The point P(6, 1) lies on the curve
y =

x − 5
.
(a) If Q is the point
(x,

x − 5
),
use your calculator to find the slope of the secant line PQ (correct to six decimal places) for the following values of x.
(i)    5.5

(ii)    5.9

(iii)    5.99

(iv)    5.999

(v)    6.5

(vi)    6.1

(vii)    6.01

(viii)    6.001

To find the slope of the secant line \( PQ \), we use the formula for the slope between two points \( P(x_1, y_1) \) and \( Q(x_2, y_2) \), which is:

\[
\text{slope of PQ} = \frac{y_2 - y_1}{x_2 - x_1}
\]

Here, the point \( P(6, 1) \) lies on the curve \( y = x - 5 \), meaning the y-coordinate of point P is \( 6 - 5 = 1 \).

For the point \( Q(x, y) \), the y-coordinate is given by \( y = x - 5 \). So, the formula for the slope becomes:

\[
\text{slope of PQ} = \frac{(x - 5) - 1}{x - 6} = \frac{x - 5 - 1}{x - 6} = \frac{x - 6}{x - 6} = 1
\]

However, this is valid only when \( x \neq 6 \). At \( x = 6 \), we would need to compute a limit or approach the slope using values close to 6. This is why we compute the slope for various values of \( x \) as you’ve requested.

Let’s now compute the slope for the given values of \( x \):

### Values to compute:
1. \( x = 5.5 \)
2. \( x = 5.9 \)
3. \( x = 5.99 \)
4. \( x = 5.999 \)
5. \( x = 6.5 \)
6. \( x = 6.1 \)
7. \( x = 6.01 \)
8. \( x = 6.001 \)

Let me calculate the slopes for all these values.For all the given values of \( x \), the slope of the secant line \( PQ \) is:

1. For \( x = 5.5 \), the slope is \( 1.0 \)
2. For \( x = 5.9 \), the slope is \( 1.0 \)
3. For \( x = 5.99 \), the slope is \( 1.0 \)
4. For \( x = 5.999 \), the slope is \( 1.0 \)
5. For \( x = 6.5 \), the slope is \( 1.0 \)
6. For \( x = 6.1 \), the slope is \( 1.0 \)
7. For \( x = 6.01 \), the slope is \( 1.0 \)
8. For \( x = 6.001 \), the slope is \( 1.0 \)

In this case, the slope is consistently 1.0 for all points near \( x = 6 \), indicating that the secant lines are almost horizontal.

Would you like more details on this, or do you have any other questions?

Here are 5 related questions:
1. What happens to the slope as \( x \) approaches 6?
2. How can you determine the equation of a tangent line at \( x = 6 \)?
3. Can you find the derivative of the function \( y = x - 5 \)?
4. How would the slope change if the curve were more complex, such as \( y = x^2 - 5 \)?
5. How does this method extend to finding slopes of secant lines in 3D space?

**Tip:** The slope of a secant line between two points on a curve provides an average rate of change, while the derivative gives the instantaneous rate of change at a specific point.

The point P(6, 1) lies on the curve y = x - 5. (a) If Q is the point (x, x - 5), use your calculator to find the slope of the secant line PQ (correct to six decimal places) for the following values of x: (i) 5.5 (ii) 5.9 (iii) 5.99 (iv) 5.999 (v) 6.5 (vi) 6.1 (vii) 6.01 (viii) 6.001

Find the slope of the secant line PQ for the curve y = x - 5 at various x-values near P(6, 1). Learn to apply the slope formula and explore the concept of limits. Suitable for students in Grades 10-12.

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