x))\)

Fahrenheit temperatures and Celsius temperatures are related by the formula \(C=\frac{5}{9}(F-32)\). An experiment requires that a solution be kept at \(50^{\circ} \mathrm{C}\) with an error of at most \(3 \%\) (or \(1.5^{\circ}\) ). You have only a Fahrenheit thermometer. What error are you allowed on it?

Estimate the number of cubic inches in your head.

Prove that the operation of composition of functions is associative; that is, \(f_{1} \circ\left(f_{2} \circ f_{3}\right)=\left(f_{1} \circ f_{2}\right) \circ f_{3}\).

Solve the inequalities. $$ |x-1|<2|x-3| $$

Estimate the length of the equator in feet. Assume the radius of the earth to be 4000 miles.

Consider a circle \(C\) and a point \(P\) exterior to the circle. Let line segment \(P T\) be tangent to \(C\) at \(T\), and let the line through \(P\) and the center of \(C\) intersect \(C\) at \(M\) and \(N\). Show that \((P M)(P N)=(P T)^{2} .\)

Solve the inequalities. $$ |2 x-1| \geq|x+1| $$

Solve the inequalities. $$ 2|2 x-3|<|x+10| $$

Let \(f(x)=x^{2}-3 x\). Using the same axes, draw the graphs of \(y=f(x), y=f(x-0.5)-0.6\), and \(y=f(1.5 x)\), all on the domain \([-2,5]\).

Solve the inequalities. $$ |3 x-1|<2|x+6| $$

\((-3,2) ; 3 x+4 y=6\)

Let \(f(x)=\left|x^{3}\right|\). Using the same axes, draw the graphs of \(y=f(x), y=f(3 x)\), and \(y=f(3(x-0.8))\), all on the domain \([-3,3]\).

Prove that \(|x|<|y| \Leftrightarrow x^{2}

Write the converse and the contrapositive to the following statements. (a) If it rains today, then I will stay home from work. (b) If the candidate meets all the qualifications, then she will be hired.

Let \(f(x)=2 \sqrt{x}-2 x+0.25 x^{2}\). Using the same axes, draw the graphs of \(y=f(x), y=f(1.5 x)\), and \(y=\) \(f(x-1)+0.5\), all on the domain \([1,5]\).

Write the converse and the contrapositive to the following statements. (a) If I get an \(\mathrm{A}\) on the final exam, I will pass the course. (b) If I finish my research paper by Friday, then I will take off next week.

Let \(f(x)=1 /\left(x^{2}+1\right)\). Using the same axes, draw the graphs of \(y=f(x), y=f(2 x)\), and \(y=f(x-2)+0.6\), all on the domain \([-4,4]\).

Use the properties of the absolute value to show that each of the following is true. (a) \(|a-b| \leq|a|+|b|\) (b) \(|a-b| \geq|a|-|b|\) (c) \(|a+b+c| \leq|a|+|b|+|c|\)

Write the converse and the contrapositive to the following statements. (a) (Let \(a, b\), and \(c\) be the lengths of sides of a triangle.) If \(a^{2}+b^{2}=c^{2}\), then the triangle is a right triangle. (b) If angle \(A B C\) is acute, then its measure is greater than \(0^{\circ}\) and less than \(90^{\circ}\).

Use the Triangle Inequality and the fact that \(0<|a|<|b| \Rightarrow 1 /|b|<1 /|a|\) to establish the following chain of inequalities. $$ \left|\frac{1}{x^{2}+3}-\frac{1}{|x|+2}\right| \leq \frac{1}{x^{2}+3}+\frac{1}{|x|+2} \leq \frac{1}{3}+\frac{1}{2} $$

Write the converse and the contrapositive to the following statements. (a) If the measure of angle \(A B C\) is \(45^{\circ}\), then angle \(A B C\) is an acute angle. (b) If \(a

In Problems 67 and 68, find the (perpendicular) distance between the given parallel lines. Hint: First find a point on one of the lines. \(2 x+4 y=7,2 x+4 y=5\)

In Problems 67 and 68, find the (perpendicular) distance between the given parallel lines. Hint: First find a point on one of the lines. \(7 x-5 y=6,7 x-5 y=-1\)

Show that $$ |x| \leq 2 \Rightarrow\left|\frac{x^{2}+2 x+7}{x^{2}+1}\right| \leq 15 $$

Find the equation for the line that bisects the line segment from \((-2,3)\) to \((1,-2)\) and is at right angles to this line segment.

Show that $$ |x| \leq 1 \Rightarrow\left|x^{4}+\frac{1}{2} x^{3}+\frac{1}{4} x^{2}+\frac{1}{8} x+\frac{1}{16}\right|<2 $$

Use the rules regarding the negation of statements involving quantifiers to write the negation of the following statements. Which is true, the original statement or its negation? (a) Every isosceles triangle is equilateral. (b) There is a real number that is not an integer. (c) Every natural number is less than or equal to its square.

The center of the circumscribed circle of a triangle lies on the perpendicular bisectors of the sides. Use this fact to find the center of the circle that circumscribes the triangle with vertices \((0,4),(2,0)\), and \((4,6)\).