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To find and simplify the difference quotient \(\frac{f(3+h) - f(3)}{h}\) for the function \(f(x) = x^2 - 2x + 2\), we will follow these steps:

1. **Calculate \(f(3)\):**
   \[
   f(3) = 3^2 - 2 \cdot 3 + 2 = 9 - 6 + 2 = 5
   \]

2. **Calculate \(f(3+h)\):**
   \[
   f(3+h) = (3+h)^2 - 2(3+h) + 2
   \]
   Expand and simplify:
   \[
   f(3+h) = (3+h)(3+h) - 2(3+h) + 2 = 9 + 6h + h^2 - 6 - 2h + 2 = h^2 + 4h + 5
   \]

3. **Form the difference quotient \(\frac{f(3+h) - f(3)}{h}\):**
   \[
   \frac{f(3+h) - f(3)}{h} = \frac{(h^2 + 4h + 5) - 5}{h} = \frac{h^2 + 4h}{h}
   \]

4. **Simplify the expression:**
   \[
   \frac{h^2 + 4h}{h} = h + 4
   \]

So, the simplified form of the difference quotient is:
\[
\boxed{h + 4}
\]

Would you like more details or have any questions about this process?

Here are 5 relative questions to further your understanding:
1. How would the difference quotient change if the function was \(f(x) = x^2 - 2x + 3\) instead?
2. What is the geometric interpretation of the difference quotient?
3. How does the difference quotient relate to the derivative of the function at \(x = 3\)?
4. Can you compute the difference quotient for \(f(x) = x^3 - x + 1\) at \(x = 2\)?
5. What happens to the difference quotient as \(h\) approaches zero?

**Tip:** The difference quotient is a fundamental concept that leads to the derivative, which measures how a function changes as its input changes.

Find the difference quotient and simplify your answer for the function f(x) = x^2 - 2x + 2, with the difference quotient given by (f(3 + h) - f(3)) / h, h ≠ 0.

Learn how to simplify the difference quotient for the function f(x) = x^2 - 2x + 2 using calculus concepts. This detailed step-by-step solution covers the difference quotient formula and provides insights into limits and derivatives. Ideal for high school students in Grades 10-12.

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