Question

Find the difference quotient

f(x+h)-f(x)
h
f(x)=3x²-x-5
Simplify your answer as much as possible.
ƒ (x + h) − f (x) = 【
h
"
where h‡0, for the function below.
X
Ś

Answer 1

Answer:
`\( (f(x+h) - f(x))/(h) = -1 + 3h + 6x \)` where
`\( h \\eq 0 \)`

Explanation:

To find the difference quotient for the function
`\( f(x) = 3x^2 - x - 5 \)`, we'll use the formula:

`\[ (f(x+h) - f(x))/(h) \]`

Step 1: Find
`\( f(x+h) \)`

`\[ f(x+h) = 3(x+h)^2 - (x+h) - 5 \]`

Step 2: Subtract
`\( f(x) \)` from
`\( f(x+h) \)`

`\[ f(x+h) - f(x) = 3(x+h)^2 - (x+h) - 5 - (3x^2 - x - 5) \]`

Step 3: Divide the result from Step 2 by
`\( h \)`

`\[ (f(x+h) - f(x))/(h) \]`

Let's plug in the values and simplify.

The simplified difference quotient for the function
`\( f(x) = 3x^2 - x - 5 \)` is:

`\[ (f(x+h) - f(x))/(h) = -1 + 3h + 6x \]`

So,
`\( (f(x+h) - f(x))/(h) = -1 + 3h + 6x \)` where
`\( h \\eq 0 \)`.