Chapter 5 - the derivative

Question 1

Definition - differentiable

Answer 1

Let I be a subset of R, c ∈ I and f: I →R
f is differentiable at c if the limit f’(c) = lim (x →c) f(x) - f(c) / x-c exists

Question 2

definition - local min and max

Answer 2

let I is a subset of R be an interval, f: I → R be a function and x0, δ>0 sunch that f(x) > F(x0) for all x ∈ I n (x0 - δ, x0 + δ)
f has a local extremum if it has either a local min or max at x0

Question 3

Fermats theorem

Answer 3

let a,b∈ R a<b
if f:(a,b)→ R is a differentiable function and f has a local extremum at x0, then f’(x0)=0

Question 4

rolles theorem

Answer 4

let a,b∈ R a<b
if f:[a,b]→ R is a continuous function such that f(a) = f(b)
if f is differentiable on (a,b)
then there exists x0 ∈(a,b) such that f’(x0) = 0

Question 5

mean value theorem

Answer 5

let a,b∈ R a<b
let f:[a,b]→ R be a continuous funtion such that f is differentiable on (a,b) then there exists a point x0∈ (a,b) such that f’(x0) = f(b)-f(a) / b-a