
By choosing appropriate secant lines, estimate the slope of the tangent line at *P*. (Round your answer to two decimal places.)

To estimate the slope of the tangent line at point \( P(1, 0) \), we need to choose values of \( x \) that are very close to 1 from both sides (i.e., slightly greater than 1 and slightly less than 1). This will help us observe the behavior of the slopes of the secant lines more accurately and attempt to identify a potential limiting value.

### Strategy

Let's compute the slopes of the secant lines for values of \( x \) that are even closer to 1, such as:

- \( x = 1.0001, 1.00001 \) (just slightly more than 1)
- \( x = 0.9999, 0.99999 \) (just slightly less than 1)

We will calculate these slopes and see if they appear to converge to a particular value when rounded to two decimal places.

Let's calculate these slopes.The slopes of the secant lines for values of \( x \) even closer to 1 are:

1. For \( x = 1.0001 \): Slope ≈ -43.97
2. For \( x = 1.00001 \): Slope ≈ -43.98
3. For \( x = 0.9999 \): Slope ≈ -43.99
4. For \( x = 0.99999 \): Slope ≈ -43.98

### Conclusion:

The slopes of the secant lines for \( x \) values very close to 1 seem to be approaching approximately **-43.98** when rounded to two decimal places. Thus, the estimated slope of the tangent line at point \( P(1, 0) \) is **-43.98**.

Would you like further details or have any additional questions?

#### Related Questions:
1. How do we find the derivative of trigonometric functions with unusual arguments?
2. Why do small changes in \( x \) near 1 lead to such rapid oscillations in the function?
3. What would happen if we chose \( x \) values even closer to 1?
4. How does the concept of a limit apply when estimating slopes?
5. Are there alternative methods for finding the slope of a tangent in such cases?

**Tip:** Using limits can provide a more precise estimation of the slope of the tangent, especially for functions that oscillate rapidly near a point.

Learn how to estimate the slope of the tangent line at point P(1, 0) using secant lines. Detailed explanation with calculations for values of x close to 1. The estimated slope of approximately -43.98 is derived through observing converging secant line slopes. Suitable for college-level calculus studies.

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