7) Derivative Rules Flashcards
What is a minium, maximum and extremum
- f has a minimum at a ∈ A if f(a) ≤ f(x) for all x ∈ A.
- f has a maximum at a ∈ A if f(x) ≤ f(a) for all x ∈ A.
- f has an extremum at a if a is either a minimum or maximum
What is a local minimum
We say that f has a local minimum at k ∈ A if f(k) ≤ f(x) for all
x in some open neighbourhood of k
What is a local maximum
We say that f has a local maximum at ℓ ∈ A if f(x) ≤ f(ℓ) for all
x in some open neighbourhood of ℓ
Describe that proof that shows that if f has a local extremum at a and f is differentiable at a then f’(a) = 0.
What is Rolle’s Theorem
If f is differentiable on the open interval (a, b), continuous on the closed interval [a, b] and f(a) = f(b) then there exists a c : a < c < b such that f′(c) = 0.
Describe the proof for Rolle’s Theorem
What is the Mean Value Theorem
If the function f is differentiable on the open interval (a, b) and continuous on the closed interval [a, b] then there exists c : a < c < b such that
Describe the proof of the Mean Value Theorem
What are the applications of the Mean Value Theorem
- Bounding the magnitude of |f′(c)|
- Looking at the sign of F′(x)
What is the Increasing-Decreasing Theorem
Assume that f is differentiable on (a, b) and continuous on [a, b]
* a) If f′(x) > 0 for all x ∈ (a, b) then f is strictly increasing on [a, b]
* b) If f′(x) < 0 for all x ∈ (a, b) then f is strictly decreasing on [a, b]
* c) If f′(x) = 0 for all x ∈ (a, b) then f is constant on [a, b]
Describe the proof of the Increasing-Decreasing Theorem
What is Cauchy’s Mean Value Theorem
If f and g are differentiable on (a, b), continuous on [a, b] and g′(x)≠ 0 for all x in (a, b) then there exists a c : a < c < b such that
( Warning - We cannot apply the earlier Mean Value Theorem to f and g
separately since we might get different values for c for each functio)
Describe the proof of Cauchy’s Mean Value Theorem
