2 - Derivatives

Question 1

Define differentiability at a point.

Answer 1

For f: (a, b)→R, We say that f is differentiable at c ∈ (a, b) if lim{h→0}[f(c + h) − f(c)]/h
exists.

Question 2

Define differentiability on an interval (a,b).

Answer 2

lim{h→0}[f(c + h) − f(c)]/h
exists for all c ∈ (a, b).

Question 3

What is the product rule?

Answer 3

(fg)′ = f’g + fg’

Question 4

What is the quotient rule?

Answer 4

(f/g)′ = (f’g - fg’) / (g^2)

Question 5

What is the chain rule?

Answer 5

(f◦g)′(c) = f′(g(c))g′(c)

Question 6

Define minimum, maximum, and extremum points.

Answer 6

For f: I→R and p, q ∈ I, then,
p is a maximum point of f in I if f(x) ≤ f(p), ∀x ∈ I.
q is a minimum point of f in I if f(x) ≥ f(q), ∀x ∈ I.
An extremum point is either a minimum or maximum point.

Question 7

If f: (a, b)→(α, β) is bijective and differentiable at c ∈
(a, b), f′(c) =/= 0 and f^{−1} is continuous at f(c), what is the conclusion?

Answer 7

f^{-1} is differentiable at c, and (f^{-1})’(f(c)) = 1 / f’(c)

Question 8

Define local maximum of a function f.

Answer 8

If there exists δ > 0 such that
f(x) ≤ f(p), ∀x ∈ (p − δ, p + δ) ∩ I, then p is a local maximum of f.

Question 9

Define local minimum of a function f.

Answer 9

If there exists δ > 0 such that
f(x) ≥ f(q), ∀x ∈ (p − δ, p + δ) ∩ I, then q is a local minimum of f.

Question 10

If c is a local minimum or maximum of f, what is f’(c)?

Answer 10

f’(c) = 0

Question 11

Define critical point of f

Answer 11

If for f: I→R
1. c is in the interior of I
2. f is differentiable at c
3. f′(c) = 0.
then c is a critical point of f

Question 12

State Rolle’s Theorem

Answer 12

Let f: [a, b]→R be continuous on [a, b] and differentiable on (a, b). If f(a) = f(b), then there exists c ∈ (a, b) such that f′(c) = 0.

Question 13

State the Mean Value Theorem.

Answer 13

Let f: [a, b]→R be continuous in [a, b] and differentiable on (a, b). Then, there exists c ∈ (a, b) such that
f′(c) = (f(b) − f(a)) / (b − a)

Question 14

State the Cauchy mean value theorem.

Answer 14

Let f, g: [a, b]→R be continuous on [a, b] and differentiable on (a, b). Then there exists c ∈ (a, b)
such that
(f(b) − f(a)) g′(c) = (g(b) − g(a)) f′(c).
In particular, if g′(x)=/= 0,
(f(b) − f(a)) / (g(b) − g(a)) = f’(c) / g’(c)

Question 15

State L’Hopital’s Rule and its 3 conditions for it to hold.

Answer 15

Let f, g : (a, b) → R be differentiable in (a, b). Assume that
(a) lim{x→a+}f(x) = 0 and lim{x→a+}g(x) = 0
(b) g′(x)=/= 0 for all x ∈ (a, b),
(c) lim{x→a+}[f′(x)/g′(x)] exists.
Then: lim{x→a+}[f(x)/g(x)]
= lim{x→a+}[f′(x)/g′(x)].

Question 16

Define what it means for a function f to be twice differentiable at a point c.

Answer 16

There exists a δ > 0 such that
1. (c−δ, c+δ) ⊂ (a, b)
2. f is differentiable in (c−δ, c+δ)
3. f′: (c−δ, c+δ)→R is differentiable at c

Question 17

Define what it means for a function f to be n-times differentiable at a point c.

Answer 17

There exists a δ > 0 such that
1. (c−δ, c+δ) ⊂ (a, b)
2. f is (n-1)-times differentiable in (c−δ, c+δ)
3. f^(n-1): (c−δ, c+δ)→R is differentiable at c

Question 18

State Taylor’s Theorem

Answer 18

Let f: (a, b)→R be (n+1)-times differentiable in (a, b). Let x_0 ∈ (a, b). Then for x ∈ (a, b),
f(x) = f(x_0)+f′(x_0) (x−x_0)+1/2 f′′(x_0) (x−x_0)^2+. . .+1/n!
f^(n)(x_0) (x−x_0)^n + R_n(x).
Where Rn(x) = 1/(n + 1)! f^(n+1)(c) (x − x_0)^(n+1)
and c ∈ (x_0, x) if x > x_0 and c ∈ (x, x_0) if x < x_0.

Question 19

What are the necessary second order conditions for extremum points?

Answer 19

1. p is a local maximum if f′(p) = 0 and f′′(p) ≤ 0
2. q is a local minimum if f′(q) = 0 and f′′(q) ≥ 0

Question 20

What are the necessary second order conditions for extremum points?

Answer 20

1. p is a local maximum if f′(p) = 0 and f′′(p) ≤ 0
2. q is a local minimum if f′(q) = 0 and f′′(q) ≥ 0