Given the function$\displaystyle f(x) = \frac{\ 2x}{x-4}$ , determine the coordinates of a point on f(x) where the slope of the tangent line equals the slope of the secant line that passes through A(5,10) and B(8,4)

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- Apr 10th 2011, 01:51 PMDevi09Rational average rate of change problem
Given the function$\displaystyle f(x) = \frac{\ 2x}{x-4}$ , determine the coordinates of a point on f(x) where the slope of the tangent line equals the slope of the secant line that passes through A(5,10) and B(8,4)

- Apr 10th 2011, 02:10 PMpickslides
First find the slope of the secant line that passes through A(5,10) and B(8,4)

Hint: $\displaystyle \displaystyle m = \frac{y_2-y_1}{x_2-x_1}$

What do you get? - Apr 10th 2011, 02:15 PMDevi09
m= (4 - 10)/(8-5)

m= -2 - Apr 10th 2011, 02:20 PMpickslides
So now solve $\displaystyle \displaystyle -2 = \left(\frac{2x}{x-4}\right)'$

- Apr 10th 2011, 02:24 PMDevi09
$\displaystyle -2x + 8 = 2x$

$\displaystyle 4x=8$

$\displaystyle x=2$ - Apr 10th 2011, 02:41 PMpickslides
- Apr 10th 2011, 02:55 PMDevi09
- Apr 10th 2011, 03:00 PMpickslides
You need to find $\displaystyle \displaystyle \left(\frac{2x}{x-4}\right)'$ which is the gradient function of $\displaystyle \displaystyle \frac{2x}{x-4}$

$\displaystyle \displaystyle \left(\frac{2x}{x-4}\right)'= \left(\frac{2(x-4)-2x\times 1}{(x-4)^2}\right)$ via the quotient rule for differentiation.