Estimating the Slope of the Tangent Line Using Part (b)

To estimate the slope of the tangent line to the graph of $f$ at point $P(4, 2)$, we can use the concept of secant lines that pass through the point $P$ and another point $Q(x_1, y_1)$ on the graph of $f$. The slope of the secant line is given by the difference quotient:

$\text{slope of secant} = \frac{f(x_1) - f(4)}{x_1 - 4}$

As $x_1$ gets closer to 4, the slope of the secant line approximates the slope of the tangent line at $P(4, 2)$.

Improving the Approximation

To improve the approximation of the slope of the tangent line at $P(4, 2)$:

1. Use Points Closer to $P$: Choose $Q(x_1, y_1)$ such that $x_1$ is very close to 4. The closer $x_1$ is to 4, the more accurate the approximation of the tangent line slope.

2. Calculate the Slope for Multiple Secant Lines: Compute the slope of secant lines for several values of $x_1$ that are both slightly greater and slightly less than 4. Averaging these slopes can give a better estimate of the tangent slope.

3. Use Smaller Intervals: Consider smaller intervals around $x = 4$ to reduce the error in the approximation. The smaller the interval, the closer the secant line's slope is to the actual tangent line's slope.

Defining Secant Lines Farther Away from $P$

1. Farther Away Points: Choose points $Q(x_1, y_1)$ where $x_1$ is significantly different from 4. However, this can lead to less accurate approximations because the slope of the secant line will differ more from the tangent line.

2. Nearly Vertical Secant Lines: These occur when $x_1$ is much larger or much smaller than 4, making the slope of the secant line either very steep or nearly vertical. This does not give a good approximation of the tangent slope at $P(4, 2)$.

Defining Secant Lines Closer to $P$

1. Closer Points: Choose $x_1$ values that are very close to 4. This will result in secant lines that approximate the tangent line more accurately.

2. Nearly Horizontal Secant Lines: These occur when the function $f(x)$ is relatively flat near $x = 4$. A horizontal secant line implies that the function has a very small rate of change around $P$, which is useful for estimating the tangent slope.

Summary

• The best way to estimate the slope of the tangent line at $P(4, 2)$ is to use secant lines where $x_1$ is very close to 4.
• Using points farther away from $P$ may give less accurate approximations, especially if the function is not linear.
• The slope of the tangent line can be improved by narrowing the interval between the points used to define the secant lines.

Would you like to see specific examples with calculations or have any questions?

---

1. What is the difference between the slope of a secant line and a tangent line?
2. How does choosing points closer to $P$ affect the accuracy of the slope estimate?
3. Why are nearly vertical secant lines not useful for estimating tangent slopes?
4. How do nearly horizontal secant lines help in slope estimation?
5. Can the method of secant lines be used for non-differentiable functions?

Tip: When estimating slopes using secant lines, consider symmetry by choosing points equidistant from the point of interest for more accurate results.