Analysis I (ch4-6)

Question 1

define differentiable

Answer 1

∃c∈ℝ ∀ε∈ℝ≥0 ∀y∈I{x}: |x-y|

Question 2

TRUE OR FALSE:

If a function is differentiable at x, then it is continuous at x.

Answer 2

TRUE

Question 3

How do you find the global extrema of a differentiable function f on a closed interval I?

Answer 3

(i) Find the stationary points f’(x0)=0
(ii) Evaluate f(x0)
(iii) evaluate f at the end points of I
(iv) select the points at which f is maximal/minimal

Question 4

State Rolle’s theorem

Answer 4

Let f∈C([a,b]) be differentiable on (a,b) with f(a)=f(b)=0. Then ∃c∈(a,b): f’(c)=0

Question 5

state the mean value theorem

Answer 5

Let f∈C([a,b]) be differentiable on (a,b). Then ∃c∈(a,b): f’(c)=(f(b)-f(a))/(b-a)

Question 6

state l’hopitals rule

Answer 6

Let f,g: [a,b]→ℝ be continuous, differentiable in (a,b), f(a)=g(a)=0, and ∀x∈(a,b): g’(x)≠0. If limx↘︎a(f’(x)/g’(x)) exists, then limx↘︎a(f(x)/g(x))=limx↘︎a(f’(x)/g’(x))

Question 7

Define an analytic function

Answer 7

Let I⊆ℝbe an open interval and f∈C∞(I). f is called analytic on I iff ∀a∈I ∃ε>0: B(a,ε)⊆I and ∀x∈B(a,ε): f(x)=the taylor series of f at a.

Question 8

what is a step function?

Answer 8

𝞅:[a,b]→ℝ is a step function iff there exists n∈ℕ and a=x0

Question 9

when is an integral Reimann integrable?

Answer 9

when the lower integral=upper integral. The set of all reimann integrable functions is denoted by R[a,b].

Question 10

What are piecewise continuous functions?

Are they Reimann integrable?

Answer 10

a function that is not all connected but defined for each x. Each bit of the function is well defined.
Yes is reimann integrable

Question 11

What is the first fundamental theorem of calculus?

Answer 11

The integral of f from a to b = F(b) - F(a)