To estimate the slope of the tangent line at point $P(1, 0)$ on the curve given by

$y = \frac{\sin(14 \pi / x)}{x},$

we need to choose values of $x$ that are closer to 1 than the ones provided in part (a) and calculate the slopes of the corresponding secant lines.

Explanation:

The slope of the secant line between points $P(1, 0)$ and $Q(x, \frac{\sin(14 \pi / x)}{x})$ is given by:

$m_{\text{secant}} = \frac{f(x) - f(1)}{x - 1} = \frac{\frac{\sin(14 \pi / x)}{x} - 0}{x - 1} = \frac{\sin(14 \pi / x)}{x (x - 1)}.$

To estimate the slope of the tangent line at $P$, we must take values of $x$ that are much closer to 1. As $x$ approaches 1, the oscillations in the value of $\frac{\sin(14 \pi / x)}{x}$ will become more pronounced, and we'll see whether the slopes of these secant lines approach any particular value.

Choosing Values Closer to 1:

Let's compute the slope for values of $x$ closer to 1, such as $x = 1.01, 1.001, 0.99, 0.999$, etc.

I will calculate a few of these slopes to see if they appear to approach a limit.The slopes of the secant lines for values of $x$ closer to 1 are:

1. For $x = 1.01$: Slope ≈ -41.7658
2. For $x = 1.001$: Slope ≈ -43.8803
3. For $x = 0.99$: Slope ≈ -43.4136
4. For $x = 0.999$: Slope ≈ -44.0562

Conclusion:

The slopes do not appear to be converging to a single value, even with values of $x$ closer to 1. This suggests that the slope of the tangent line at point $P(1, 0)$ is not well-defined due to the rapid oscillations in the function near $x = 1$.

Explanation for Part (b):

The frequent oscillations of the function around point $P$ cause the slopes of the secant lines to vary significantly, preventing a clear limit from emerging.

If you have any further questions or need more details, feel free to ask!

Related Questions:

1. How does the behavior of trigonometric functions affect the estimation of slopes?
2. Why do oscillations around a point lead to difficulties in defining a tangent slope?
3. What other methods can be used to estimate slopes when oscillations are present?
4. How would using a smaller step size between $x$ values help in numerical differentiation?
5. Can we use a different mathematical approach to find the limit of the slopes?

Tip: Try plotting the function near $x = 1$ to visually see the oscillations and understand why the slopes do not converge.