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Chapter 1: Problem 89

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

Short Answer

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

Step by step solution

01

Calculate f(x+h) for f(x) = sqrt(x)

In this step, we will replace 'x' in the function by 'x+h' to find the expression for f(x+h). \(f(x) = \sqrt{x}\) \(f(x+h) = \sqrt{x+h}\)
02

Write down the expressions for the difference quotients

Now that we have f(x+h), write down the difference quotients given in the exercise. \(\frac{f(x+h)-f(x)}{h} = \frac{\sqrt{x+h}-\sqrt{x}}{h}\) \(\frac{f(x)-f(a)}{x-a} = \frac{\sqrt{x}-\sqrt{a}}{x-a}\)
03

Rationalize the numerator of the first difference quotient

In this step, we'll multiply the numerator and denominator of the first difference quotient by the conjugate of the numerator. The conjugate of \(\sqrt{x+h}-\sqrt{x}\) is \(\sqrt{x+h}+\sqrt{x}\). \(\frac{\sqrt{x+h}-\sqrt{x}}{h} \cdot \frac{\sqrt{x+h}+\sqrt{x}}{\sqrt{x+h}+\sqrt{x}} = \frac{((\sqrt{x+h}+\sqrt{x})(\sqrt{x+h}-\sqrt{x}))}{(h(\sqrt{x+h}+\sqrt{x}))}\)
04

Simplify the first difference quotient

Now, we must simplify the fraction obtained in the previous step: \(\frac{((\sqrt{x+h}+\sqrt{x})(\sqrt{x+h}-\sqrt{x}))}{(h(\sqrt{x+h}+\sqrt{x}))} = \frac{(x+h)-x}{h(\sqrt{x+h}+\sqrt{x})} = \frac{h}{h(\sqrt{x+h}+\sqrt{x})} = \frac{1}{\sqrt{x+h}+\sqrt{x}}\)
05

Rationalize the numerator of the second difference quotient

In this step, we'll multiply the numerator and denominator of the second difference quotient by the conjugate of the numerator. The conjugate of \(\sqrt{x}-\sqrt{a}\) is \(\sqrt{x}+\sqrt{a}\). \(\frac{\sqrt{x}-\sqrt{a}}{x-a} \cdot \frac{\sqrt{x}+\sqrt{a}}{\sqrt{x}+\sqrt{a}} = \frac{((\sqrt{x}+\sqrt{a})(\sqrt{x}-\sqrt{a}))}{((x-a)(\sqrt{x}+\sqrt{a}))}\)
06

Simplify the second difference quotient

Now, we must simplify the fraction obtained in the previous step: \(\frac{((\sqrt{x}+\sqrt{a})(\sqrt{x}-\sqrt{a}))}{((x-a)(\sqrt{x}+\sqrt{a}))} = \frac{x-a}{(x-a)(\sqrt{x}+\sqrt{a})} = \frac{1}{\sqrt{x}+\sqrt{a}}\) Now that the expressions for the difference quotients are simplified, we can see that: \(\frac{f(x+h)-f(x)}{h} = \frac{1}{\sqrt{x+h}+\sqrt{x}}\) \(\frac{f(x)-f(a)}{x-a} = \frac{1}{\sqrt{x}+\sqrt{a}}\)

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