To estimate the slope of the tangent line to the graph of $f$ at $P(4, 2)$, we can use the slopes of the secant lines calculated in part (a). The idea is that as the points $Q(x, f(x))$ get closer to $P(4, 2)$, the slope of the secant line should approach the slope of the tangent line at $P$.

Recap of Secant Line Slopes

From part (a), the slopes of the secant lines were:

• Slope through $Q(3, f(3))$: $1.000$
• Slope through $Q(6, f(6))$: $1.000$
• Slope through $Q(8, f(8))$: $1.000$

Estimating the Tangent Line Slope at $P(4, 2)$

Given that the slopes of the secant lines for different values of $x$ (i.e., 3, 6, 8) are all the same and equal to 1, we can conclude that the slope of the tangent line at $P(4, 2)$ is also $1.000$.

Conclusion

The slope of the tangent line to the graph of $f(x)$ at the point $P(4, 2)$ is estimated to be 1.000.

This result is consistent with the assumption that $f(x) = x - 2$, which is a linear function. For a linear function, the slope of the tangent line at any point is constant and equal to the slope of the line itself.

Would you like more details or have any questions?

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Here are 5 related questions to consider:

1. What does the tangent line represent geometrically at the point $P(4, 2)$?
2. How would the tangent slope estimation change if we had used secant lines with points closer to $P(4, 2)$?
3. How does this estimation method work if the function $f(x)$ is non-linear?
4. Can you derive the exact formula for the slope of the tangent line using calculus?
5. What is the relationship between the derivative of $f(x)$ and the slope of the tangent line at any point on the curve?

Tip: The tangent line at a point on a curve gives the best linear approximation of the curve at that point.