Chapter 1

Sketch the region in the plane satisfying the given conditions. \(x \leq 3\) and \(y \leq 2\)

The Tee-rific Company produces 100,000 golf tees daily and sells them for \(5 \phi\) apiece. Assume that the total cost of producing one tee is \(2 \phi .\) Find the company's profit \(P\) (in cents) in terms of the number of working days.

Postage for domestic first class letters is \(37 \phi\) for the first ounce or part ounce and \(23 \phi\) for each additional ounce or part ounce up to 14 ounces. Let \(P(x)\) denote the postage for a letter weighing \(x\) ounces, and assume that the domain of \(P\) is \((0,14)\). Express \(P(x)\) in terms of the greatest integer function.

Solve the inequality. $$ |x+1|<0.01 $$

Sketch the region in the plane satisfying the given conditions. \(x \geq-1\) and \(y \geq \frac{1}{2}\)

Starting at noon, \(A\) flies 2400 miles from New York to San Francisco at a velocity of 400 miles per hour. \(B\) starts the same trip at \(2: 00\) P.M. the same day with a velocity of 800 miles per hour. Express the distance \(D\) between \(A\) and \(B\) at any instant between noon and \(5: 00 \mathrm{P} . \mathrm{M}\). in terms of the time in hours elapsed after noon.

Solve the inequality. $$ \left|x+\frac{1}{2}\right| \leq 2 $$

Sketch the region in the plane satisfying the given conditions. \(x>2\) and \(y<1\)

Two cars depart from the same location at the same time. One travels north at 40 miles per hour and the other travels east at 50 miles per hour. Find a formula for the function \(D\) that expresses in terms of \(t\) the distance between the cars \(t\) hours after departure.

Let \(f(x)=4 x(1-x)\). Calculate the first 100 iterates of \(f\) at \(0.3\), and see if you observe any pattern or repetitions in the iterates.

Solve the inequality. $$ |x+3| \geq 3 $$

a. Describe the region consisting of all \((x, y)\) for which $$ (x, y)=(x,-y) . $$ b. Describe the region consisting of all \((x, y)\) for which $$ (x, y)=(-x, y) . $$

In the study of the response to acetylcholine by a frog's heart, the formula $$ R(x)=\frac{x}{c+d x} $$ arises, where \(x\) denotes the concentration of the drug and \(c\) and \(d\) are positive constants. a. Find \(R(0)\) and \(R\) (2). What is the physical significance of \(R(0)\) ? b. Find a formula that expresses the concentration \(x\) in terms of \(R(x)\).

The revenue function \(R\) for a certain product is given by $$ R(x)=5 x^{2}-\frac{x^{4}}{10} $$ The cost function \(C\) is given by $$ C(x)=4 x^{2}-24 x+38 $$ The profit function \(P\) is defined as the difference \(R-C\). Find the equation that describes \(P\). Then find \(P(1)\) and \(P(2)\), and show that it is possible to lose money and also possible to make a profit.

Solve the inequality. $$ |x-0.3|>1.5 $$

Suppose two vertices of a rectangle \(R\) are \((2,5)\) and \((7,1)\), and the sides of \(R\) are parallel to the coordinate axes. Determine the other vertices of \(R\).

Recall that the volume \(V(r)\) of a spherical balloon of radius \(r\) is given by the formula $$ V(r)=\frac{4}{3} \pi r^{3} \quad \text { for } r \geq 0 $$ Suppose the radius is given by \(r(t)=3 \sqrt{t} .\) Write a formula for the volume in terms of \(t\).

Solve the inequality. $$ |2 x+1| \geq 1 $$

Suppose the sides of a square \(S\) are 4 units long and are parallel to the coordinate axes. If \((-3,3)\) is the vertex of \(S\) closest to the origin, find the other vertices of \(S\).

A sphere with surface area \(s\) has a radius \(r(s)\) given by $$ r(s)=\frac{1}{2} \sqrt{\frac{s}{\pi}} $$ a. Using the formula in Exercise 64 , find a formula for the volume of a sphere in terms of its surface area. b. Determine the volume corresponding to a surface area of 6 .

Solve the inequality. $$ |3 x-5| \leq 2 $$

Let \((2,1),(-3,-2)\), and \((a, b)\) form a triangle. Show that the collection of points \((a, b)\) for which the triangle is isosceles, and for which \((a, b)\) is the vertex common to the two sides of equal length, is a line (with one point deleted). Find an equation of that line.

