Problem 30 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

True or False: If the statement is always true, write "True;" if the statement is not always true, produce a counterexample. (a) \(\left(a^{2}+b^{2}\right)^{1 / 2}=a+b\) (b) \((a+b)^{-1}=\frac{1}{a+b}, \quad a, b \neq 0\) (c) \((a+b)^{-1}=\frac{1}{a}+\frac{1}{b}, \quad a, b \neq 0\) (d) \(R^{-1 / 2}=-\frac{1}{\sqrt{R}}, \quad R>0\) (e) \(x^{z}+x^{z}=2 x^{z}\) (f) \(x^{z} x^{z}=x^{\left(z^{2}\right)}\) (g) \(x^{z} x^{z}=x^{2 z}\)

Step-by-Step Solution

Answer

(a) False, (b) True, (c) False, (d) False, (e) True, (f) False, (g) True

Step 1: Problem (a)

The left side of the equation is the square root of the sum of squares, while the right side is a direct sum of two numbers. The equality \(\left(a^{2}+b^{2}\right)^{1 / 2}=a+b\) does not hold universally. Consider the simple counterexample: let a and b both equal 1. Then we find \(\left(1^2 + 1^2\right)^{1/2} = \sqrt{2} \neq 1 + 1 = 2\). Therefore, the statement is false.

Step 2: Problem (b)

Considering the equality \((a+b)^{-1}=\frac{1}{a+b}\), this indeed holds true, as \((a+b)^{-1}\) is a shorthand for expressing the reciprocal of the sum a + b, which can be written as \(\frac{1}{a+b}\). Thus, the statement is true.

Step 3: Problem (c)

For the given equation \((a+b)^{-1}=\frac{1}{a}+\frac{1}{b}\) let's inspect it closely. Set again a and b to 1. Then we find \( (1+1)^{-1} = 0.5 \neq \frac{1}{1} + \frac{1}{1} = 2\), hence the equation does not hold always true and the statement is false.

Step 4: Problem (d)

Let's focus on the equation \(R^{-1 / 2}=-\frac{1}{\sqrt{R}}\). This suggests that the negative square root of a number is equivalent to the negative reciprocal of its square root, this is not accurate. Let R = 4, then \(4^{-1/2} = \frac{1}{2} \neq -\frac{1}{\sqrt{4}} = -\frac{1}{2}\). Thus, the statement is false.

Step 5: Problem (e)

Looking at the equality \(x^{z}+x^{z}=2 x^{z}\), this holds true:) since adding two of the same quantities (x^z) is indeed equivalent to multiplying that quantity by 2. Thus, the statement is true.

Step 6: Problem (f)

For the equation \(x^{z} x^{z}=x^{\left(z^{2}\right)}\) let's pick x = 2 and z = 2. Then \(2^{2} * 2^{2} = 16\) while \(2^{2^2} = 2^{4} = 16\). Hence, this relation does not always hold true, and the statement is false.

Step 7: Problem (g)

Addressing the equation \(x^{z} x^{z}=x^{2 z}\), we know from the exponent multiplication rule that when you multiply two powers with the same base, the exponents get added, so \(x^{z} x^{z} = x^{z + z} = x^{2z}\). Thus, the statement is true.

Key Concepts

Counterexamples in Mathematics

Counterexamples play a crucial role in mathematical problem-solving, primarily because they provide a straightforward means of disproving a statement's universality. A counterexample is a specific case or instance that shows a general statement to be false. It's important to remember that to prove a statement false, you only need to find one counterexample.

For instance, in our original exercise, problem (a) proposed that \(\left(a^{2}+b^{2}\right)^{1 / 2}=a+b\) holds true for all values of a and b. However, by choosing both a and b as 1, we get a counterexample, as \(\sqrt{2}\) is not equal to 2, showing that the square root of the sum of two squares doesn't necessarily equal the sum of those two numbers. This exercise demonstrates the importance of checking more than one example before concluding the truth of a statement and how a single counterexample can be sufficient to disprove a claim.

Exponent Rules

Understanding exponent rules, also known as laws of exponents, is essential for simplifying and manipulating expressions involving powers. Some of the fundamental exponent rules include:

• The product rule: \(x^{a} \cdot x^{b} = x^{a+b}\) implies when multiplying like bases, add their exponents.
• The power rule: \(\left(x^{a}\right)^{b} = x^{a \cdot b}\) indicates that when raising a power to another power, multiply the exponents.
• The quotient rule: \(\frac{x^{a}}{x^{b}} = x^{a-b}\) tells us when dividing like bases, subtract the exponents.

In problems (f) and (g) from our original exercise, the product rule is applied. The correct application in problem (g) \(x^{z} x^{z}=x^{2 z}\) reflects the rule accurately, showing that when you multiply like bases, you add the exponents \(z + z\) to get \(2z\). However, problem (f) \(x^{z} x^{z}=x^{\left(z^{2}\right)}\) incorrectly applies the rule; it's misleading to raise the exponent to a power when you should be adding them. These distinctions are crucial for mastering algebraic expressions and equations.

Properties of Equality

The properties of equality are basic rules that guide the manipulation of equations, ensuring that they remain balanced. Some critical properties include the Additive Property (if \(a = b\), then \(a + c = b + c\)), the Multiplicative Property (if \(a = b\), then \(a \cdot c = b \cdot c\)), and the Substitution Property (if \(a = b\), then a can be replaced with b in any expression).

For example, in our textbook solution problem (e), the statement \(x^{z}+x^{z}=2 x^{z}\) showcases the Additive Property; adding the same quantity to itself is equivalent to multiplying it by 2. This is a simple yet powerful property that upholds the equality of the given equation. Understanding these properties is fundamental not just for solving equations, but for all areas of mathematics that require equation manipulation and balancing.