Analysis Term 2
If f is a power series then it has a continuous derivative over...
(-R,R)
Integrability of function which is discontinuous at finitely many points
(ignore discontinuities in partition)
Proof of Cauchy's mean value theorem
- Define new function h(x) which is a linear combination of f(x) and g(x) - Show that h(a)=h(b) - Apply Rolle's theorem
Rewriting defintion of convex function
- Write t as a wighted average of c and d since it lies between them: let t=sc+(1-s)d for some s∈[0,1]
A power series converges __________ anywhere inside radius of convergence
A power series converges absolutely anywhere inside radius of convergence
Idea of proof for mean value theorem
Add/subtract linear terms from the function, f, so that the red lines become horizontal, then apply Rolle's theorem
Does a set have to be either open of closed?
No can be both or neither
Mean value theorem
Note that we require the function to be continuous at the end points but not differentiable
Link between Taylor's theorem and MVT
When we take n=0 in Taylor's theorem, we get the MVT
How can we write the set S={y>0|sin(y)=0} in terms of π?
S={nπ|n∈N}
Is the empty set open or closed?
Both (vacuously)
Is R closed? Is it open? It is compact?
Closed and also open, but not compact
Definition of uniform continuity
Given an epsilon, and have to choose a delta which works with all a at the same time
Continuous functions send a closed and bounded interval to....
another closed and bounded interval
Uniform continuity depends on...
the function itself but also on the domain
Definition of limit
x close to a (but not equal to) implies f(x) close to b
Proof that if f'(x)=g'(x) then there exists some c∈R such that f(x)=g(x)+c
- Define h(x)=f(x)-g(x) - Differentiate - Use result that derivative of 0 means the function is constant
Idea of proof that composing an integrable and continuous function gives integrable function
- Find partition, P, with U(h,P)-L(h,P)<ε - split intervals of P into two types, either (i) f doesn't change much (so neither does gf by uniform continuity of g), or (ii) f changes a lot, which is rare - this will give us bounds on U(h,P)-L(h,P)
How to show that f is NOT uniformly continuous
- Fix epsilon - Choose x and y depending on delta, such that |x-y|<delta is satisfied, but |f(x)-f(y)|≥epsilon i.e find sequences x_n and y_n where |s_n-y_n| converges to 0 (THIS DOES NOT MEAN THAT THE CONVERGE TO THE SAME THING) but |f(x_n)-f(y_n)| does not? -Convergent sequences will not work so think about 1/n, 1/(n+1) etc...
Proof that there exists y>0 s.t. sin(y)>0
- Suppose not - Then cos(x)≠0 ∀x>0 also - By IVT, we have cos(x)>0 ∀x>0 - This means derivative of sin is positive and so sin is strictly monotone increasing, and sin(x)>0 ∀x>0 also - Apply MVT to cos between (1,x) for any x>1 - Rearrange for contradiction
Second derivative test proof (minimum case)
- Use limit definition of derivative - Since we assume f''(a)>0, we have that the expression inside limit >0 also within some δ neighbourhood of a - Then can show f is strictly decreasing on first bit of interval, and strictly decreasing on the second - Therefore, local minimum
Proof that sin and cos are periodic functions
- We know π is first root of sin so sin(π/2)>0 and cos(x)=1-2sin²(π/2)<1 - Can deduce cos(2π)=0 - Use double angle formulae to show sin(x+2π)=sin(x) and cos(x+2π)=cos(x) - Suppose there is a smaller period and come to contradiction
Idea of proof that we require uniform convergence for integral of limit to equal limit of integrals
- fₙ is very close to f when n is large - so upper and lower darboux sums for f must be very close to those for fₙ - so the upper and lower darboux integrals get arbitrarily close to the integral of fₙ as n→∞
Idea of proof of integration by parts
- start with product rule - f' and g' are continuous so f'g and fg' are continuous - and so (fg)'=f'g+fg' is integrable - apply 2nd FTC
Sine and cos are periodic with period of...
2π
Alternative graphical interpretation of convex (regions)
A line joining any two points above the function doesn't cross below the function
A sequence of continuous functions needs to be ______ in order to converge to something continuous
A sequence of continuous functions needs to be uniformly continuous in order to converge to something continuous
All ________ functions are integrable
All continuous functions are integrable
Image of continuous function on closed bounded interval
Also a closed bounded interval
Proof that f'(x)=0 implies constant
Alternative proof: if x<y in [a,b], then by MVT we know ∃c∈(x,y) s.t: (f(y)-f(x))/(y-x)=f'(c)=0. This implies f(y)=f(x)
Definition of bounded
Both bounded above and bounded below
What do we know about the set {L(f,P)|P is a partition of [a,b]}
Bounded above by any UD sum, U(f,Q) (any Q works)
What do we know about the set {U(f,P)|P is a partition of [a,b]}
Bounded below by any LD sum, L(f,Q) (any Q works)
Why are sine and cosine continuous?
