Chapter 2: Q. 81 (page 70)

In Problems 75– 82, find the difference quotient of $f$ ; that is, find $\frac{f (x + h) - f (x)}{h}, h \neq 0,$ for each function. Be sure to simplify.
$f (x) = \sqrt{x}$

Short Answer

Expert verified

The value of $\frac{f (x + h) - f (x)}{h} = \frac{1}{\sqrt{x + h} + \sqrt{x}}$

Step by step solution

Step 1. Given Information

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

The given function is
$f (x) = \sqrt{x}$

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.

$f (x + h) = \sqrt{x + h}$

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

$\frac{f (x + h) - f (x)}{h} = \frac{\sqrt{x + h} - \sqrt{x}}{h}$

To rationalize we multiply and divide with $\sqrt{x + h} + \sqrt{x}$

$\frac{f (x + h) - f (x)}{h} = \frac{\sqrt{x + h} - \sqrt{x}}{h} \cdot \frac{\sqrt{x + h} + \sqrt{x}}{\sqrt{x + h} + \sqrt{x}}$

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

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

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In Problems 75– 82, find the difference quotient of $f$ ; that is, find $\frac{f (x + h) - f (x)}{h}, h \neq 0,$ for each function. Be sure to simplify.
$f (x) = \sqrt{x}$

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

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

Probability and Statistics

Decision Maths

Discrete Mathematics

Logic and Functions

Pure Maths

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

In Problems 75– 82, find the difference quotient of $f$ ; that is, find $\frac{f (x + h) - f (x)}{h}, h \neq 0,$ for each function. Be sure to simplify.
$f (x) = \sqrt{x}$