Chapter 1
Calculus Early Transcendentals: Pearson New International Edition · 441 exercises
Problem 64
Write the converse and the contrapositive to the following statements. (a) If I get an 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.
7 step solution
Problem 64
Use a computer or a graphing calculator 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 \([1,5]\).
6 step solution
Problem 65
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 \(90^{\circ}\)
4 step solution
Problem 65
Use a computer or a graphing calculator 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 \([-4,4]\).
4 step solution
Problem 66
Use this result to find the distance from the given point to the given line. (3,-1) ; y=2 x-5
6 step solution
Problem 66
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) \(|a+b+c| \leq|a|+|b|+|c|\)
3 step solution
Problem 66
Use the Triangle Inequality and the fact that \(0<|a|<|b| \Rightarrow 1 /|b|<1 /|a|\) to establish the following chain of inequalities. $$ \left|\frac{1}{x^{2}+3}-\frac{1}{|x|+2}\right| \leq \frac{1}{x^{2}+3}+\frac{1}{|x|+2} \leq \frac{1}{3}+\frac{1}{2} $$
3 step solution
Problem 66
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
6 step solution
Problem 67
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
6 step solution
Problem 68
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
3 step solution
Problem 68
Show that $$ |x| \leq 2 \Rightarrow\left|\frac{x^{2}+2 x+7}{x^{2}+1}\right| \leq 15 $$
6 step solution
Problem 69
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.
4 step solution
Problem 69
Show that $$ |x| \leq 1 \Rightarrow\left|x^{4}+\frac{1}{2} x^{3}+\frac{1}{4} x^{2}+\frac{1}{8} x+\frac{1}{16}\right|<2 $$
5 step solution
Problem 69
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.
4 step solution
Problem 70
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 \((4,6)\)
4 step solution
Problem 70
Show each of the following:
(a) \(x
5 step solution
Problem 70
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 natural number is rational. (b) There is a circle whose area is larger than \(9 \pi\). (c) Every real number is larger than its square.
4 step solution
Problem 71
Find the radius of the circle that is inscribed in a triangle with sides of lengths 3,4 , and 5 (see Figure 21).
5 step solution
Problem 71
Show that \(a \neq 0 \Rightarrow a^{2}+1 / a^{2} \geq 2\). Hint: Consider \((a-1 / a)^{2}\)
4 step solution
Problem 71
Which of the following are true? Assume that \(x\) and \(y\) are real numbers.
(a) For every \(x, x>0 \Rightarrow x^{2}>0\).
(b) For every \(x, x>0 \Leftrightarrow x^{2}>0\).
(c) For every \(x, x^{2}>x\).
(d) For every \(x\), there exists a \(y\) such that \(y>x^{2}\).
(e) For every positive number \(y\), there exists another positive number \(x\)
such that \(0
5 step solution
Problem 72
Suppose that \((a, b)\) is on the circle \(x^{2}+y^{2}=r^{2}\). Show that the line \(a x+b y=r^{2}\) is tangent to the circle at \((a, b)\).
5 step solution
Problem 72
The number \(\frac{1}{2}(a+b)\) is called the average, or arithmetic mean, of \(a\) and \(b\). Show that the arithmetic mean of two numbers is between the two numbers; that is, prove that $$ a
4 step solution
Problem 72
Which of the following are true? Unless it is stated otherwise, assume that
\(x, y\), and \(\varepsilon\) are real numbers.
(a) For every \(x, x
5 step solution
Problem 73
The number \(\sqrt{a b}\) is called the geometric mean of two positive numbers
\(a\) and \(b\). Prove that
$$
0
4 step solution
Problem 73
. Prove the following statements. (a) If \(n\) is odd, then \(n^{2}\) is odd. (Hint: If \(n\) is odd, then there exists an integer \(k\) such that \(n=2 k+1 .\) ) (b) If \(n^{2}\) is odd, then \(n\) is odd. (Hint: Prove the contrapositive.)
7 step solution
Problem 74
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 \(y=m x+B-b\)
5 step solution
Problem 74
For two positive numbers \(a\) and \(b\), prove that $$ \sqrt{a b} \leq \frac{1}{2}(a+b) $$
6 step solution
Problem 74
Prove that \(n\) is odd if and only if \(n^{2}\) is odd. (See Problem \(73 .\) )
4 step solution
Problem 75
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 \((b, c)\).
4 step solution
Problem 75
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 .
6 step solution
Problem 75
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
3 step solution
Problem 76
Show that the line segments joining the midpoints of adjacent sides of any quadrilateral (four-sided polygon) form a parallelogram.
5 step solution
Problem 76
. 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, \((45)^{2}=3 \cdot 3 \cdot 3 \cdot 3 \cdot 5 \cdot 5\).
5 step solution
Problem 77
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.
4 step solution
Problem 77
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 \(R\)
5 step solution
Problem 78
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.
5 step solution
Problem 79
79\. Show that the sum of two rational numbers is rational.
5 step solution
Problem 80
Show that the product of a rational number (other than 0 ) and an irrational number is irrational. Hint: Try proof by contradiction.
7 step solution
Problem 81
. Which of the following are rational and which are irrational? (a) \(-\sqrt{9}\) (b) \(0.375\) (c) \((3 \sqrt{2})(5 \sqrt{2})\) (d) \((1+\sqrt{3})^{2}\)
4 step solution
Problem 82
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 \(\\}\)
6 step solution
Problem 83
. 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?
3 step solution