Spencer Morrison
10/26/2023 · Elementary School
Determine the difference quotient \( \frac{f(x+h)-f(x)}{h} \) \( f(x)=2 x^{2}-3 \)
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Tutor-Verified Answer
Step-by-step Solution
To determine the difference quotient \( \frac{f(x+h)-f(x)}{h} \) for the function \( f(x) = 2x^2 - 3 \), we first need to calculate \( f(x+h) \).
1. **Calculate \( f(x+h) \)**:
\[
f(x+h) = 2(x+h)^2 - 3
\]
Expanding \( (x+h)^2 \):
\[
(x+h)^2 = x^2 + 2xh + h^2
\]
Therefore,
\[
f(x+h) = 2(x^2 + 2xh + h^2) - 3 = 2x^2 + 4xh + 2h^2 - 3
\]
2. **Calculate \( f(x+h) - f(x) \)**:
Now we need to find \( f(x+h) - f(x) \):
\[
f(x) = 2x^2 - 3
\]
Thus,
\[
f(x+h) - f(x) = (2x^2 + 4xh + 2h^2 - 3) - (2x^2 - 3)
\]
Simplifying this expression:
\[
f(x+h) - f(x) = 2x^2 + 4xh + 2h^2 - 3 - 2x^2 + 3
\]
\[
= 4xh + 2h^2
\]
3. **Calculate the difference quotient**:
Now we can substitute this result into the difference quotient:
\[
\frac{f(x+h) - f(x)}{h} = \frac{4xh + 2h^2}{h}
\]
We can simplify this by dividing each term by \( h \) (assuming \( h \neq 0 \)):
\[
= 4x + 2h
\]
Thus, the difference quotient is:
\[
\frac{f(x+h) - f(x)}{h} = 4x + 2h
\]
Quick Answer
The difference quotient is \( 4x + 2h \).
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