1. The point P = (1/4,1/ 2) lies on the curve y = cos(πx) where x is in radians, as shown below.
y
P
−2 −1 0 1 2
x
(a) If Q = (x, cos(πx)) then use your calculator to find the slope of the secant line P Q, rounded to four digits after the decimal point, for the following values of x:
(i) 0.24,
(ii) 0.249,
(iii) 0.2499,
(iv) 0.26,
(v) 0.251,
(vi) 0.2501.

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!

1. The point P = (1/4,1/ 2) lies on the curve y = cos(πx) where x is in radians, as shown below. y P −2 −1 0 1 2 x (a) If Q = (x, cos(πx)) then use your calculator to find the slope of the secant line P Q, rounded to four digits after the decimal point, for the following values of x: (i) 0.24, (ii) 0.249, (iii) 0.2499, (iv) 0.26, (v) 0.251, (vi) 0.2501.

Find the slope of the secant line between the points P = (1/4, 1/2) and Q = (x, cos(πx)) for specific values of x. Learn how to apply the slope formula and trigonometric functions to solve this calculus problem. Suitable for high school calculus students.

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