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3. The point [tex][tex]$P(2, -1)$[/tex][/tex] lies on the curve [tex][tex]$y = \frac{1}{1-x}$[/tex][/tex].

(a) If [tex]Q[/tex] is the point [tex][tex]$(x, \frac{1}{1-x})$[/tex][/tex], find the slope of the secant line [tex]PQ[/tex] (correct to six decimal places) for the following values of [tex]x[/tex]:

(i) 1.5
(ii) 1.9
(iii) 1.99
(iv) 1.999
(v) 2.5
(vi) 2.1
(vii) 2.01
(viii) 2.001



Answer :

GinnyAnswer
To solve this problem, let's break it down step by step.

Given:
- Point [tex]\( P \)[/tex] is [tex]\( (2, -1) \)[/tex].
- The function [tex]\( y = \frac{1}{1 - x} \)[/tex] defines the curve on which point [tex]\( Q \)[/tex] lies.
- Point [tex]\( Q \)[/tex] is [tex]\( (x, \frac{1}{1 - x}) \)[/tex].

The slope of the secant line passing through points [tex]\( P \)[/tex] and [tex]\( Q \)[/tex] can be calculated using the following formula for the slope:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]

Here, point [tex]\( P \)[/tex] is [tex]\( (2, -1) \)[/tex] and point [tex]\( Q \)[/tex] is [tex]\( (x, \frac{1}{1 - x}) \)[/tex].

Let's calculate the slopes for the given values of [tex]\( x \)[/tex] to six decimal places.

### (i) For [tex]\( x = 1.5 \)[/tex]:
- [tex]\( Q \)[/tex] is [tex]\( (1.5, \frac{1}{1 - 1.5}) \)[/tex]
- [tex]\( Q \)[/tex] is [tex]\( (1.5, \frac{1}{-0.5}) \)[/tex]
- [tex]\( Q \)[/tex] is [tex]\( (1.5, -2) \)[/tex]

Slope:
[tex]\[ m = \frac{-2 - (-1)}{1.5 - 2} = \frac{-2 + 1}{1.5 - 2} = \frac{-1}{-0.5} = 2.0 \][/tex]

### (ii) For [tex]\( x = 1.9 \)[/tex]:
- [tex]\( Q \)[/tex] is [tex]\( (1.9, \frac{1}{1 - 1.9}) \)[/tex]
- [tex]\( Q \)[/tex] is [tex]\( (1.9, \frac{1}{-0.9}) \)[/tex]
- [tex]\( Q \)[/tex] is [tex]\( (1.9, -1.\overline{1}) \)[/tex]

Slope:
[tex]\[ m = \frac{-1.\overline{1} - (-1)}{1.9 - 2} = \frac{-1.\overline{1} + 1}{1.9 - 2} = \frac{-0.111111}{-0.1} \approx 1.111111 \][/tex]

### (iii) For [tex]\( x = 1.99 \)[/tex]:
- [tex]\( Q \)[/tex] is [tex]\( (1.99, \frac{1}{1 - 1.99}) \)[/tex]
- [tex]\( Q \)[/tex] is [tex]\( (1.99, \frac{1}{-0.99}) \)[/tex]
- [tex]\( Q \)[/tex] is [tex]\( (1.99, -1.010101) \)[/tex]

Slope:
[tex]\[ m = \frac{-1.010101 - (-1)}{1.99 - 2} = \frac{-1.010101 + 1}{1.99 - 2} = \frac{-0.010101}{-0.01} \approx 1.010101 \][/tex]

### (iv) For [tex]\( x = 1.999 \)[/tex]:
- [tex]\( Q \)[/tex] is [tex]\( (1.999, \frac{1}{1 - 1.999}) \)[/tex]
- [tex]\( Q \)[/tex] is [tex]\( (1.999, \frac{1}{-0.999}) \)[/tex]
- [tex]\( Q \)[/tex] is [tex]\( (1.999, -1.001001) \)[/tex]

