To find the slope of the secant line $PQ$ for each given value of $x$, we will use the formula for the slope between two points, $P = \left(\frac{1}{4}, \frac{1}{2}\right)$ and $Q = (x, \cos(\pi x))$.

The slope formula between the two points is:

$\text{slope} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\cos(\pi x) - \frac{1}{2}}{x - \frac{1}{4}}$

We will calculate this slope for each value of $x$:

1. $x = 0.24$
2. $x = 0.249$
3. $x = 0.2499$
4. $x = 0.26$
5. $x = 0.251$
6. $x = 0.2501$

Let's calculate each slope rounded to four decimal places.Here are the slopes of the secant line $PQ$ for each given value of $x$, rounded to four decimal places:

1. $x = 0.24$: $-22.8969$
2. $x = 0.249$: $-209.3247$
3. $x = 0.2499$: $-2073.2889$
4. $x = 0.26$: $18.4547$
5. $x = 0.251$: $204.8819$
6. $x = 0.2501$: $2068.8460$

These values indicate how the slope of the secant line changes as $x$ approaches $\frac{1}{4}$ (or $0.25$).

Would you like to see more details or have any questions?

Related Questions:

1. What is the significance of the slope values when $x$ is close to 0.25?
2. How do you interpret the change in the slope of the secant line as $x$ approaches 0.25?
3. Can we find the derivative of $\cos(\pi x)$ at $x = 0.25$ using limits?
4. What happens to the slope of the secant line if $x$ moves further away from 0.25?
5. How can this method be used to estimate the tangent line's slope at a point?

Tip: For small increments in $x$, the secant line slope approximates the derivative at a point!