Chapter 3 Applications of Differentiation

Question 0

Mean Value Theorem

Answer 0

1. f is continuous on [a,b]
2. f is differentiable on (a,b)
   If all conditions are true then there exists some number c in (a,b) such that f’(c) = ( f(b) - f(a) ) / ( b - a )

Question 1

Rolles Theorem

Answer 1

1. f is continuous on [a,b]
2. f is differentiable on (a,b)
3. f(a) = f(b)
   If all conditions are true then
   there is at least one number c in (a,b) such that f’(c) = 0

Question 2

Definition of Critical Number

Answer 2

Let f be defined at c. If f’(c) = 0 or f’ is undefined at c, then c is a critical number.

Question 3

First Derivative Test

Answer 3

c is a critical number.
1. If f’(x) changes from - to + then f(c) is a relative minimum
2. If f’(x) changes from + to - then f(c) is a relative maximum
(Use sign chart when solving)

Question 4

Points of Inflection

Answer 4

If ( c, f(c) ) is a point of inflection of the graph of f, then either f’‘(c) = 0 or f’’ is undefined at x = c. c is a critical number.

Question 5

Second Derivative Test

Answer 5

Let f be a function such that f’(c) = 0 and the second derivative of f exists on an open interval containing c.
1. If f’‘(c) > 0, then f(c) is a relative minimum (concave up)
2. If f’‘(c) < 0, then f(c) is a relative maximum (concave down)
If f’‘(c) = 0 or f’‘(c) is undefined use First Derivative Test