The **difference quotient** is a formula used to find the average rate of change of a function between two points. It is given by:

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

For the function \( f(x) = x^2 \), the difference quotient is calculated as follows:

1. **Substitute \( f(x) = x^2 \) into the formula:**
   \[
   \frac{(x+h)^2 - x^2}{h}
   \]
   
2. **Expand \( (x+h)^2 \):**
   \[
   (x+h)^2 = x^2 + 2xh + h^2
   \]
   
   So the difference quotient becomes:
   \[
   \frac{x^2 + 2xh + h^2 - x^2}{h}
   \]

3. **Simplify the expression:**
   \[
   \frac{2xh + h^2}{h}
   \]

4. **Factor out \( h \):**
   \[
   \frac{h(2x + h)}{h}
   \]

5. **Cancel \( h \) (as long as \( h \neq 0 \)):**
   \[
   2x + h
   \]

Thus, the **difference quotient** of \( x^2 \) is \( 2x + h \).

---

Now, if \( x = 3 \), substitute \( x = 3 \) into the difference quotient:

\[
2(3) + h = 6 + h
\]

So, the difference quotient at \( x = 3 \) is \( 6 + h \).

---

Would you like more details or have any questions?

### 5 Related Questions:
1. What is the difference quotient of other polynomial functions?
2. How do you interpret the difference quotient geometrically?
3. What happens as \( h \) approaches 0 in the difference quotient?
4. How does the difference quotient relate to the derivative?
5. Can you apply the difference quotient to non-polynomial functions?

### Tip:
The difference quotient is the foundation for understanding **derivatives**, which measure the instantaneous rate of change at a point.

Learn how to calculate the difference quotient for the function x^2 when x=3. This step-by-step guide explains key algebra and calculus concepts, using the formula for the average rate of change. Understand the basics of the difference quotient and its relation to the derivative, suitable for high school calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation