Chapter 2: Q. 78 (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.
$f (x) = 3 x^{2} - 2 x + 6$

Short Answer

Expert verified

The value of $\frac{f (x + h) - f (x)}{h} = 6 x + 3 h - 2$

Step by step solution

Step 1. Given Information

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

The given function is
$f (x) = 3 x^{2} - 2 x + 6$

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.

$f (x + h) = 3 {(x + h)}^{2} - 2 (x + h) + 6$

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

$f (x + h) = 3 ({x^{2} + 2 x h + h}^{2}) - 2 (x + h) + 6 f (x + h) = 3 {x^{2} + 6 x h + 3 h}^{2} - 2 x - 2 h + 6$

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

$\frac{f (x + h) - f (x)}{h} = \frac{3 {x^{2} + 6 x h + 3 h}^{2} - 2 x - 2 h + 6 - (3 x^{2} - 2 x + 6)}{h} \frac{f (x + h) - f (x)}{h} = \frac{3 {x^{2} + 6 x h + 3 h}^{2} - 2 x - 2 h + 6 - 3 x^{2} + 2 x - 6}{h} \frac{f (x + h) - f (x)}{h} = \frac{{6 x h + 3 h}^{2} - 2 h}{h} \frac{f (x + h) - f (x)}{h} = \frac{h (6 x + 3 h - 2)}{h} \frac{f (x + h) - f (x)}{h} = 6 x + 3 h - 2$

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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.
$f (x) = 3 x^{2} - 2 x + 6$

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)=3x2-2x+6

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

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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.
$f (x) = 3 x^{2} - 2 x + 6$