FINAL QUIZZZ

Question 1

continuity

Answer 1

A function f : A → R is continuous at a point c ∈ A if, for all ε>0, there exists δ>0 such that whenever |x-c|< δ it follows that |f(x) - f(c)| < ε.

Question 2

uniform continuity

Answer 2

A function f : A → R is uniformly continuous on A if for every ε>0, there exists δ>0 such that for all x,y ∈ A, |x-y|< δ implies |f(x) - f(y)| < ε.

Question 3

derivative

Answer 3

the derivative of g at c is defined by g’(c) = limx→c([g(x) -g(c)] / [x-c])

Question 4

Intermediate Value Theorem

Answer 4

Let f : [a,b] → R be continuous. If L is a real number satisfying f(a) < L < f(b) or f(a) > L > f(b), then there exists a point c ∈ (a,b) where f(c) = L

Question 5

Mean Value Theorem

Answer 5

If f : [a,b] → R is continuous on [a,b] and differentiable on (a,b), then there exists a point c ∈ (a,b) where

f’(c) = f(b) - f(a) / b - a