Test 3 Questions Flashcards
Let f: (-1, 1) -> R be given. If f’ exists on (-1,1), then f continuous at 0.
True
Let (fn) be a sequence of functions defined on A ⊂ R. If (fn) converges uniformly on A to a function f, then f is continuous
False
The radius of convergence of the power series for f (x) = 2x / (1 + 4x) is R = 1/4
True
Suppose f : [a, b] -> R is differentiable. If f(a) = f(b) = 0, f’(a) > 0 and f’(b) > 0, then there is a c ∈ (a, b) with f(c) = 0
True
The series sqrt(x^2 + 1/n) / n^2 converges uniformly on [0,2]
True
Let f : [0, 1] -> R. If f is differentiable on [0,1], then f is integrable on [0,1]
True
Suppose f : R -> R is infinitely differentiable. It is possible for the Taylor series of f (centered at 0) to converge only on [0, 1), and diverge elsewhere.
False
Let f : [0, 1] ⊂ R, and assume f has countably many discontinuities. Then f is integrable
False
Let f : R -> R be differentiable. If f(0) = 0 and f’(x) < 1 for all x ∈ R, then f(1) < 1
True
How do you show that a derivative and a second derivative exists at a point a?
Set up:
lim (x->a)
f(x) - f(a) / x - a
Simplify
Find limit
If f’(x) exists, do the same thing but with
f’(x) - f’(a) / x - a
And then solve
How do you show that a sequence of functions converges uniformly?
|fn(x) - f(x)| < ϵ
use inequalities so that you get ~ < ϵ
Assume n >= N
Choose N = ~
Work backwards until you get to fn(x)
(you can use x ∈ [a, b] in the inequalities)
How do you find the radius of convergence if you already have the series function?
R is where
|Fn+1(x) / Fn(x)| < 1
The center is where the expression containing x is equal to zero
If you want the interval you must check the end points
How do you find the power series of a function?
alter the function until it is of the form
1 / (1 - x)
if there is an x on the outside, put it in front of the normal geometric series Σx^n
if there is a 1 / (1 - x)^2 , you simply take the derivative of the original geometric series function (Σnx^n-1)
plug in - (x) into the geometric series
Suppose f : A → R and g : B → R are differentiable on their domains and that f (A) ⊆ B. Then g ◦ f is differentiable on its domain
True
Let (fn) be a sequence of functions defined on [a, b] and suppose
(f’n) converges uniformly on [a, b]. Then (fn) converges uniformly on [a, b].
False,
Every Taylor series converges on all of R.
False
A power series that converges on (R, R] is continuous on(R, R]
True
The sequence of functions
1/(1 + x^n)
converges uniformly on
[0, 1]
False, this is only if |x| < 1
It is possible for a function to be discontinuous at countably
many points and still be integrable.
True, this is false for Reimann integrability, but true for Lebesgue integrability. It does not mean that every function is, only that it is possible.
Suppose f is bounded on [a, b]. If f is integrable on [a, c] for every c ∈ (a, b), then f is integrable on [a, b].
True
If f is a bounded function on [a, b], then the upper sum satisfies
U(f + g, P) = U(f, P) + U(g, P) for all partitions P
False on many levels, first of all g is not bounded and thus it can go to infinity, but additionally the supremums across the sub intervals may be in different places for both f and g, thus
U(f+g,P) <= U(f,P) + U(g,P)
Suppose g : [a, b] ! R is differentiable, g(a) > 0 and g(y) > g(x) for all x, y ∈ [a, b] with y > x. Then there is a c ∈ (a, b) with
(inverse derivitive = inverse average derivative)
True, g(y) > g(x) just gaurentees that g’(x) > 0, thus the inverse cannot be undefined, and the rest of this just fits the mean value theorem
How to show that functions are differentiable on a range?
If it is piecewise and contains polynomials, you only need to check that the derivatives from both sides are equal using the definition of a derivative
How to show that sum functions are differentiable and continuous
Use comparison test to other functions to show that the function must converge, thus it must converge uniformly
Let fn be differentiable functions defined on an interval A, and assume ∞
n=1 fn(x) converges uniformly to a limit
g(x) on A. If there exists a point x0 ∈ [a, b] where ∞
n=1 fn(x0) converges, then the series ∞
n=1 fn(x) converges uniformly to a
differentiable function f(x) satisfying f
(x) = g(x) on A
Cauchy Criterion for Uniform Convergence of Series
A series Σfn converges uniformly on A ⊆ R if and only if for every ϵ > 0
there exists an N ∈ N such that
Weierstrass M-Test
Basically term by term |fn(x)|<= M^n and ΣM^n converging means that the Σfn converges uniformly
Generalized Mean Value Theorem
If f and g are continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists a point c ∈ (a, b) where
[f(b) − f(a)]g’(c) = [g(b) − g(a)]f’(c)
Thus if g’(c) =/= 0
f’(c)/g’(c) = f(b) − f(a) / g(b) − g(a)
Lagrange’s Remainder Theorem
Basically, if
f^(N+1)(c)/(N + 1)! * x^N+1
converges to zero, then the Taylor series converges across the range
This is used to prove that Taylor series converges uniformly across a range by showing that En (the function above) never gets out of control
Darboux’s Theorem
Basically intermediate value theorem but for differentiability
If f is differentiable on an interval
[a, b], and if α satisfies f’(a) < α <f’(b) (or f’(a) > α > f’(b)), then there exists a point c ∈ (a, b) where f’(c) = α
Term-by-term Continuity Theorem
Continuous Limit Theorem but for series
Let fn be continuous functions defined on a set A ⊆ R, and assume Σfn converges uniformly on A to a function f. Then, f is continuous on A.
