Q. 79 In Problems 75鈥� 82, find the d... [FREE SOLUTION]

91影视

Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 2: Q. 79 (page 70)

In Problems 75鈥� 82, find the difference quotient of $f$ ; that is, find $\frac{f (x + h) - f (x)}{h}, h 鈮� 0,$ for each function. Be sure to simplify.
$f (x) = \frac{1}{x^{2}}$

Short Answer

Expert verified

The value of $\frac{f (x + h) - f (x)}{h} = \frac{- 2 x - h}{x^{2} ({x^{2} + 2 x h + h}^{2})}$

Step by step solution

Step 1. Given Information

In this problems we have to find the difference quotient of $f$ ; that is, find $\frac{f (x + h) - f (x)}{h}, h 鈮� 0,$ for given function. Be sure to simplify.

The given function is
$f (x) = \frac{1}{x^{2}}$

Step 2. Firstly we find the value of f(x+h)We find the value of f(x+h) by putting x=x+h in given function.

$f (x + h) = \frac{1}{{(x + h)}^{2}}$

Using the formula ${(a + b)}^{2} = a^{2} + 2 a b + b^{2}$

$f (x + h) = \frac{1}{{x^{2} + 2 x h + h}^{2}}$

Step 3. Putting the value of f(x) and f(x+h) in the f(x+h)- f(x)h

$\frac{f (x + h) - f (x)}{h} = \frac{\frac{1}{{x^{2} + 2 x h + h}^{2}} - \frac{1}{x^{2}}}{h} \frac{f (x + h) - f (x)}{h} = \frac{1}{h} (\frac{1}{{x^{2} + 2 x h + h}^{2}} - \frac{1}{x^{2}})$

The LCD of ${x^{2} + 2 x h + h}^{2} and x^{2} is x^{2} ({x^{2} + 2 x h + h}^{2})$

$\frac{f (x + h) - f (x)}{h} = \frac{1}{h} (\frac{1}{{x^{2} + 2 x h + h}^{2}} 路 \frac{x^{2} ({x^{2} + 2 x h + h}^{2})}{x^{2} ({x^{2} + 2 x h + h}^{2})} - \frac{1}{x^{2}} 路 \frac{x^{2} ({x^{2} + 2 x h + h}^{2})}{x^{2} ({x^{2} + 2 x h + h}^{2})}) \frac{f (x + h) - f (x)}{h} = \frac{1}{h} (\frac{x^{2}}{x^{2} ({x^{2} + 2 x h + h}^{2})} - \frac{{x^{2} + 2 x h + h}^{2}}{x^{2} ({x^{2} + 2 x h + h}^{2})}) \frac{f (x + h) - f (x)}{h} = \frac{1}{h} (\frac{x^{2} - ({x^{2} + 2 x h + h}^{2})}{x^{2} ({x^{2} + 2 x h + h}^{2})}) \frac{f (x + h) - f (x)}{h} = \frac{1}{h} (\frac{x^{2} - {x^{2} - 2 x h - h}^{2}}{x^{2} ({x^{2} + 2 x h + h}^{2})}) \frac{f (x + h) - f (x)}{h} = \frac{1}{h} (\frac{{- 2 x h - h}^{2}}{x^{2} ({x^{2} + 2 x h + h}^{2})}) \frac{f (x + h) - f (x)}{h} = \frac{1}{h} 路 \frac{h (- 2 x - h)}{x^{2} ({x^{2} + 2 x h + h}^{2})} \frac{f (x + h) - f (x)}{h} = \frac{- 2 x - h}{x^{2} ({x^{2} + 2 x h + h}^{2})}$

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91影视

In Problems 75鈥� 82, find the difference quotient of $f$ ; that is, find $\frac{f (x + h) - f (x)}{h}, h 鈮� 0,$ for each function. Be sure to simplify.
$f (x) = \frac{1}{x^{2}}$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Firstly we find the value of f(x+h)We find the value of f(x+h) by putting x=x+h in given function.

Step 3. Putting the value of f(x) and f(x+h) in the f(x+h)- f(x)h

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91影视

In Problems 75鈥� 82, find the difference quotient of f; that is, find f(x+h)-f(x)h,h鈮�0,for each function. Be sure to simplify.f(x)=1x2

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Firstly we find the value of f(x+h)We find the value of f(x+h) by putting x=x+h in given function.

Step 3. Putting the value of f(x) and f(x+h) in the f(x+h)- f(x)h

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Logic and Functions

Mechanics Maths

Discrete Mathematics

Pure Maths

Geometry

Study anywhere. Anytime. Across all devices.

In Problems 75鈥� 82, find the difference quotient of $f$ ; that is, find $\frac{f (x + h) - f (x)}{h}, h 鈮� 0,$ for each function. Be sure to simplify.
$f (x) = \frac{1}{x^{2}}$