Foundations II Flashcards
Definition of a limit
Given a function f : A -> R and cluster point x0 of A we can say that the limit of f at x0 is equal to the real number L, Ɛ >0 there exists a ∂ > 0.
we write limx->x0 f(x) = L
Applying definition of a limit
We need to show that for all Ɛ > 0 there exists a ∂ > 0 such that: |f(x) - L |< Ɛ ∀x in A with 0 < |x-x0| < ∂
Algebra of limits
If limx->x0 f(x) = L and limx->x0 g(x) = T then
1. limx->x0 cf(x) = cL for a constant c
2. limx->x0 (f(x)+g(x)) = L+T
3. limx->x0 (f(x)g(x)) = LT
4. for T ≠ 0, limx->x0 f(x)/g(x) = L/T
Continuity at a point
We can say a function f is continuous at a point x0 if and only if limx->x0 f(x) = f(x0)
|f(x) - f(x0)| < Ɛ ∀x0 with 0 < |x-x0 | < ∂
it is required that as the function approaches the point from both the positive and negative side it gives the same value at the same point.
Definition of the derivative
Given function f and point x0 we say that f is differentiable at x0 if the following limit exists and is finite:
f’(x0) = limx->x0 (f(x)-f(x0))/(x-x0)
alternatively we may write
f’(x) = limx->x0 (f(x+h) - f(x)/h
Sum of derivatives
if
h(x) = f(x) + g(x)
then
h’(x) = f’(x) + g’(x)
Product rule
Let h(x) = f(x)g(x)
then h’(x) = f’(x)g(x) + f(x)g’(x)
Quotient rule
Let h(x) = f(x)/g(x)
then h’(x) = (f’(x)g(x) - f(x)g’(x))/(g(x))²
Chain rule
Let h(x) = f(g(x))
then h’(x) = f’(g(x)) g’(x)
Stationary point
Given a function f is differentiable at point x0. x0 is a stationary point on the function f if f’(x0) = 0
L’ Hopital rule
Let f and g be continuous functions on a closed interval [a,b] and differentiable on the open interval (a,b) and have limx→x0 f(x) = limx→x0 g(x) = 0 if the limit of f’ and g’ exists at x0 then the limx→x0 (f(x)/g(x)) = limx→x0 (f’(x)/g’(x))
Intermediate value theorem
If a continuous function f(x) is defined on a closed interval [a,b], then for any value c between f(a) and f(b), there exists at least one value x in the interval [a,b] such that f(x) = c
Mean value theorem
If f(x) is continuous on the closed interval [a,b] and differentiable on the open interval (a,b) then there exists at least one number c in the open interval (a,b) such that:
f’(c) = (f(b) - f(a))/ (b-a)
Extreme value theorem
If f(x) is a continuous function on a closed interval [a,b], then f(x) has both a maximum and a minimum value in that interval
Rolle’s theorem
If f(x) is a continuous function on a closed interval [a,b] and differentiable on a open interval (a,b) such that f(a) = f(b). Then there exists at least one point c such that f’(c) = 0.
Cauchy mean value theorem
Let f and g be functions defined in a closed interval [a,b] and differentiable in an open interval (a,b). For g’(x) ≠ 0. we have
f’(x)/g’(x) = (f(b)-f(a))/(g(b)-g(a))
Taylors polynomial
Pₙ(x) = the sum of (fⁿ(a))/n! * (x-a)ⁿ
Taylors expansion
f(x) = Pₙ(x;x0) + Rₙ(x;x0)
Algebra of differential functions ( Multiplication of scalar)
h(x) = af(x) where a is a real number is differentiable n times
then
hⁿ(x0) = afⁿ(x0)
Algebra of differential functions (Sum)
If h(x) = f(x) + g(x)
then hⁿ(x0) = fⁿ(x0) + gⁿ(x0)
Algebra of differential functions (product)
if h(x) = f(x)g(x)
then hⁿ(x0) = ∑(ni) f ͥ (x0)g^(n-i)(x0)
where (ni) are the binomial coefficients
Antiderivatives
for functions f and F, F is the antiderivative of f if F is continuous on [a,b] and differentiable on (a,b). Such that F’(x) = f(x) and F(x) = ∫f(x) dx
Fundamental theorem of calculus
suppose f : [a,b] → R, is integrable and has antiderivative F(x)
then a to b of ∫ f(x)dx = F(b) - F(a)
Integration by parts
∫f(t)g’(t)dt = [f(t)g(t)] - ∫f’(t)g(t)
or
∫u(x)v’(x)= u(x)v(x) - ∫v(x)u’(x)
Integration by substitution
∫f(u(t))u’(t) dt = ∫ f(u)
change the difficult part of a function to a u then sub back in changing for u’s.
Improper integrals
Let f : [a,+∞) → R the improper integral is defined by ∫f(x) dx = limb→+∞ a∫b f(x) dx
