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**Estimate f(2+h)-f(2)/h for h+-1**
**FInd the difference quotient**

The graph shows the function \( f(x) \), and you're being asked to estimate the difference quotient, \( \frac{f(2+h) - f(2)}{h} \), for \( h = 1 \) and \( h = -1 \). This expression approximates the derivative of the function at \( x = 2 \).

### Step-by-Step Process:

1. **When \( h = 1 \):**
   - We need to find \( f(2+h) = f(3) \) and \( f(2) \).
   - From the graph, we estimate:
     - \( f(3) \approx 3.0 \) (where the blue curve at \( x = 3 \)).
     - \( f(2) \approx 1.5 \) (from the blue curve at \( x = 2 \)).

   Now, substitute into the difference quotient formula:
   \[
   \frac{f(2+1) - f(2)}{1} = \frac{f(3) - f(2)}{1} = \frac{3.0 - 1.5}{1} = 1.5
   \]

2. **When \( h = -1 \):**
   - We need to find \( f(2+h) = f(1) \) and \( f(2) \).
   - From the graph, we estimate:
     - \( f(1) \approx 2.0 \) (from the blue curve at \( x = 1 \)).
     - \( f(2) \approx 1.5 \) (from the blue curve at \( x = 2 \)).

   Now, substitute into the difference quotient formula:
   \[
   \frac{f(2+(-1)) - f(2)}{-1} = \frac{f(1) - f(2)}{-1} = \frac{2.0 - 1.5}{-1} = \frac{0.5}{-1} = -0.5
   \]

### Summary of Results:
- For \( h = 1 \), the difference quotient is \( 1.5 \).
- For \( h = -1 \), the difference quotient is \( -0.5 \).

Would you like further details on the steps or have any questions?

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Here are 5 follow-up questions for further practice:
1. What is the general formula for the difference quotient, and how does it relate to the derivative?
2. How would you calculate the derivative of the function \( f(x) \) at \( x = 2 \) using these estimates?
3. Can you derive an approximation for \( f'(2) \) using symmetric difference (using both \( h = 1 \) and \( h = -1 \))?
4. How would the difference quotient change if we used smaller values of \( h \)?
5. How can you improve the accuracy of your estimate for the derivative from a graph?

**Tip**: The difference quotient formula is a crucial tool for estimating the slope of the tangent line to the graph at a point and is the foundation of the concept of derivatives in calculus.

Estimate f(2+h)-f(2)/h for h = ±1 and find the difference quotient.

Learn how to estimate the difference quotient (f(2+h) - f(2)) / h for h = ±1 using graph interpretation of the function f(x). This example covers the key concepts of calculus and helps students understand how to approximate the derivative of a function using a graph. Suitable for Grades 10-12.

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