According to Newton's Law of Gravitation, if two bodies are a distance \(r\) apart, then the gravitational force \(F(r)\) exerted by one body on the other is given by $$ F(r)=\frac{k}{r^{2}} \quad \text { for } r>0 $$ where \(k\) is a positive constant. Suppose that as a function of time \(t\), the distance between the two bodies is given by $$ r(t)=4000\left(\frac{1+t}{1+t^{2}}\right) \text { for } t \geq 0 $$ Find a formula for the force in terms of time.

Solve the inequality. $$ \left|2 x-\frac{1}{3}\right|>\frac{2}{3} $$

Solve the inequality. $$ 0<|x-1|<0.5 $$

Show that the midpoints of the sides of any rectangle are the vertices of a rhombus (a quadrilateral with all sides of equal length). (Hint: Let the vertices of the rectangle be \((0,0),(a, 0),(0, b)\), and \((a, b) .)\)

Solve the inequality. $$ -1<|4-2 x|<1 $$

Show that in any triangle the sum of the squares of the lengths of the medians (the line segments joining the vertices to the midpoints of the opposite sides) is equal to three fourths the sum of the squares of the lengths of the sides. (Hint: Pick the vertices of the triangle judiciously.)

Solve the inequality. $$ |x-a| \leq d $$

Show that the sum of the squares of the lengths of the sides of a parallelogram is equal to the sum of the squares of the lengths of the diagonals.

Modify the expression, and then find its approximate value by calculator or computer. $$ \frac{69^{800}}{59^{800}} $$

Modify the expression, and then find its approximate value by calculator or computer. $$ \frac{221^{907}}{221^{897}} $$

A bookcase is 7 feet tall, 4 feet wide, and 1 foot deep. What is the minimum vertical distance that is necessary in order to raise the bookcase from a horizontal position on the floor to a vertical position against the wall?

Modify the expression, and then find its approximate value by calculator or computer. $$ \frac{(0.123)^{9000}}{(0.125)^{9000}} $$

a. Show that \(1 /(\sqrt{25,000}-\sqrt{24,998})=\frac{1}{2}(\sqrt{25,000}\) \(+\sqrt{24,998})\) b. Compare the value your calculator gives for \(1 /(\sqrt{25,000}-\sqrt{24,998})\) with the value it gives for \(\frac{1}{2}(\sqrt{25,000}+\sqrt{24,998})\). If they are different, which do you think is the more accurate?

Find all numbers \(x\) the sum of whose distances from 12 and from 13 exceeds 4 . Draw a figure to illustrate your solution.

Find all numbers \(x\) with the property that the distance from \(x\) to 2 is less than twice the distance from \(x\) to 3 .

If \(x^{2} \leq 25\), is it necessarily true that \(x \leq 5 ?\) Explain.

If \(x^{3}>125\), is it necessarily true that \(x>5\) ? Explain.

Is \(1 / x

a. Show that \(x1\). b. Show that \(x^{2}

Use the definition of absolute value to prove the following. a. \(|a b|=|a||b|\) b. \(-|b| \leq b \leq|b|\) c. \(|a-b|=|b-a|\)

Show that \(|a+b|=|a|+|b|\) if and only if \(a b \geq 0\) (which means that \(a=0, b=0\), or \(a\) and \(b\) have the same sign).

Prove that if \(a

Prove that if \(00 .\right)\)

Let \(0

Prove that \(\sqrt{2}\) is irrational. (Hint: Assume that \(\sqrt{2}=\) \(p / q\), where \(p\) and \(q\) are integers such that at most one of them is divisible by \(2 .\) It can be shown that a square integer is divisible by 2 only if it is also divisible by 4 . Use this fact to show first that \(p\) is divisible by 2 and then that \(q\) is also divisible by \(2 .\) This contradicts the assumption.)

A rectangle \(R\) has length \(x\) and width \(y\). a. Write an inequality that expresses the condition that the area of \(R\) is less than 10 . b. Write an inequality that expresses the condition that the perimeter of \(R\) is at least 47 .

Show that if a square and a circle have equal perimeters, then the circle has a larger area than the square. (Hint: Show first that a circle of perimeter \(P\) has area \(\left.P^{2} / 4 \pi .\right)\)