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Chapter 1: Problem 90

Simplify the difference quotients \(\frac{f(x+h)-f(x)}{h}\) and \(\frac{f(x)-f(a)}{x-a}\) by rationalizing the numerator. $$f(x)=\sqrt{1-2 x}$$

Short Answer

Expert verified
Question: Simplify the given difference quotients for the function \(f(x) = \sqrt{1-2x}\): (1) \(\frac{f(x+h)-f(x)}{h}\) and (2) \(\frac{f(x)-f(a)}{x-a}\). Answer: (1) \(\frac{-2}{\sqrt{1-2(x+h)} + \sqrt{1-2x}}\) (2) \(\frac{-2}{\sqrt{1-2x} + \sqrt{1-2a}}\)

Step by step solution

01

Write down the given function and difference quotients

The given function is \(f(x) = \sqrt{1-2x}\). We want to simplify two difference quotients: 1. \(\frac{f(x+h)-f(x)}{h}\) 2. \(\frac{f(x)-f(a)}{x-a}\)
02

Substitute the given function into the first difference quotient

Substitute \(f(x+h)\) and \(f(x)\) into the first difference quotient \(\frac{f(x+h)-f(x)}{h}\): $$\frac{\sqrt{1-2(x+h)} - \sqrt{1-2x}}{h}$$
03

Rationalize the numerator of the first difference quotient

Rationalizing the numerator involves multiplying both the numerator and denominator by the conjugate of the numerator. For \(\frac{\sqrt{1-2(x+h)} - \sqrt{1-2x}}{h}\), the conjugate is \(\sqrt{1-2(x+h)} + \sqrt{1-2x}\): $$\frac{\sqrt{1-2(x+h)} - \sqrt{1-2x}}{h} \cdot \frac{\sqrt{1-2(x+h)} + \sqrt{1-2x}}{\sqrt{1-2(x+h)} + \sqrt{1-2x}}$$
04

Simplify the first difference quotient

Multiply the numerators and denominators from Step 3: $$\frac{(1-2(x+h))-(1-2x)}{h(\sqrt{1-2(x+h)} + \sqrt{1-2x})}$$ Now, simplify the expression: $$\frac{-2h}{h(\sqrt{1-2(x+h)} + \sqrt{1-2x})} = \frac{-2}{\sqrt{1-2(x+h)} + \sqrt{1-2x}}$$ The simplified first difference quotient is: $$\frac{-2}{\sqrt{1-2(x+h)} + \sqrt{1-2x}}$$
05

Substitute the given function into the second difference quotient

Substitute \(f(x)\) and \(f(a)\) into the second difference quotient \(\frac{f(x)-f(a)}{x-a}\): $$\frac{\sqrt{1-2x} - \sqrt{1-2a}}{x-a}$$
06

Rationalize the numerator of the second difference quotient

Rationalizing the numerator involves multiplying both the numerator and denominator by the conjugate of the numerator. For \(\frac{\sqrt{1-2x} - \sqrt{1-2a}}{x-a}\), the conjugate is \(\sqrt{1-2x} + \sqrt{1-2a}\): $$\frac{\sqrt{1-2x} - \sqrt{1-2a}}{x-a} \cdot \frac{\sqrt{1-2x} + \sqrt{1-2a}}{\sqrt{1-2x} + \sqrt{1-2a}}$$
07

Simplify the second difference quotient

Multiply the numerators and denominators from Step 6: $$\frac{(1-2x)-(1-2a)}{(x-a)(\sqrt{1-2x} + \sqrt{1-2a})}$$ Now, simplify the expression: $$\frac{-2(x-a)}{(x-a)(\sqrt{1-2x} + \sqrt{1-2a})} = \frac{-2}{\sqrt{1-2x} + \sqrt{1-2a}}$$ The simplified second difference quotient is: $$\frac{-2}{\sqrt{1-2x} + \sqrt{1-2a}}$$

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