Question

Hi, my name is Carlos. I do need help with the following: given the equation f(x)= 5x^2-4x, find the slope at (a, f(a)) and the equation at (2, f(2)) -
The answer is: m = 10a – 4 and the equation at (2, f(2)) is : y=16x – 20. But I cannot get the same result using the the following:
m= f(x) – f(a) / x – a
So, using that formula, it begins like this: (5x^2-4x) – (5a^2-4a) / x – a .. .
Any idea ?
Thank you very much

Answer 1

to get the slope at point (a,f(a)) you need to take limit of "slope formula" as x approaches a:
\[\lim_{x \rightarrow a} \frac{f(x) - f(a)}{x-a}\]
this is easier to do if you make the limit approaching zero.... sub in a new variable
h = x-a ...... a = x-h
\[\rightarrow \lim_{h \rightarrow 0} \frac{f(x) - f(x-h)}{h}\]
\[\lim_{h \rightarrow 0} \frac{(5x^2 -4x) - (5(x-h)^2 -4(x-h))}{h}\]
\[\lim_{h \rightarrow 0} \frac{10xh - 5h^2 -4h}{h}\]
Factor out an "h"
\[\lim_{h \rightarrow 0} (10x -4 - 5h) = 10x - 4\]
we are evaluating slope at x = a,
\[m = 10a - 4\]

Answer 2

equation of line comes from using point-slope form
\[y - f(2) = m (x - 2)\]
\[y = mx -2m + f(2)\]

Answer 3

dumbcow.. . THANK YOU VERY MUCH !! Appreciated