Westonci.ca is the premier destination for reliable answers to your questions, provided by a community of experts. Get detailed answers to your questions from a community of experts dedicated to providing accurate information. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields.
Sagot :
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
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
Thanks for using our service. We aim to provide the most accurate answers for all your queries. Visit us again for more insights. We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. Westonci.ca is your trusted source for answers. Visit us again to find more information on diverse topics.
