Clicker Question 1

1
Clicker Question 1

• What is an equation of the tangent line to the
  curve f (x ) x 2 at the point (1, 1)?
• A. y 2x
• B. y 2
• C. y 2x 2
• D. y 2x 1
• E. y 2x 1

2
The Derivative Function (2/13/09)

• Since we know how to compute (or at least
  estimate) the derivative (i.e., the instantaneous
  rate of change) of a given function f at any
  fixed point, we can then view the derivative
  itself as a function .
• We denote this function by f ' (we say f
  prime).
• Given an input x , it outputs the rate of change
  at x .

3
The Derivative Function

• Definition
• The derivative function can be
• Graphed, given a graph of f
• Estimated, given numerical info about f
• Computed algebraically, given a formula for f

4
Derivatives of very simple functions

• What is the derivative of any constant function f
  (x) c at any point x ?
• What is the derivative of any linear function f
  (x) m x b at any point x ?
• That is, again, you dont need calculus to
  understand the rate of change of linear functions.

5
Derivatives of Some Algebraic Functions

• Use the definition of the derivative to find the
  derivative of the function f (x) x 3
• Use the definition to find the derivative of f
  (x) 1 /x .
• Use the definition to find the derivative of f
  (x) ?x .

6
Clicker Question 2

• Using what we just learned, what is the
  instantaneous rate of change of the function f (x
  ) x 3 at the point (2, 8)?
• A. 3x 2
• B. 8
• C. 12
• D. 12x
• E. 12x 2

7
Differentiability

• A function f is said to be differentiable at x
  a if f '(a ) exists.
• How can f not be differentiable at a ??
• If the graph of f breaks apart at a (we say f
  is not continuous at a ).
• If the graph has a sharp bend at a .
• If the graph becomes vertical at a .

8
Clicker Question 3

• The function f (x ) x is not differentiable
• A. at x -1
• B. at x 0
• C. at x 1
• D. at all three of these points
• E. everywhere

9
Assignment

• We will have Lab 3 on Monday on rates of change
  of functions.
• For Wednesday, please read Section 2.8 and do
  Exercises 1, 3, 5, 9, 14, 16, 19, 21, 25, 27, 29,
  33a, 35, and 51.