To estimate the value of \( f'(8) \), you generally use a method that approximates the derivative at that point. This is typically done using numerical methods such as the difference quotient, especially if you do not have an explicit formula for \( f(x) \). The derivative \( f'(x) \) represents the slope of the tangent line to the curve at a given point, and estimating it often involves looking at the values of \( f(x) \) near \( x = 8 \).

### Estimating \( f'(8) \) Using the Difference Quotient

The most common way to estimate \( f'(8) \) is by using a difference quotient, either:

1. **Forward Difference Approximation**:
   \[
   f'(8) \approx \frac{f(8+h) - f(8)}{h}
   \]

2. **Backward Difference Approximation**:
   \[
   f'(8) \approx \frac{f(8) - f(8-h)}{h}
   \]

3. **Central Difference Approximation** (most accurate):
   \[
   f'(8) \approx \frac{f(8+h) - f(8-h)}{2h}
   \]

Here, \( h \) is a small number (e.g., 0.1 or 0.01), and the closer \( h \) is to zero, the better the estimate.

### Example Scenario:

- If you have specific values for \( f(x) \) near \( x = 8 \), such as \( f(8) \), \( f(8.1) \), and \( f(7.9) \), you can plug those into the difference quotient formulas above to estimate \( f'(8) \).

### Steps to Estimate \( f'(8) \):

1. **Choose a small \( h \)** value, like \( 0.1 \).
2. **Calculate \( f(8+h) \), \( f(8) \), and \( f(8-h) \)** using your function or given data.
3. **Plug into the central difference formula** (recommended for accuracy):
   \[
   f'(8) \approx \frac{f(8+0.1) - f(8-0.1)}{2 \times 0.1}
   \]

### Do you have specific function values or need more help choosing \( h \) and applying the formulas?

### Related Questions:

1. What are the values of \( f(8+h) \) and \( f(8-h) \) if you use \( h = 0.1 \)?
2. How does the choice of \( h \) affect the accuracy of the derivative estimate?
3. What is the difference between using a forward difference and a central difference method?
4. How can we use higher-order derivatives to improve the estimation of \( f'(8) \)?
5. What are some real-world applications where estimating a derivative is useful?

### Tip:

Always try to use the central difference approximation when possible, as it provides a more accurate estimation of the derivative by averaging the rates of change from both sides of the point.

Learn how to estimate the derivative f'(8) using difference quotient methods. Understand forward, backward, and central difference approximations to approximate the slope of the tangent line at x = 8. Suitable for advanced high school and college-level students.

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