differentiable functions Flashcards
differentiation by first principles:
lim(x->x0)(f(x)-f(x0))/(x-x0)
aka difference quotient, newton quotient
easy differentiables:
constant functions have derivative 0
f(x)=x has derivative 1
f(x)=x^n has derivative nx^(n-1)
differentiability and continuity:
if f:S->R is differentiable at x0 in S, then f is continuous at x0
chain rule:
(g’ o f)(x0)=g’(f(x0))*f’(x0)
corollary of the chain rule:
if f and g are differentiable, so is g o f with (g o f)’=(g’ o f)*f’
addition of functions differentiation:
(f+g)’(x0)=f’(x0)+g’(x0)
multiplication of functions differentiation:
(fg)’(x0)=(f’(x0)g(x0))+(f(x0)*g’(x0))
division of functions differentiation:
if g(x)!=0 for all x
(f/g)’(x0)=(g(x0)f’(x0)-g’(x0)f(x0))/g^(2)(x0)
local maximum:
a point x0 is a local maximum of a function f:S->R if there is some h>0 such that for all x in S, x0-h<x<x0+h => f(x0)>=f(x)
local minimum:
a point x0 is a local minimum of a function f:S->R if there is some h>0 such that for all x in S, x0-h<x<x0+h => f(x0)<=f(x)
local extremum:
local minimum or maximum
if f:S->R is differentiable at x0 in S and x0 is a local extremum:
f’(x0)=0 (cause it’s yknow. flat there)
rolle’s theorem:
let f:[a,b]->R be a continuous function that’s differentiable in (a,b) - if f(a)=f(b), then there is some x0 in (a,b) with f(x0)=0
basically if a and b have the same y value, there must be a local extremum in between (dw about constant functions cause they’re not differentiable)
let f,g:[a,b]->R be continuous functions which are differentiable in (a,b) - then there’s some x0 in (a,b) with:
f’(x0)(g(b)-g(a))=g’(x0)(f(b)-f(a))
mean value theorem:
let f:[a,b]->R be a continuous function differentiable in (a,b) - there’s some x0 in (a,b) with f’(x0)=(f(b)-f(a))/b-a
