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SCALCET8 2.1.502.XP.

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PRACTICE ANOTHER
The point P(7, 1) lies on the curve
y =

x − 6
.
(a) If Q is the point
(x,

x − 6
),
use your calculator to find the slope of the secant line PQ (correct to six decimal places) for the following values of x.
(i)    6.5

(ii)    6.9

(iii)    6.99

(iv)    6.999

(v)    7.5

(vi)    7.1

(vii)    7.01

(viii)    7.001

To solve this problem, you need to find the slope of the secant line \( PQ \) between the fixed point \( P(7, 1) \) on the curve \( y = x - 6 \) and a point \( Q(x, y) \) where \( y = x - 6 \). The slope of the secant line between points \( P(x_1, y_1) \) and \( Q(x_2, y_2) \) is given by the formula:

\[
m_{\text{secant}} = \frac{y_2 - y_1}{x_2 - x_1}
\]

In this case, \( P(7, 1) \) corresponds to \( (x_1, y_1) = (7, 1) \), and \( Q(x, x-6) \) corresponds to \( (x_2, y_2) = (x, x-6) \). So, the slope of the secant line \( PQ \) is:

\[
m_{\text{secant}} = \frac{(x - 6) - 1}{x - 7} = \frac{x - 7}{x - 7} = 1
\]

However, let's confirm this by calculating the slope for each given value of \( x \):

### Calculations:

1. **For \( x = 6.5 \):**
\[
m_{\text{secant}} = \frac{(6.5 - 6) - 1}{6.5 - 7} = \frac{0.5 - 1}{-0.5} = \frac{-0.5}{-0.5} = 1.000000
\]

2. **For \( x = 6.9 \):**
\[
m_{\text{secant}} = \frac{(6.9 - 6) - 1}{6.9 - 7} = \frac{0.9 - 1}{-0.1} = \frac{-0.1}{-0.1} = 1.000000
\]

3. **For \( x = 6.99 \):**
\[
m_{\text{secant}} = \frac{(6.99 - 6) - 1}{6.99 - 7} = \frac{0.99 - 1}{-0.01} = \frac{-0.01}{-0.01} = 1.000000
\]

4. **For \( x = 6.999 \):**
\[
m_{\text{secant}} = \frac{(6.999 - 6) - 1}{6.999 - 7} = \frac{0.999 - 1}{-0.001} = \frac{-0.001}{-0.001} = 1.000000
\]

5. **For \( x = 7.5 \):**
\[
m_{\text{secant}} = \frac{(7.5 - 6) - 1}{7.5 - 7} = \frac{1.5 - 1}{0.5} = \frac{0.5}{0.5} = 1.000000
\]

6. **For \( x = 7.1 \):**
\[
m_{\text{secant}} = \frac{(7.1 - 6) - 1}{7.1 - 7} = \frac{1.1 - 1}{0.1} = \frac{0.1}{0.1} = 1.000000
\]

7. **For \( x = 7.01 \):**
\[
m_{\text{secant}} = \frac{(7.01 - 6) - 1}{7.01 - 7} = \frac{1.01 - 1}{0.01} = \frac{0.01}{0.01} = 1.000000
\]

8. **For \( x = 7.001 \):**
\[
m_{\text{secant}} = \frac{(7.001 - 6) - 1}{7.001 - 7} = \frac{1.001 - 1}{0.001} = \frac{0.001}{0.001} = 1.000000
\]

### Conclusion:

For all the values of \( x \), the slope of the secant line \( PQ \) is consistently 1.000000, correct to six decimal places.

---

Would you like more details on any part of this explanation? Here are some related questions to consider:

1. What is the significance of the slope of the secant line in calculus?
2. How does the slope of the secant line relate to the concept of a derivative?
3. How do we calculate the slope of a secant line if the function is more complex?
4. What happens to the slope of the secant line as \( Q \) approaches \( P \)?
5. Can you explain the difference between a secant line and a tangent line?

**Tip:** When calculating slopes, always be cautious about subtracting very close numbers, as this can lead to significant rounding errors in practical calculations.

The point P(7, 1) lies on the curve y = x - 6. If Q is the point (x, x - 6), use your calculator to find the slope of the secant line PQ (correct to six decimal places) for the following values of x: (i) 6.5, (ii) 6.9, (iii) 6.99, (iv) 6.999, (v) 7.5, (vi) 7.1, (vii) 7.01, (viii) 7.001.

Discover how to calculate the slope of the secant line PQ for the curve y = x - 6 using specific x-values. This problem involves key concepts in analytic geometry, such as the slope of secant lines. Suitable for students in Grades 10-12, the solution provides step-by-step calculations correct to six decimal places.

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