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.