Can be defined in terms of e which we know is continuous
Approximating sin using Taylor's Theorem
Can find upper bound for remainder term
The stringer version of the mean value theorem is called...
Cauchy's mean value theorem
is [13,∞) closed? Is it compact?
Closed but not compact
Useful tool for proving things about two functions
Define a new function as the sum/difference of them
Definition of continuity everywhere
Delta can depend on a (but not always necessary)
What requirement do we not require for Taylor's theorem?
Don't need (n+1)th derivative to be differentiable
If f is strictly increasing, then the supremum is located where on the interval?
End
How to show a function has a root using IVT?
Find a point ≤ 0 and another point ≥ 0. Then apply IVT
How can we show a function is integrable, without knowing the integral
Find upper bound for the integrand which is integrable
Definition of open set
For all x in S, the delta neighbourhood of x is also in S
Useful think to thing about when asked for example of convergent sequences with discontinuities
For x∈(0,1), xⁿ→0, but 1ⁿ→1
Graphical interpretation of convex (lines)
Function lies below line joining the two end points
Sequential continuity
Given a sequence that lies completely in the domain of a function with a limit of a, f(x_n) will have a limit of f(a)
Link between analytic functions and Taylor series
If a function is equal to its own Taylor series, then we say that it is analytic
If a sequence of continuous functions converges uniformly, then they must converge to a _______ function
If a sequence of continuous functions converges uniformly, then they must converge to a continuous function
When is the second derivative test inconclusive?
If f(x) satisfies f'(a)=f''(a)=0. Then it does not tell us whether f has a local maximum or local minimum at x=a.
When can we differentiate a sum term by term?
If it is finite, or if we have proven convergence
First fundamental theorem of calculus
If you integrate a function that is continuous, then (i) the integral is a continuous function (ii) its derivative exists and is the original function
Integration is a ______ operator
Integration is a linear operator
We say a function f:[a,b]→R is convex is for all c<t<d in the domain, we have...
LHS is a linear function in t which passes through (c,f(c)) at t=0, and (d,f(d)) at t=d.
If we have a sequence of functions that converge pointwise, what can we say about the limits of the integrals
Limit of integrals not necessarily equal to integral of limits
Definition of a closed set
Limits of all convergent subsequences in set are also in set
how do we check that f is continuous everywhere?
Must satisfy condition of continuity for all a. This is difficult to check so we use sequential continuity instead
Can we use the MVT to prove L'Hoptial?
NO since L'Hopital doesn't require g to be a continuous function
For a Taylor series of P(x) of f(x) about a=0, is it always true that we have f(x)=P(x) on (-1,1)?
No, consider 'bump function' as counterexample
Does the Taylor series of a function at a=0, P(x), have an infinite radius of convergence?
No, consider f(x)=1/(1-x) as counterexample
Is every function equal to its own Taylor series at a point?
No, consider this counterexample where f(x)=P(x) only when x=0
Are all integrable functions continuous?
No, counterexample:
When is a power series is integrable?
On a closed subinterval of [-R,R] where R is radius of convergence
If a set S is open, then what can we say about its compliment?
R\S is closed
Taylor's Series definition
Requires function to be infinitely differentiable
The mean value theorem is a more general case of...
Rolle's theorem where we do not necessarily have f(a)=f(b)
Sequences converge _______. Functions converge _______.
Sequences converge absolutely. Functions converge uniformly.
Using IVT how can we prove there exists more than one x s.t f(x)=c?
Split the function up and apply IVT to the different sections
If f is strictly increasing, then the infimum is located where on the interval?
Start
Definition of partition
Strictly increasing sequence of real numbers
Cauchy's mean value theorem
Stronger than mean value theorem
Taylor polynomial for a function which has its (n+1)th derivative equal to 0 at all x for some n
Taylors theorem says that f(x)=P_n(x) and so f must be a polynomial of degree at most n
How to integrate power series
Term by term
What do we require to define the infimum and supremum over an interval?
That the function is bounded on that interval (but doesn't have to be continuous!)
What requirements do we need in order for nth derivative of f to exist at x=a?
The (n-1)th derivative must exist in a neighbourhood of a and be differentiable at x=a
What is the partition of an interval?
The partition splits up an interval into closed intervals
What condition do we require for the limit of the derivatives of a sequence of functions to converge to the derivative of the limit of the functions?
The sequence of the derivatives of the functions converge uniformly. Note: we only need the sequence of functions to converge pointwise
What function do we define in order to prove the mean value theorem?
Then differentiate and apply Rolle's theorem
Example of function where neither upper or lower Darboux sum exist
Unbounded on every interval
What condition do we need on a sequence of functions, fₙ, for the limit of the integral to be the integral of the limit?
Uniform continuity
Proof that cos²x+sin²x=1
Use difference of two squares and the fact that cos is even and sin is odd
Proof of L'Hopital's rule from one-sided version
We have proven L'Hopital for x↑a and analogously for x↓a so we can just combine these
If a function is differentiable and we know the derivative, then what do we know about the function?
