Fichas
Math 341 Definitions -- Chapter 5
Fichas del conjunto (13)
- 1Derivativeg'(c) = lim as x→c of [g(x) - g(c)] / [x - c]
- 2Differentiable at c:derivative as x → c exists
- 3Differentiable on AFor all c ∈ A, differentiable at c
- 4Theorem 5.2.3g'(c) exists ⇒ g is continuous at c
or
f is differentiable at a ⇒ f is continuous at a
(Same thing just two ways of writing it) - 5Theorem 5.2.4: Algebraic Differentiability TheoremLet f, g be differentiable at c.
Then
(i) (f + g)'(c) = f'(c) + g'(c)
(ii) (kf)'(c) = kf'(c)
(iii) (fg)'(c) = f'(c)g(c) + g'(c)f(c)
(iv) (f/g)'(c) = [f'(c)g(c) - g'(c)f(c)] / [g(c)^2] - 6Chain Ruleg: A→ B, f: B → R both differentiable
⇒
f(g(x))' = f'(g)g' - 7Interior Extremum TheoremAssume f: [a, b] → R is differentiable.
If ∋ c ∈ [a, b] s.t. f(c) = supf(x) or f(c) = inff(x) then
f'(c) = 0 - 8Rolle's TheoremLet
f: [a, b] → R be continuous and
f: (a, b) → R be differentiable.
If f(a) = f(b),
then
there exists c ∈ (a, b) s.t. f'(c) = 0 - 9Mean Value Theoremf: [a, b] → R is continuous.
f: (a, b) → R is differentiable.
⇒
there exists c ∈ (a, b) s.t.
f(b) - f(a) = f'(c)(b - a) - 10Generalized Mean Value Theorem[f(b) - f(a)] / [g(b) - g(a)] = [f'(c) / g'(c)]
- 11L'Hospital's Rule (0/0)f, g, continuous, differentiable on interval [a, b] w/ a ∈ [a, b]. If f(a) = g(a) = 0 and g'(a) ≠ 0,
⇒
lim as x → a f'(x) / g'(x) = lim f(x) / g(x) - 12L'Hospital's Rule (∞ / ∞)f, g, continuous, differentiable on interval [a, b] w/ a ∈ [a, b]. If lim g(x) → ∞ or -∞,
⇒
lim as x → a f'(x) / g'(x) = lim f(x) / g(x) - 13Darboux's TheoremIf f is differentiable on [a, b] and f'(a) < α < f'(b)
then
∃ c ∈ (a, b) s.t. f'(c) = α
Información
- Descripción
- Mathematics Fichas sobre Math 341 Definitions -- Chapter 5, creado por Jake Callahan el 18/04/2018.
- Creado por
- Jake Callahan · abr 2018
- Actualizado
- 18 abr 2018
- Visitas
- 0
- Valoración