Test 3 Questions

Question 1

Let f: (-1, 1) -> R be given. If f’ exists on (-1,1), then f continuous at 0.

Answer 1

True

Question 2

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

Answer 2

False

Question 3

The radius of convergence of the power series for f (x) = 2x / (1 + 4x) is R = 1/4

Answer 3

True

Question 4

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

Answer 4

True

Question 5

The series sqrt(x^2 + 1/n) / n^2 converges uniformly on [0,2]

Answer 5

True

Question 6

Let f : [0, 1] -> R. If f is differentiable on [0,1], then f is integrable on [0,1]

Answer 6

True

Question 7

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.

Answer 7

False

Question 8

Let f : [0, 1] ⊂ R, and assume f has countably many discontinuities. Then f is integrable

Answer 8

False

Question 9

Let f : R -> R be differentiable. If f(0) = 0 and f’(x) < 1 for all x ∈ R, then f(1) < 1

Answer 9

True

Question 10

How do you show that a derivative and a second derivative exists at a point a?

Answer 10

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

Question 11

How do you show that a sequence of functions converges uniformly?

Answer 11

|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)

Question 12

How do you find the radius of convergence if you already have the series function?

Answer 12

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

Question 13

How do you find the power series of a function?

Answer 13

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

Question 14

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

Answer 14

True

Question 15

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].

Answer 15

False,

Question 16

Every Taylor series converges on all of R.

Answer 16

False

Question 17

A power series that converges on (R, R] is continuous on(R, R]

Answer 17

True

Question 18

The sequence of functions
1/(1 + x^n)
converges uniformly on
[0, 1]

Answer 18

False, this is only if |x| < 1

Question 19

It is possible for a function to be discontinuous at countably
many points and still be integrable.

Answer 19

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.

Question 20

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].

Answer 20

True

Question 21

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

Answer 21

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)

Question 22

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)

Answer 22

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

Question 23

How to show that functions are differentiable on a range?

Answer 23

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

Question 24

How to show that sum functions are differentiable and continuous

Answer 24

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

Question 25

Cauchy Criterion for Uniform Convergence of Series

Answer 25

A series Σfn converges uniformly on A ⊆ R if and only if for every ϵ > 0
there exists an N ∈ N such that

Question 26

Weierstrass M-Test

Answer 26

Basically term by term |fn(x)|<= M^n and ΣM^n converging means that the Σfn converges uniformly

Question 27

Generalized Mean Value Theorem

Answer 27

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)

Question 28

Lagrange’s Remainder Theorem

Answer 28

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

Question 29

Darboux’s Theorem

Answer 29

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

Question 30

Term-by-term Continuity Theorem

Answer 30

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.