We know the function up to some additive constant
When you see the sum of a family of functions, think about...
Weierstrass M-Test
What is an improper integral?
When the integral is not defined at some point
Idea of proof that the product of two integrable functions is integrable
Write fg as a linear combination/composition of integrable/continuous functions
Is f(x)=x convex?
Yes
Integrability of monotone increasing functions
always integrable
Integrability of strictly monotone increasing function, f (inverse)
both f and f⁻¹ are integrable
The intersection of (possibly infinitely many) closed sets is...
closed
The union of finitely many closed sets is...
closed
Definition of compact
closed and bounded
NOT SURE IF THIS IS TRUE: If the upper and lower darboux sums are equal on all partitions then the function is...
constant
What condition is not required for L'Hopital?
continuity
If f is continuous and injective, the f⁻¹ is...
continuous
f'(x) is integrable if it is...
continuous
If f is differentiable at a, then it is also...
continuous at a
Define cos in terms of the real part of another function
cos(x)=Re(E(ix))
L'Hopital's Rule
don't actually understand!!
If a function is constant then its upper and lower darboux sums are...
equal over any partition
If we bound something between 0 and ε, then it must be...
equal to 0
Sequential compactness: A subset of R, S, is compact iff...
every sequence which is a subset of S has a convergent subsequence which converges to a limit in S
For a bounded function, how do we know and inf and sup exist over an interval?
extreme value theorem
Three equivalent statements about a continuous f:[a,b]→R
f injective ↔f strictly monotonic ↔f is a bijection
If a sequence of (uniformly) continuous functions f_n:S→R converges uniformly to f, then...
f is (uniformly) continuous
If f:[a,b]→R is continuous (on closed bounded invterval), then...
f is bounded
For f continuous on [a,b] and differentiable on (a,b). If f'(x)=0 for all x∈(a,b), then...
f is constant on [a,b]
Definition of continuity at point a
f is continuous at a point a, if and only if for all epsilon there exists a delta such that for all x in a delta neighbourhood of a, f(x) is in an epsilon neighbourhood of f(a)
If f⁻¹ is integrable then...
f is integrable
For f continuous on [a,b] and differentiable on (a,b). If f'(x)≥0 then...
f is monotone increasing on [a,b]
For f continuous on [a,b] and differentiable on (a,b). If f'(x)>0 then...
f is strictly monotone increasing on [a,b]
If f'(x) is monotone increasing then...
f''(x)≥0
If the lower and upper Darboux integrals are not equal, then...
f(x) is not integrable on [a,b]
If f and g are continuous at a, then the following are also continuous at a... What do we use to prove these?
f+g, f-g, fg, f/g (iff g(a)≠0) Proven by sequential continuity and algebra of limits
If strictly monotonic increasing then...
for any distinct points x<y in the domain we have f(x)<f(y)
If strictly monotonic decreasing then...
for any distinct points x<y in the domain we have f(x)>f(y)
The original MVT is the Cauchy MVT with...
g(x)=x
A bounded function f:[a,b]→R is integrable iff...
i.e: as long as we can get lower and upper sums arbitrarily close for a subset of partitions, then integrable
If f:[a,b]→R or f:R→R is continuous, the f is injective
iff f is strictly monotonic
If f:[a,b]→R is a continuous function, then f is...
integrable
If we compose an integrable (not necessarily continuous) function with a continuous function then the result is...
integrable
Linear combinations of integrable functions are...
integrable
The product of two integrable functions is...
integrable
The antiderivative of f(x) is the...
integral of f(x)
If f is bijective then f has an...
inverse
f:[a,b]→R continuous means that...
it is uniformly continuous since we are going from a compact set
f'(a)=
lim (f(x)-f(a))/(x-a) or...
What inequality do we always know is true for upper and lower darboux sums
lower≤upper
Rolle's theorem is a special case of the...
mean value theorem, where f(a)=f(b)
A half open interval is...
neither open nor closed
Any lower Darboux sum for f is ≤ any upper Darboux sum of f, regardless...
of the partitions used
The intersection of a finite number of open sets is...
open
The preimage of an open set is...
open
The union of of open subsets (possibly infinitely many) of R is...
open
Integral test for convergence of a series
ps7
Define sin in terms of the imaginary part of another function
sin(x)=Im(E(ix))
If f is convex on (x,y), and we take some c,d∈(x,y), then what can we say about the slopes between various points?
slope between x and x ≤ slope between x and y ≤ slope between d and y
What do we do when finding an integral from a to b where f is unbounded at some interior point c∈(a,b) or where both a=-∞ and b=∞
split the integral!
Why/when is tan continuous?
tan=sin/cos and we know sin and cos are continuous. This is only when cos(x)≠0, x≠(2k+1)π/2
If S is compact and f:S→R is continuous, then f is...
uniformly continuous
If the upper and lower Darboux sums integrals of f on [a,b] are equal, then...
we say f is (Darboux) integrable on [a,b] and define:
