Today in Calculus

1
Today in Calculus

• Go over homework
• Derivative Theorems
• Derivative Rules
• Homework

2
When f'(a) fails to exist

• Where the tangent does not exist
• points of discontinuity (function does not
  exist)
• at corners
• at cusps
• Also at vertical tangents (slope is undefined)

3
Theorems

• If f has a derivative at x a, then f is
  continuous at x a.
• Intermediate Value Theorem for derivatives If a
  b are any two points in an interval on which f
  is differentiable, then f ' takes on every value
  between f '(a) f '(b).

4

• Note A function can be continuous but not
  differentiable but if it is differentiable it has
  to be continuous

5
Derivative Rules

• Constant rule If f is a function with a
  constant value c, then f'(x)0
• Example f(x) 3

6
Derivative Rules

• Power rule
• f(x) f'(x)
• 3x
• 7x
• x2
• x3

7
Derivative Rules

• Sum and difference rules If u and v are
  differentiable functions of x, then
• f(x) 3x25x 12

8
Derivative Rules

• Product rule If u and v are differentiable
  functions of x, then
• f(x) (x21)(x33)

9
Derivative Rules

• Higher order derivatives
• f(x) 5x6 4x3 3x
• f'(x)
• f"(x)
• f?(x)
• f4(x)

10
Derivative Rules

• y x-4 x2

11
Examples
12
Examples
13
Homework

• Pg 111 5-10
• Pg 120 1-11