Chapter 1

Which of the following are true? Unless it is stated otherwise, assume that \(x, y\), and \(\varepsilon\) are real numbers. (a) For every \(x, x0\), there exists a \(y\) such that \(y>\frac{1}{x}\). (d) For every positive \(x\), there exists a natural number \(n\) such that \(\frac{1}{n}

Find the equations of the two tangent lines to the circle \(x^{2}+y^{2}=36\) that go through \((12,0)\). Hint: See Problem 72

The number \(\sqrt{a b}\) is called the geometric mean of two positive numbers \(a\) and \(b\). Prove that $$ 0

Prove the following statements. (a) If \(n\) is odd, then \(n^{2}\) is odd. (b) If \(n^{2}\) is odd, then \(n\) is odd.

Express the perpendicular distance between the parallel lines \(y=m x+b\) and \(y=m x+B\) in terms of \(m, b\), and \(B\). Hint: The required distance is the same as that between \(y=m x\) and \(y=m x+B-b\).

For two positive numbers \(a\) and \(b\), prove that $$ \sqrt{a b} \leq \frac{1}{2}(a+b) $$ This is the simplest version of a famous inequality called the geometric mean-arithmetic mean inequality.

Prove that \(n\) is odd if and only if \(n^{2}\) is odd.

Show that the line through the midpoints of two sides of a triangle is parallel to the third side. Hint: You may assume that the triangle has vertices at \((0,0),(a, 0)\), and \((b, c)\).

Show that, among all rectangles with given perimeter \(p\), the square has the largest area. Hint: If \(a\) and \(b\) denote the lengths of adjacent sides of a rectangle of perimeter \(p\), then the area is \(a b\), and for the square the area is \(a^{2}=[(a+b) / 2]^{2}\). Now see Problem 74 .

According to the Fundamental Theorem of Arithmetic, every natural number greater than 1 can be written as the product of primes in a unique way, except for the order of the factors. For example, \(45=3 \cdot 3 \cdot 5\). Write each of the following as a product of primes. (a) 243 (b) 124 (c) 5100

Show that the line segments joining the midpoints of adjacent sides of any quadrilateral (four-sided polygon) form a parallelogram.

Use the Fundamental Theorem of Arithmetic (Problem 75) to show that the square of any natural number greater than 1 can be written as the product of primes in a unique way, except for the order of the factors, with each prime occurring an even number of times. For example, \((45)^{2}=3 \cdot 3 \cdot 3 \cdot 3 \cdot 5 \cdot 5\).

A wheel whose rim has equation \(x^{2}+(y-6)^{2}=25\) is rotating rapidly in the counterclockwise direction. A speck of dirt on the rim came loose at the point \((3,2)\) and flew toward the wall \(x=11\). About how high up on the wall did it hit? Hint: The speck of dirt flies off on a tangent so fast that the effects of gravity are negligible by the time it has hit the wall.

The formula \(\frac{1}{R}=\frac{1}{R_{1}}+\frac{1}{R_{2}}+\frac{1}{R_{3}}\) gives the total resistance \(R\) in an electric circuit due to three resistances, \(R_{1}, R_{2}\), and \(R_{3}\), connected in parallel. If \(10 \leq R_{1} \leq 20,20 \leq R_{2} \leq 30\), and \(30 \leq R_{3} \leq 40\), find the range of values for \(R\).

The radius of a sphere is measured to be about 10 inches. Determine a tolerance \(\delta\) in this measurement that will ensure an error of less than \(0.01\) square inch in the calculated value of the surface area of the sphere.

Show that \(\sqrt{3}\) is irrational

Show that the sum of two rational numbers is rational.

Show that the product of a rational number (other than 0 ) and an irrational number is irrational.

Which of the following are rational and which are irrational? (a) \(-\sqrt{9}\) (b) \(0.375\) (c) \((3 \sqrt{2})(5 \sqrt{2})\) (d) \((1+\sqrt{3})^{2}\)

A number \(b\) is called an upper bound for a set \(S\) of numbers if \(x \leq b\) for all \(x\) in \(S\). For example \(5,6.5\), and 13 are upper bounds for the set \(S=\\{1,2,3,4,5\\}\). The number 5 is the least upper bound for \(S\) (the smallest of all upper bounds). Similarly, 1.6, 2, and \(2.5\) are upper bounds for the infinite set \(T=\\{1.4,1.49,1.499,1.4999, \ldots\\}\), whereas \(1.5\) is its least upper bound. Find the least upper bound of each of the following sets. (a) \(S=\\{-10,-8,-6,-4,-2\\}\) (b) \(S=\\{-2,-2.1,-2.11,-2.111,-2.1111, \ldots\\}\) (c) \(S=\\{2.4,2.44,2.444,2.4444, \ldots\\}\) (d) \(S=\left\\{1-\frac{1}{2}, 1-\frac{1}{3}, 1-\frac{1}{4}, 1-\frac{1}{5}, \ldots\right\\}\) (e) \(S=\left\\{x: x=(-1)^{n}+1 / n, n\right.\) a positive integer \(\\}\); that is, \(S\) is the set of all numbers \(x\) that have the form \(x=(-1)^{n}+1 / n\), where \(n\) is a positive integer. (f) \(S=\left\\{x: x^{2}<2, x\right.\) a rational number \(\\}\)

The Axiom of Completeness for the real numbers says: Every set of real numbers that has an upper bound has a least upper bound that is a real number. (a) Show that the italicized statement is false if the word real is replaced by rational. (b) Would the italicized statement be true or false if the word real were replaced by natural?