Chapter 1

Circular motion can be modeled by using the parametric representations of the form \(x(t)=\sin t\) and \(y(t)=\cos t\) (A parametric representation means that a variable, \(t\) in this case, determines both \(x(t)\) and \(y(t) .)\) This will give the full circle for \(0 \leq t \leq 2 \pi .\) If we consider a 4 -foot- diameter wheel making one complete rotation clockwise once every 10 seconds, show that the motion of a point on the rim of the wheel can be represented by \(x(t)=2 \sin (\pi t / 5)\) and \(y(t)=2 \cos (\pi t / 5)\) (a) Find the positions of the point on the rim of the wheel when \(t=2\) seconds, 6 seconds, and 10 seconds. Where was this point when the wheel started to rotate at \(t=0\) ? (b) How will the formulas giving the motion of the point change if the wheel is rotating counterclockwise. (c) At what value of \(t\) is the point at \((2,0)\) for the first time?

To use technology effectively, you need to discover its capabilities, its strengths, and its weaknesses. We urge you to practice graphing functions of various types using your own computer package or calculator. Graph \(f(x)=(3 x-4) /\left(x^{2}+x-6\right)\) on the domain \([-6,6]\) (a) Determine the \(x\) - and \(y\) -intercepts. (b) Determine the range of \(f\) for the given domain. (c) Determine the vertical asymptotes of the graph. (d) Determine the horizontal asymptote for the graph when the domain is enlarged to the natural domain.

Show that the indicated implication is true $$ |x-3|<0.5 \Rightarrow|5 x-15|<2.5 $$

\(\approx\) 50. A belt fits tightly around the two circles, with equations \((x-1)^{2}+(y+2)^{2}=16\) and \((x+9)^{2}+(y-10)^{2}=16\) How long is this belt?

After being in business for \(t\) years, a manufacturer of cars is producing \(120+2 t+3 t^{2}\) units per year. The sales price in dollars per unit has risen according to the formula \(6000+700 t\). Write a formula for the manufacturer's yearly revenue \(R(t)\) after \(t\) years.

. The circular frequency \(v\) of oscillation of a point is given by \(v=\frac{2 \pi}{\text { period }}\). What happens when you add two motions that have the same frequency or period? To investigate, we can graph the functions \(y(t)=2 \sin (\pi t / 5)\) and \(y(t)=\sin (\pi t / 5)+\) \(\cos (\pi t / 5)\) and look for similarities. Armed with this information. we can investigate by graphing the following functions over the interval \([-5,5]:\) (a) \(y(t)=3 \sin (\pi t / 5)-5 \cos (\pi t / 5)+2 \sin ((\pi t / 5)-3)\) (b) \(y(t)=3 \cos (\pi t / 5-2)+\cos (\pi t / 5)+\cos ((\pi t / 5)-3)\)

Show that the indicated implication is true $$ |x+2|<0.3 \Rightarrow|4 x+8|<1.2 $$

$$ \text {. Find two irrational numbers whose sum is rational. } $$

Show that the midpoint of the hypotenuse of any right triangle is equidistant from the three vertices.

Starting at noon, airplane A flies due north at 400 miles per hour. Starting 1 hour later, airplane \(\mathrm{B}\) flies due east at 300 miles per hour. Neglecting the curvature of the Earth and assuming that they fly at the same altitude, find a formula for \(D(t)\), the distance between the two airplanes \(t\) hours after noon. Hint: There will be two formulas for \(D(t)\), one if \(0

. We now explore the relationship between \(A \sin (\omega t)+\) \(B \cos (\omega t)\) and \(C \sin (\omega t+\phi)\) (a) By expanding \(\sin (\omega t+\phi)\) using the sum of the angles formula, show that the two expressions are equivalent if \(A=C \cos \phi\) and \(B=C \sin \phi\) (b) Consequently, show that \(A^{2}+B^{2}=C^{2}\) and that \(\phi\) then satisfies the equation \(\tan \phi=\frac{B}{A}\). (c) Generalize your result to state a proposition about \(A_{1} \sin \left(\omega t+\phi_{1}\right)+A_{2} \sin \left(\omega t+\phi_{2}\right)+A_{3} \sin \left(\omega t+\phi_{3}\right)\) (d) Write an essay, in your own words, that expresses the importance of the identity between \(A \sin (\omega t)+B \cos (\omega t)\) and \(C \sin (\omega t+\phi) .\) Be sure to note that \(|C| \geq \max (|A|,|B|)\) and that the identity holds only when you are forming a linear combination (adding and/or subtracting multiples of single powers) of sine and cosine of the same frequency.

Show that the indicated implication is true $$ |x-2|<\frac{\varepsilon}{6} \Rightarrow|6 x-12|<\varepsilon $$

\(\approx\)s 51-56, find the best decimal approximation that your calculator allows. Begin by making a mental estimate. $$ (\sqrt{3}+1)^{3} $$

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

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.

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$$ (\sqrt{2}-\sqrt{3})^{4} $$

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 \(h^{-1}=g^{-1} \circ f^{-1}\).

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?

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

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?

. 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 \(j(x)=(g \circ f)(x)\). $$ \begin{array}{l} \text { (a) Use functional composition to form } h(x)=(f \circ g)(x), \text { as }\\\ \text { well as } j(x)=(g \circ f)(x) \text { . } \end{array} $$

Find \(\delta\) (depending on \(\varepsilon\) ) so that the given implication is true. $$ |x-2|<\delta \Rightarrow|4 x-8|<\varepsilon $$

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 \(f=f^{-1}\) ?

Suppose that a continuous function is periodic with period 1 and is linear between 0 and \(0.25\) and linear between \(-0.75\) and \(0 .\) In addition, it has the value 1 at 0 and 2 at \(0.25 .\) Sketch the function over the domain \([-1,1]\), and give a piecewise definition of the function.

Find \(\delta\) (depending on \(\varepsilon\) ) so that the given implication is true. $$ |x+6|<\delta \Rightarrow|6 x+36|<\varepsilon $$

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.

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 \(f(x)=\frac{1}{x-1}\). 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?

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. $$ 2|2 x-3|<|x+10| $$

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

Use a computer or a graphing calculator in Problems \(62-65 .\) 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 \([-2,5]\).

Use this result to find the distance from the given point to the given line. (-3,2) ; 3 x+4 y=6

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

63\. 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.

Use a computer or a graphing calculator 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 \([-3,3]\)

Use this result to find the distance from the given point to the given line. (4,-1) ; 2 x-2 y+4=0