Slope:
[tex]\[ m = \frac{-1.001001 - (-1)}{1.999 - 2} = \frac{-1.001001 + 1}{1.999 - 2} = \frac{-0.001001}{-0.001} \approx 1.001001 \][/tex]

### (v) For [tex]\( x = 2.5 \)[/tex]:
- [tex]\( Q \)[/tex] is [tex]\( (2.5, \frac{1}{1 - 2.5}) \)[/tex]
- [tex]\( Q \)[/tex] is [tex]\( (2.5, \frac{1}{-1.5}) \)[/tex]
- [tex]\( Q \)[/tex] is [tex]\( (2.5, -\frac{2}{3}) \)[/tex]

Slope:
[tex]\[ m = \frac{-\frac{2}{3} - (-1)}{2.5 - 2} = \frac{-\frac{2}{3} + 1}{2.5 - 2} = \frac{\frac{1}{3}}{0.5} = 0.666667 \][/tex]

### (vi) For [tex]\( x = 2.1 \)[/tex]:
- [tex]\( Q \)[/tex] is [tex]\( (2.1, \frac{1}{1 - 2.1}) \)[/tex]
- [tex]\( Q \)[/tex] is [tex]\( (2.1, \frac{1}{-1.1}) \)[/tex]
- [tex]\( Q \)[/tex] is [tex]\( (2.1, -0.909091) \)[/tex]

Slope:
[tex]\[ m = \frac{-0.909091 - (-1)}{2.1 - 2} = \frac{-0.909091 + 1}{2.1 - 2} = \frac{0.090909}{0.1} = 0.909091 \][/tex]

### (vii) For [tex]\( x = 2.01 \)[/tex]:
- [tex]\( Q \)[/tex] is [tex]\( (2.01, \frac{1}{1 - 2.01}) \)[/tex]
- [tex]\( Q \)[/tex] is [tex]\( (2.01, \frac{1}{-1.01}) \)[/tex]
- [tex]\( Q \)[/tex] is [tex]\( (2.01, -0.990099) \)[/tex]

Slope:
[tex]\[ m = \frac{-0.990099 - (-1)}{2.01 - 2} = \frac{-0.990099 + 1}{2.01 - 2} = \frac{0.009901}{0.01} \approx 0.990099 \][/tex]

### (viii) For [tex]\( x = 2.001 \)[/tex]:
- [tex]\( Q \)[/tex] is [tex]\( (2.001, \frac{1}{1 - 2.001}) \)[/tex]
- [tex]\( Q \)[/tex] is [tex]\( (2.001, \frac{1}{-1.001}) \)[/tex]
- [tex]\( Q \)[/tex] is [tex]\( (2.001, -0.999001) \)[/tex]

Slope:
[tex]\[ m = \frac{-0.999001 - (-1)}{2.001 - 2} = \frac{-0.999001 + 1}{2.001 - 2} = \frac{0.000999}{0.001} \approx 0.999001 \][/tex]

Thus, the slopes of the secant line for the given values of [tex]\( x \)[/tex] are:

(i) [tex]\( x = 1.5 \rightarrow m \approx 2.000000 \)[/tex]

(ii) [tex]\( x = 1.9 \rightarrow m \approx 1.111111 \)[/tex]

(iii) [tex]\( x = 1.99 \rightarrow m \approx 1.010101 \)[/tex]

(iv) [tex]\( x = 1.999 \rightarrow m \approx 1.001001 \)[/tex]

(v) [tex]\( x = 2.5 \rightarrow m \approx 0.666667 \)[/tex]

(vi) [tex]\( x = 2.1 \rightarrow m \approx 0.909091 \)[/tex]

(vii) [tex]\( x = 2.01 \rightarrow m \approx 0.990099 \)[/tex]

(viii) \( x = 2.001 \rightarrow m \approx 0.999001

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