- #1
Hernaner28
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Generic "definition" of derivative?
Hi. This is a theoric doubt I have since I went to class today. The professor "redifined" the derivative at point a. He draw a curve (the function) and the tangent at point a. Then he draw another two lines in the same point.
Well, then he said that the error is the difference between the lines and the function or I understood that (if it's not that way please clarify that to me).
So:
$$\eqalign{
& Error = f(x) - y \cr
& Error = f(x) - [m(x - a) + f(a)] \cr
& Error = f(x) - f(a) - m(x - a) \cr
& Error = \underbrace {(x - a)}_{\scriptstyle 0 \atop
\scriptstyle x \to a} \underbrace {\underbrace {\left( {\frac{{f(x) - f(a)}}{{x - a}} - m} \right)}_{\scriptstyle f'(a) - m \atop
\scriptstyle x \to a} }_{r(x)} \cr
& {\text{The error is minimum if m = f'(a)}} \cr
& {\text{so}} \cr
& \underbrace {f(x) = f(a) + f'(a)(x - a) + (x - a)r(x)}_{{\text{With }}r(x) \to 0{\text{ when }}x \to a} \cr} $$
Now... this is really interesting but I wonder what this is for... is this useful for sth in particular?
Thanks!
Hi. This is a theoric doubt I have since I went to class today. The professor "redifined" the derivative at point a. He draw a curve (the function) and the tangent at point a. Then he draw another two lines in the same point.
Well, then he said that the error is the difference between the lines and the function or I understood that (if it's not that way please clarify that to me).
So:
$$\eqalign{
& Error = f(x) - y \cr
& Error = f(x) - [m(x - a) + f(a)] \cr
& Error = f(x) - f(a) - m(x - a) \cr
& Error = \underbrace {(x - a)}_{\scriptstyle 0 \atop
\scriptstyle x \to a} \underbrace {\underbrace {\left( {\frac{{f(x) - f(a)}}{{x - a}} - m} \right)}_{\scriptstyle f'(a) - m \atop
\scriptstyle x \to a} }_{r(x)} \cr
& {\text{The error is minimum if m = f'(a)}} \cr
& {\text{so}} \cr
& \underbrace {f(x) = f(a) + f'(a)(x - a) + (x - a)r(x)}_{{\text{With }}r(x) \to 0{\text{ when }}x \to a} \cr} $$
Now... this is really interesting but I wonder what this is for... is this useful for sth in particular?
Thanks!