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The problem asks to estimate the slope of the tangent line to the curve \( y = \cos(\pi x) \) at the point \( P = \left(\frac{1}{4}, \frac{1}{\sqrt{2}}\right) \). Here’s a breakdown of the steps to approach this problem:

### Part (a)
You're asked to find the slope of the secant line \( PQ \), where \( Q = (x, \cos(\pi x)) \), for various values of \( x \). To find the slope of the secant line between points \( P \) and \( Q \), use the slope formula:

\[
\text{slope of PQ} = \frac{y_Q - y_P}{x_Q - x_P} = \frac{\cos(\pi x) - \frac{1}{\sqrt{2}}}{x - \frac{1}{4}}
\]

You need to compute this for the following values of \( x \):
- \( x = 0.24 \)
- \( x = 0.249 \)
- \( x = 0.2499 \)
- \( x = 0.25 \)
- \( x = 0.2501 \)
- \( x = 0.251 \)

These values will help you estimate the slope of the tangent line at \( x = \frac{1}{4} \).

### Part (b)
After calculating the slopes of the secant lines, you will estimate the slope of the tangent line at \( x = \frac{1}{4} \). The tangent line’s slope at a point is essentially the limit of the secant line slopes as \( x \) approaches \( \frac{1}{4} \).

### Steps for Calculation:
1. **Evaluate** \( \cos(\pi x) \) for each value of \( x \).
2. **Calculate** the slope of the secant line for each \( x \).
3. **Analyze** the trend in slopes to estimate the slope of the tangent line.

Would you like me to perform these calculations now?

The point P = (1/4, 1/√2) lies on the curve y = cos(πx), where x is in radians. Find the slope of the secant line PQ for various values of x and estimate the slope of the tangent line to the curve at P.

This problem involves finding the slope of the secant line PQ for different values of x and estimating the slope of the tangent line to the curve y = cos(πx) at the point P = (1/4, 1/√2). Suitable for high school students in Grades 11-12, this problem covers trigonometry, calculus, and the application of the limit theorem.

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