Tangent to a a curve, something seems wrong (Calculus)

[christian0710]

I'm studying Calculus and i can see that the definition of the tangent to a point on a curve is

y= f'(a)(x-a)+b this must mean that

f'(a) = (y-b)/(x-a)

But that to me seems troubeling, because f'(a) is the slope at ONE point, while (y-b)/(x-a) is a quotient with the difference between 2 points. Is there a better explanation?

 

[HallsofIvy]

No, that's perfectly correct. However, "f'(a)= (y- b)/(x- a)" requires that x and y be the coordinates of a point on the tangent line, not the curve. A line has the same slope at any point.

 

[RUber]

Comparing this form to y = mx+k, you have f'(a) = m giving you the slope of the tangent at x=a. Your y- intercept of the line is b-f'(a) a, this is assuming that f(a) = b. Essentially, you are tracing back the line along the slope to find the y-intercept.

(y-b)/(x-a) is 0/0 at (a,b), so might not be the best form to use...although it does give some insight to L'Hopital's rule.