Answered: Evaluate and simplify the difference…

College Algebra (MindTap Course List)

12th Edition

ISBN:9781305652231

Author:R. David Gustafson, Jeff Hughes

Publisher:R. David Gustafson, Jeff Hughes

Chapter2: Functions And Graphs

Section2.CT: Chapter Test

Problem 5CT: Find the difference quotient. f(x)=x2x+5

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Question

Evaluate and simplify the difference quotients ƒ1x + h2- ƒ1x2 / h and ƒ1x2- ƒ1a2 / x - a for each function.

ƒ1x2= 7 / x + 3

Expert Solution

Step 1

Given:

$f (x) = \frac{7}{x + 3}$

To Evaluate and Simplify:

$Difference Quotients: \frac{f (x + h) - f (x)}{h}, and \frac{f (x) - f (a)}{x - a}$

Step 2

Explanation:

$We have f (x) = \frac{7}{x + 3} (a) . Now, we have to find \frac{f (x + h) - f (x)}{h} \Rightarrow \frac{f (x + h) - f (x)}{h} = \frac{\frac{7}{(x + h) + 3} - \frac{7}{x + 3}}{h} \Rightarrow \frac{f (x + h) - f (x)}{h} = \frac{\frac{7 (x + 3) - 7 ((x + h) + 3)}{((x + h) + 3) (x + 3)}}{h} = \frac{7 x + 21 - 7 x - 7 h - 21}{h [((x + h) + 3) (x + 3)]} \Rightarrow \frac{f (x + h) - f (x)}{h} = \frac{- 7 h}{h [((x + h) + 3) (x + 3)]} = \frac{- 7}{[((x + h) + 3) (x + 3)]} \Rightarrow \frac{f (x + h) - f (x)}{h} = \frac{- 7}{[((x + h) + 3) (x + 3)]} Now, if we take \lim_{h \to 0} both the side we get \lim_{h \to 0} \frac{f (x + h) - f (x)}{h} = \lim_{h \to 0} \frac{- 7}{[((x + h) + 3) (x + 3)]} = \frac{- 7}{{(x + 3)}^{2}} \Rightarrow f' (x) = \lim_{h \to 0} \frac{f (x + h) - f (x)}{h} = \frac{- 7}{{(x + 3)}^{2}}$

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