Lecture Notes

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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)

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):

¹⁾

f(x) - f(p