Foundations II

Question 1

Definition of a limit

Answer 1

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

Question 2

Applying definition of a limit

Answer 2

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| < ∂

Question 3

Algebra of limits

Answer 3

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

Question 4

Continuity at a point

Answer 4

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.

Question 5

Definition of the derivative

Answer 5

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

Question 6

Sum of derivatives

Answer 6

if
h(x) = f(x) + g(x)
then
h’(x) = f’(x) + g’(x)

Question 7

Product rule

Answer 7

Let h(x) = f(x)g(x)
then h’(x) = f’(x)g(x) + f(x)g’(x)

Question 8

Quotient rule

Answer 8

Let h(x) = f(x)/g(x)
then h’(x) = (f’(x)g(x) - f(x)g’(x))/(g(x))²

Question 9

Chain rule

Answer 9

Let h(x) = f(g(x))
then h’(x) = f’(g(x)) g’(x)

Question 10

Stationary point

Answer 10

Given a function f is differentiable at point x0. x0 is a stationary point on the function f if f’(x0) = 0

Question 11

L’ Hopital rule

Answer 11

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))

Question 12

Intermediate value theorem

Answer 12

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

Question 13

Mean value theorem

Answer 13

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)

Question 14

Extreme value theorem

Answer 14

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

Question 15

Rolle’s theorem

Answer 15

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.

Question 16

Cauchy mean value theorem

Answer 16

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))

Question 17

Taylors polynomial

Answer 17

Pₙ(x) = the sum of (fⁿ(a))/n! * (x-a)ⁿ

Question 18

Taylors expansion

Answer 18

f(x) = Pₙ(x;x0) + Rₙ(x;x0)

Question 19

Algebra of differential functions ( Multiplication of scalar)

Answer 19

h(x) = af(x) where a is a real number is differentiable n times
then
hⁿ(x0) = afⁿ(x0)

Question 20

Algebra of differential functions (Sum)

Answer 20

If h(x) = f(x) + g(x)
then hⁿ(x0) = fⁿ(x0) + gⁿ(x0)

Question 21

Algebra of differential functions (product)

Answer 21

if h(x) = f(x)g(x)
then hⁿ(x0) = ∑(ni) f ͥ (x0)g^(n-i)(x0)
where (ni) are the binomial coefficients

Question 22

Antiderivatives

Answer 22

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

Question 23

Fundamental theorem of calculus

Answer 23

suppose f : [a,b] → R, is integrable and has antiderivative F(x)
then a to b of ∫ f(x)dx = F(b) - F(a)

Question 24

Integration by parts

Answer 24

∫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)

Question 25

Integration by substitution

Answer 25

∫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.

Question 26

Improper integrals

Answer 26

Let f : [a,+∞) → R the improper integral is defined by ∫f(x) dx = limb→+∞ a∫b f(x) dx