
Use the results of part (b) to estimate the slope of the tangent line to the graph of *f* at *P*(4, 2).

To estimate the slope of the tangent line to the graph of \( f \) at \( P(4, 2) \), I'll need to know the results from part (b) of the problem. However, since you haven't provided those results, I can guide you through the process assuming that part (b) involved calculating the slope of secant lines.

### Steps to Estimate the Slope of the Tangent Line

1. **Recall the Secant Lines from Part (b):**  
   If part (b) involved calculating the slopes of secant lines using points near \( P(4, 2) \), these slopes can help estimate the slope of the tangent line.

   Let's assume you have the following secant line slopes from part (b) using points \( Q(x_1, y_1) \) on the graph of \( f \):
   
   - Slope using \( Q(3.9, f(3.9)) \)
   - Slope using \( Q(4.1, f(4.1)) \)

   The slope of the secant line between \( P(4, 2) \) and \( Q(x_1, f(x_1)) \) is given by:
   \[
   \text{Slope of secant line} = \frac{f(x_1) - 2}{x_1 - 4}
   \]

2. **Calculate the Slopes (if values are known):**  
   Substitute the function values from part (b) into the formula above to find the slope of each secant line.

3. **Estimate the Tangent Slope:**
   - If the slope at \( x_1 = 3.9 \) is, say, \( m_1 \), and the slope at \( x_1 = 4.1 \) is \( m_2 \), then an estimate for the slope of the tangent line at \( P(4, 2) \) could be the average of these slopes:
     \[
     \text{Estimated slope of tangent} = \frac{m_1 + m_2}{2}
     \]

### Example (Assuming Hypothetical Values)

Let's suppose you found the following slopes in part (b):

- Slope of secant line between \( P(4, 2) \) and \( Q(3.9, f(3.9)) \): \( m_1 = 1.8 \)
- Slope of secant line between \( P(4, 2) \) and \( Q(4.1, f(4.1)) \): \( m_2 = 2.2 \)

Then the estimated slope of the tangent line at \( P(4, 2) \) would be:

\[
\text{Estimated slope of tangent} = \frac{1.8 + 2.2}{2} = 2.0
\]

### Conclusion

The estimated slope of the tangent line at \( P(4, 2) \) is \( 2.0 \), assuming the hypothetical values used here. If you have the actual results from part (b), you can substitute them into the process above for a precise estimate.

Would you like to go through the actual values, or do you have any questions?

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1. What should you do if the secant slopes are not symmetric around \( x = 4 \)?
2. How does the distance between points affect the accuracy of the slope estimate?
3. What is the difference between averaging secant slopes and using a limit to find the derivative?
4. Why might the slopes of secant lines differ significantly from the tangent line slope for non-linear functions?
5. How can you ensure that your estimate of the tangent slope is as accurate as possible?

**Tip:** When calculating secant slopes, ensure that the interval is small enough to minimize error, but large enough to avoid numerical instability.

Learn how to estimate the slope of the tangent line to the graph of a function at point (4, 2) using secant lines. This problem involves applying the concept of secant lines and the Mean Value Theorem in calculus.

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