https://file.aiquickdraw.com/mathbot/file/17264721057050wcc5wy2.jpg

The given problem is asking to find and simplify the following expression:

\[
\frac{f(a+h) - f(a)}{h}
\]

where the function \( f(x) = 2x^2 - 1 \).

### Step 1: Find \( f(a+h) \)
We substitute \( x = a+h \) into the function \( f(x) = 2x^2 - 1 \):

\[
f(a+h) = 2(a+h)^2 - 1
\]

Now expand \( (a+h)^2 \):

\[
f(a+h) = 2(a^2 + 2ah + h^2) - 1 = 2a^2 + 4ah + 2h^2 - 1
\]

### Step 2: Find \( f(a) \)
Substitute \( x = a \) into the function \( f(x) = 2x^2 - 1 \):

\[
f(a) = 2a^2 - 1
\]

### Step 3: Subtract \( f(a) \) from \( f(a+h) \)
Now subtract \( f(a) \) from \( f(a+h) \):

\[
f(a+h) - f(a) = (2a^2 + 4ah + 2h^2 - 1) - (2a^2 - 1)
\]

Simplify the expression:

\[
f(a+h) - f(a) = 4ah + 2h^2
\]

### Step 4: Divide by \( h \)
Now divide the result by \( h \):

\[
\frac{f(a+h) - f(a)}{h} = \frac{4ah + 2h^2}{h}
\]

Simplify the expression:

\[
\frac{f(a+h) - f(a)}{h} = 4a + 2h
\]

### Final Answer:
\[
\frac{f(a+h) - f(a)}{h} = 4a + 2h
\]

Would you like more details on any part of the solution?

Here are 5 related questions to expand on this topic:
1. How do you interpret the expression \(\frac{f(a+h) - f(a)}{h}\) in the context of calculus?
2. Can you explain how to apply the limit as \( h \to 0 \) to this expression?
3. What is the significance of the difference quotient in terms of derivatives?
4. How would this problem change if the function \( f(x) \) had a higher degree, say \( f(x) = 3x^3 - 1 \)?
5. How does expanding \( (a+h)^2 \) apply to other quadratic expressions?

**Tip:** The expression \(\frac{f(a+h) - f(a)}{h}\) is the foundation of finding derivatives, which measure the rate of change of a function at a point.

Untuk f(x) = 2x^2 - 1, cari dan sederhanakan [f(a + h) - f(a)] / h.

Learn how to find and simplify the difference quotient for the quadratic function f(x) = 2x^2 - 1. This problem demonstrates key algebraic and calculus concepts, including the difference quotient formula. Suitable for students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation