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Lecture 4/6
Differentiability (f: [a, b] → R)
1) f is differentiable at p ∈ [a, b] if:
lim ((f(x) - f(p))/(x - p)) as x -> p exists.
2) f'(p) = lim 1)/(x - p)) as x → p
If f is differentiable at p ∈ [a, b]:
3) f is continuous at p.
4) ∃ u(x) s.t. f(x) = f(p) + (x - p)f'(p) + (x - p)u(x)
- lim u(x) = 0 as x → p
Let g: [a, b] → R also be differentiable at p ∈ [a, b]:
5) (f + g)'(p) = f'(p) + g'(p)
6) (f⋅g)'(p) = f'(p)g(p) + f(p)g'(p)
7) (f/g)'(p) = (f'g - fg')/(g^2) if g(p) ≠ 0
Let f, g: R → R, f(a) = b, and f is differentiable at a, g is differentiable at b:
8) h(x) = g(f(x)) is differentiable at a; h'(a) = g'(f(a))·f'(a)
Let f, g: [a, b] → R:
9) p ∈ [a, b] is a local maximum (local minimum) of f if there is a δ > 0 s.t.
∀ x ∈ [a, b] ∩ Ball with center at p, we have f(x) ≤ f(p) (**f(x) ≥ f(p)**)
10) If p is a local maximum (or local minimum) of f, p ∈ (a, b), f'(p) exists:
f'(p) = 0
11) If f, g are differentiable on (a, b), ∃ c ∈ (a, b) s.t.
[f(b) - f(a)]g'(c) = [g(b) - g(a)]f'(c) * If g(x) = x, f(b) - f(a) = (b-a)f'(c)
Let f: [a, b] → R, f is continuous, f'(x) exists ∀ x ∈ (a, b), and
f(a) = f(b):
12) ∃ c ∈ (a, b) s.t. f'© = 0
Lecture 4/8
1) If f'(x) > 0, ∀ x ∈ (a, b):
f is a strictly increasing function on [a, b]
2) If f: R → R, f is continuous and differentiable, f'(x) is bounded:
f is uniformly continuous
3) If f: [a, b] → R, f is differentiable, f'(a) < f'(b):
∀ u ∈ R s.t. f'(a) < u < f'(b), ∃ c ∈ (a, b) s.t. f'(c) = u
4) If f, g: (a, b) → R, are differentiable, g(x) ≠ 0, g'(x) ≠ 0 over (a, b), and
a) lim (f'(x)/g'(x)) as x -> a = C ∈ R ∪ {+ inf, - inf}, and
b) lim f(x) as x -> a = lim g(x) as x -> a = 0
**or**
c) lim g(x) as x -> a = + inf, then
lim (f(x)/g(x)) as x -> a = C
Lecture 4/13
1) f^n(x) = (f^(n-1))'(x)
2)
