The point P(8, 1) lies on the curve
y =
x − 7
.
(a) If Q is the point
(x,
x − 7
),
use your calculator to find the slope of the secant line PQ (correct to six decimal places) for the following values of x.
(i) 7.5
(ii) 7.9
(iii) 7.99
(iv) 7.999
(v) 8.5
(vi) 8.1
(vii) 8.01
(viii) 8.001
the point with coordinates (1+√3,5√3) lies on the curve with the equation 2x^2+px+Q. find values of p
To find the slope of the secant line PQ, we need to calculate the difference in y-coordinates divided by the difference in x-coordinates.
Let's calculate the slope for each given value of x.
(i) For x = 7.5:
Coordinates of Q: (7.5, 7.5 - 7) = (7.5, 0.5)
Slope = (0.5 - 1) / (7.5 - 8) = -0.5 / -0.5 = 1
(ii) For x = 7.9:
Coordinates of Q: (7.9, 7.9 - 7) = (7.9, 0.9)
Slope = (0.9 - 1) / (7.9 - 8) = -0.1 / -0.1 = 1
(iii) For x = 7.99:
Coordinates of Q: (7.99, 7.99 - 7) = (7.99, 0.99)
Slope = (0.99 - 1) / (7.99 - 8) = -0.01 / -0.01 = 1
(iv) For x = 7.999:
Coordinates of Q: (7.999, 7.999 - 7) = (7.999, 0.999)
Slope = (0.999 - 1) / (7.999 - 8) = -0.001 / -0.001 = 1
(v) For x = 8.5:
Coordinates of Q: (8.5, 8.5 - 7) = (8.5, 1.5)
Slope = (1.5 - 1) / (8.5 - 8) = 0.5 / 0.5 = 1
(vi) For x = 8.1:
Coordinates of Q: (8.1, 8.1 - 7) = (8.1, 1.1)
Slope = (1.1 - 1) / (8.1 - 8) = 0.1 / 0.1 = 1
(vii) For x = 8.01:
Coordinates of Q: (8.01, 8.01 - 7) = (8.01, 1.01)
Slope = (1.01 - 1) / (8.01 - 8) = 0.01 / 0.01 = 1
(viii) For x = 8.001:
Coordinates of Q: (8.001, 8.001 - 7) = (8.001, 1.001)
Slope = (1.001 - 1) / (8.001 - 8) = 0.001 / 0.001 = 1
So, the slope of the secant line PQ for all given values of x is 1.
To find the slope of the secant line PQ, we need to calculate the difference in y-coordinates divided by the difference in x-coordinates (Δy/Δx).
Given that point P is (8, 1) and point Q is (x, x - 7), the y-coordinate of point Q is (x - 7).
The difference in y-coordinates (Δy) for PQ is (x - 7) - 1 = (x - 8).
Now, let's calculate the difference in x-coordinates (Δx) for PQ. The x-coordinate of point Q is x, so Δx = x - 8.
To find the slope, we divide Δy by Δx: slope = (x - 8) / (x - 8).
Using a calculator, we can substitute the given values of x to find the corresponding slopes:
(i) x = 7.5
slope = (7.5 - 8) / (7.5 - 8) = -0.5 / -0.5 = 1.000000
(ii) x = 7.9
slope = (7.9 - 8) / (7.9 - 8) = -0.1 / -0.1 = 1.000000
(iii) x = 7.99
slope = (7.99 - 8) / (7.99 - 8) = -0.01 / -0.01 = 1.000000
(iv) x = 7.999
slope = (7.999 - 8) / (7.999 - 8) = -0.001 / -0.001 = 1.000000
(v) x = 8.5
slope = (8.5 - 8) / (8.5 - 8) = 0.5 / 0.5 = 1.000000
(vi) x = 8.1
slope = (8.1 - 8) / (8.1 - 8) = 0.1 / 0.1 = 1.000000
(vii) x = 8.01
slope = (8.01 - 8) / (8.01 - 8) = 0.01 / 0.01 = 1.000000
(viii) x = 8.001
slope = (8.001 - 8) / (8.001 - 8) = 0.001 / 0.001 = 1.000000
Therefore, the slope of the secant line PQ, for all the given values of x, is 1.000000.
If Q is where the slope of PQ is just
f(x)-f(8)
--------------
x-8
Now just plug and chug