Continuous Functions:

1
Section 9.2

• Continuous Functions
• Limits at Infinity

2
Continuous Functions

• Identify each function as continuous or
  discontinuous.

(a)
(d)
(c)
(b)
3
Pencil Test

• If you can draw the graph of a function without
  lifting your pencil, the function is continuous.

4
Requirements for Continuity

• The function f is continuous at x c if all of
  the following conditions are satisfied.
• f(c) exists
• exists
• f(c)
• If one or more of the conditions do not hold, we
  say the function is discontinuous at x c.

5
Discontinuous Functions

• Describe why each of the functions is
  discontinuous. Are there any points where these
  functions are continuous?

(3,2)
2
(a)
(b)
(c)
6
Polynomial Functions

• Every polynomial function is continuous for all
  real numbers.
• f(x) 3x2 2x - 5
• g(x) 4x 2
• h(x) x3
• The functions f, g, and h are each continuous for
  all real numbers.

7
Rational Functions

• Every rational function is continuous at all
  values of x except those that make the
  denominator 0.
• The function C is continuous at all values of x
  except x 7. Equivalently, we say C is
  discontinuous at x 7.

8
Example 1

• Determine the values of x, if any, for which the
  following function is discontinuous.

9
Example 2

• Determine the values of x, if any, for which the
  following function is discontinuous.

10
Limits at Infinity

• Find

11
Limits at Infinity

• If c is any constant, then

12
Example 3

• Find

f (x) 2
13
Example 4

• Find

y 1/x3
14
Example 5

• Find the limit, if it exists.

15
Example 5

• Find the limit, if it exists.
• We have to rewrite the function to apply the
  properties from the previous examples.

16
Example 5

• Find the limit, if it exists.
• Start by dividing the numerator and denominator
  by the largest power of x in the denominator.

17
Example 5

• Find the limit, if it exists.

18
Example 5

• Find the limit, if it exists.

19
Example 5

• Find the limit, if it exists.

20
Example 5

• Find the limit, if it exists.

21
Example 6

• Find the limit, if it exists.

22
Example 6

• Find the limit, if it exists.

23
Example 6

• Find the limit, if it exists.

24
Example 6

• Find the limit, if it exists.

25
Example 6

• Find the limit, if it exists.

26
Example 7

• Suppose that the average number of minutes M that
  it takes a new employee to assemble one unit of a
  product is given by
• where t is the number of days on the job. Is
  this function continuous
• for all values of t ?
• at t 14?
• for all t gt 0?
• What is the domain of this application?

27
Section 9.3

• Average and Instantaneous Rates of Change
• The Derivative

28
Average Rate of Change

• Bert traveled 200 miles in 4 hours.
• What was his average speed?

29
Average Rate of Change

• Bert traveled 200 miles in 4 hours.
• Did he necessarily drive 50 mph for the entire
  time?

30
Average Rate of Change

• The average rate of change of a function y f(x)
  from xa to xb is defined by

The average rate of change is the slope, m, of
this line.
y f(x)
(a, f(a))
(b, f(b))
a
b
31
Average Rate of Change

• Consider the function f(x) -(x-3)216.

32
Average Rate of Change

• Consider the function f(x) -(x-3)216.

Find the average rate of change on the interval
0, 5.
16
(5, 12)
(0, 7)
7
33
Average Rate of Change

• Consider the function f(x) -(x-3)216.

Find the average rate of change on the interval
0, 5.
By definition, the average rate of change on 0,
5 is
34
Secant Lines

• A line that intersects a curve at two points is
  called a secant line.

The slope of the secant line is the average rate
of change that we just found.
35
Tangent Lines

• A line that intersects a curve at one
• point and has the same slope as the
• curve at that point is called a tangent
• line.

How well does the slope of the secant line
approximate that of the tangent line?
36
Secant and Tangent Lines

• Suppose that we want to find the
• instantaneous rate of change when
• x5.

The instantaneous rate of change is the slope of
the tangent line.
37
Secant and Tangent Lines

• Find the instantaneous rate of change
• when x5.

The closer the secant line is to the tangent
line, the better the approximation.
38
Secant and Tangent Lines

• Find the instantaneous rate of change
• when x5.

Thus we want the second point to have an
x-coordinate very close to 5.
39
Secant and Tangent Lines

• Find the instantaneous rate of change
• when x5.

We can find the slope of the secant line passing
through the points (5, 12) and (5h, f(5h)),
where h is very close to zero.
40
Derivative

• If f is a function defined by y f(x), then the
  derivative of f(x) at any value x, denoted by f
  '(x), is
• if this limit exists. If f '(c) exists, we say
  that f is differentiable at c.
• The derivative is the slope of the tangent line.

41
Derivatives
See the procedure on page 652.

• Find the derivative of f(x) x2 - 3x.

42
Derivatives

• Find the instantaneous rate of change of
• f(x) x2 - 3x at any value and at the
• value x 2.

43
Derivatives

• Find the slope of the tangent to
• f(x) x2 - 3x at the value x 2.

44
Marginal Revenue

• Suppose that the total revenue function
• for a product is given by R R(x),
• where x is the number of units sold.
• Then the marginal revenue at x units is
• R(x) provided that the derivative exists.

45
Marginal Revenue

• Suppose the total revenue function for a
• textbook is R(x) 24x - 0.01x2 where x is
• the number of textbooks sold.
• What function gives the marginal revenue?
• What is the marginal revenue when 100
• units are sold, and what does it mean?

46
Marginal Revenue

• R(x) 24x - 0.01x2

47
Homework

• Section 9.2
• 1, 2, 3, 5, 6, 7, 10-14 all, 17, 19, 21,
• 31-35 all, 43, 45, 51, 56
• Section 9.3
• 1-9 odd, 15, 17, 23, 27 - 37 odd, 45,
• 47, 50