Checklist Flashcards
The completeness axiom
Every set S of real numbers which is bounded above has a supremum B, there exists a real number B s.t sub S = B
Epsolin-Delta definition
For every Epsilon>0 there exists a Delta>0 such that …
If you can keep making N(f(p)) and N(p) smaler
See image on page 128
Definition of continuity of a function at a point (Epsilon-Delta)
lim f(x) = f(x) at any x
There is also always an epsilon for every delta which satisfies the condition of the epsilon-delta definition
Intermediate value theorem for continuous functions
First imagine figure 3.8, with f(x)=k
g(x1)=f(x)-k<0 and g(x2)=f(x)-k>0
Applying bolzanos, we see that g(c)=0 at some point on the intarval [x1,x2].
Extreme value theorem for continious functions
Assume f is continious on a closed intaval [a,b], then we will have a maximum and a minimum in that intaval meaning there wil be some numbers c and d such that
f(c)=sup f and f(d)=inf f
a and b might be the inf and sup, if fx. the graph was a linear function
If (a,b) then there is no specific max or minimum values, it will have sup and inf though
Epsilon - Delta definition of uniform continuity
Use |f(x1) - f(x2)|<Epsilon to make |x1-x2|<Delta appeare. See easy example in “Prove Uniformly Continuous”
There is also another way to prove it, see Trench
Continuity on a closed bounded interval implies uniform continuity
Conceptually: if a set is closed and bounded, then there will always be a delta wich can contain a given epsilon.
Bounded derivative and uniform continuity
If f has a dirivative smaler than a given K>0 then the function is continious
|f’(x)|< K (bigger than or equal to)
Proof by mean value theorem
f(b) - f(a) = f’(c) (b-a)
Definition of differentiability at a point
h’erne går ud med hinanden i gymformlen
Differentiability implies continuity
Remember easy example from book (page 163 - example 7) where we use the identity (gymformel)
f(x+h) = f(x) + h(gymformel)
Mean value theorem for derivatives
The tangent at a given point and the secant between the two endpoints, will at some point be parallel on the graph. Given the Intaval is closed and f is continious.
Use Rolle’s theorem (wiki) on
f(b) - f(a) = f’(c)(b-a)
Definition of a (infinite) sequence (Turing Machine)
A function f with a domain spanining all the positive intergeres, it has a clear starting point, and no specific last term, f(n) is the n’th term of the function.
Turring machine.
Arithmetic infinite sequence - same intaval
Geometric - common ratio, fx multiply by 2
Epsilon - N definition of convergence of a sequence
There exists an epsilon and N so that for every |f(n)-L|<Epsilon>N</Epsilon>
After the N’th value on the x axis, the function can no longer leave the restrictic Epsilon area
Definition of a (infinite) series
See book, very simple, just adding a_n…..
s_1=a_1 s_2=a_1+a_2 and so on
Ad from series, remember the adding together to give 1/2
Definition of convergence of a series
Epsilon - N definition. Imagine the Epsilon band in which the function has to be after the N’th value
A series is the sum of the terms of an infinite sequence of numbers.
If we get closer and closer to a given number
If there exists a number n>N such that |f(n)-L|<Epsilon (L is the sum of all sequences)
