Derivative

Question 1

When is function f differentiable at c

Answer 1

Function f is differentiable at c when:

Lim h tends to 0 f(c+h)-f(c) /h exists

Question 2

Why can we define lim h tends to 0 f(c+h)-f(c) /h

Answer 2

We can define lim h tends to 0 f(c+h)-f(c)/h because g(h)=f(c+h) -f(c)/h is defined in a punctured neighbourhood of 0

Question 3

What are definitions of max and min points

Answer 3

Definitions of max point and min point are:
Max p E D such that for all x in D, f(x) <= f(p)
Min q E D s.t for all x in d f(x)>= f(q)

Question 4

What is definition of local max point p of function f

Answer 4

Definition of a local max point p of function f is:

Edelta s.t for all x in D mod(x-p) < delta implies f(x) <= f(p)

Question 5

What is definition of local min point q of function f

Answer 5

Definition of local min point q of function f is:

E delta s.t for all x in D mod(x-q) < delta implies f(x) >= f(q)

Question 6

If f is a function (a,b) and c E (a,b) and is a max/min point, what is f’(c)

Answer 6

If f is a function (a,b) and c E (a,b) and is a max/min point:
F’(c) =0
Doesn’t apply if f is a function [a,b] and max/min point is a.

Question 7

When is c a critical point of f: I to R where I = (a,b) or [a,b]

Answer 7

C is a critical point of f : I to R where I = (a,b) if:
C E (a,b)
F is differentiable at c
F’(c)=0

Question 8

What does Rolles theorem state

Answer 8

Rolles theorem states that:

If f is continuous on [a,b] and differentiable in (a,b). If f(a) = f(b) there is c E (a,b) such that f’(c)=0

Question 9

What does the mean value theorem state

Answer 9

Mean value theorem states that:
If f is continuous on [a,b] and differentiable in (a,b) then there is c E (a,b) such that
F’(c) = f(b)-f(a) /b-a

Question 10

What does de l’hopital’s theorem state

Answer 10

De l’hôpitals theorem states that!
If f(c)=g(c)=0 and f’(c), g’(c) exist and g’(c) doesn’t =0
Then lim tends to c f(x)/g(x) = f’(c)/g’(c)

Question 11

What does Cauchy mean value theorem state

Answer 11

Cauchy mean value theorem states that:
If f,g from [a,b] to R are continuous and differentiable on (a,b), then there exists c in (a,b) such that:
(F(b)-f(a))g’(c) = (g(b)-g(a))f’(c)

Question 12

What is f if it is differentiable at c

Answer 12

If f is differentiable at c, then it’s continuous at c