Find the slope of the secant to the curve f(x) = -2logx + 3 between: (1 each)
i) x = 1 and x = 2
ii) x = 1 and x = 1.5
iii) x = 1 and x = 1.1
vi) x = 1 and x = 1.01
b) Extend the result from part a to determine the slope of the tangent to the curve at x = 1 accurate to 3 decimal places. (2 marks)
i) f(1) = -2log(1) + 3 = 3
f(2) = -2log2 + 3
slope of secant = (-2log2 + 3 - 3)/1 = -2log2
= appr -.602
....
iii) f(1) = 3
f(1.1) = -2log 1.1 + 3
slope = (-2log 1.1 + 3 - 3)/(1.1-1) = -2log 1.1 /.1
= appr -.828
iv)
...
slope = -2log1.01 /.01
= appr -.864
etc.
To find the slope of a secant to the curve f(x) = -2logx + 3 between two points, you can use the formula:
Slope = (f(x2) - f(x1)) / (x2 - x1)
where x1 and x2 are the x-coordinates of the two points and f(x) represents the equation of the curve.
i) For x = 1 and x = 2:
Plug these values into the formula:
Slope = (-2log2 + 3 - (-2log1 + 3)) / (2 - 1)
Simplify the equation to find the slope.
ii) For x = 1 and x = 1.5:
Plug these values into the formula:
Slope = (-2log(1.5) + 3 - (-2log1 + 3)) / (1.5 - 1)
Simplify the equation to find the slope.
iii) For x = 1 and x = 1.1:
Plug these values into the formula:
Slope = (-2log(1.1) + 3 - (-2log1 + 3)) / (1.1 - 1)
Simplify the equation to find the slope.
iv) For x = 1 and x = 1.01:
Plug these values into the formula:
Slope = (-2log(1.01) + 3 - (-2log1 + 3)) / (1.01 - 1)
Simplify the equation to find the slope.
b) To determine the slope of the tangent to the curve at x = 1 accurate to 3 decimal places, you need to take the limit as the difference in x-values approaches zero.
Use the formula:
Slope of tangent = lim (x -> 1) ((f(x) - f(1)) / (x - 1))
Substitute the equation f(x) = -2logx + 3 and calculate the limit to find the slope of the tangent at x = 1.