Splash Screen

1
Splash Screen
2
Lesson Menu
Five-Minute Check Then/Now New Vocabulary Key
Concept Increasing, Decreasing, and Constant
Functions Example 1 Analyze Increasing and
Decreasing Behavior Key Concept Relative and
Absolute Extrema Example 2 Estimate and Identify
Extrema of a Function Example 3 Real-World
Example Use a Graphing Calculator to Approximate
Extrema Example 4 Use Extrema for
Optimization Key Concept Average Rate of
Change Example 5 Find Average Rates of
Change Example 6 Real-World Example Find
Average Speed
3
5Minute Check 1
Determine whether the function y x 2 x 5 is
continuous at x 7.
A. yes B. no
4
5Minute Check 1
Determine whether the function y x 2 x 5 is
continuous at x 7.
A. yes B. no
5
5Minute Check 2
A. yes B. no
6
5Minute Check 2
A. yes B. no
7
5Minute Check 3
A. yes B. no
8
5Minute Check 3
A. yes B. no
9
5Minute Check 4
Describe the end behavior off (x) 6x 4 3x 3
17x 2 5x 12.
10
5Minute Check 4
Describe the end behavior off (x) 6x 4 3x 3
17x 2 5x 12.
11
5Minute Check 5
Determine between which consecutive integers the
real zeros of f (x) x 3 x 2 2x 5 are
located on the interval 4, 4.
A. 2 lt x lt 1 B. 3 lt x lt 2 C. 0 lt x lt 1 D.
4 lt x lt 3
12
5Minute Check 5
Determine between which consecutive integers the
real zeros of f (x) x 3 x 2 2x 5 are
located on the interval 4, 4.
A. 2 lt x lt 1 B. 3 lt x lt 2 C. 0 lt x lt 1 D.
4 lt x lt 3
13
Then/Now
You found function values. (Lesson 1-1)

• Determine intervals on which functions are
  increasing, constant, or decreasing, and
  determine maxima and minima of functions.
• Determine the average rate of change of a
  function.

14
Vocabulary

• increasing
• decreasing
• constant
• maximum
• minimum
• extrema
• average rate of change
• secant line

15
Key Concept 1
16
Example 1
Analyze Increasing and Decreasing Behavior
A. Use the graph of the function f (x) x 2 4
to estimate intervals to the nearest 0.5 unit on
which the function is increasing, decreasing, or
constant. Support the answer numerically.
17
Example 1
Analyze Increasing and Decreasing Behavior
Support Numerically Create a table using
x-values in each interval.
The table shows that as x increases from negative
values to 0, f (x) decreases as x increases from
0 to positive values, f (x) increases. This
supports the conjecture.
18
Example 1
Analyze Increasing and Decreasing Behavior
Answer
19
Example 1
Analyze Increasing and Decreasing Behavior
20
Example 1
Analyze Increasing and Decreasing Behavior
B. Use the graph of the function f (x) x 3 x
to estimate intervals to the nearest 0.5 unit on
which the function is increasing, decreasing, or
constant. Support the answer numerically.
21
Example 1
Analyze Increasing and Decreasing Behavior
Support Numerically Create a table using
x-values in each interval.
22
Example 1
Analyze Increasing and Decreasing Behavior
23
Example 1
Analyze Increasing and Decreasing Behavior
Answer
24
Example 1
Analyze Increasing and Decreasing Behavior
25
Example 1
Use the graph of the function f (x) 2x 2 3x
1 to estimate intervals to the nearest 0.5 unit
on which the function is increasing, decreasing,
or constant. Support the answer numerically.
A. f (x) is increasing on (8, 1) and (1,
8). B. f (x) is increasing on (8, 1) and
decreasing on (1, 8). C. f (x) is decreasing on
(8, 1) and increasing on (1, 8). D. f (x) is
decreasing on (8, 1) and decreasing on (1, 8).
26
Example 1
Use the graph of the function f (x) 2x 2 3x
1 to estimate intervals to the nearest 0.5 unit
on which the function is increasing, decreasing,
or constant. Support the answer numerically.
A. f (x) is increasing on (8, 1) and (1,
8). B. f (x) is increasing on (8, 1) and
decreasing on (1, 8). C. f (x) is decreasing on
(8, 1) and increasing on (1, 8). D. f (x) is
decreasing on (8, 1) and decreasing on (1, 8).
27
Key Concept 2
28
Example 2
Estimate and Identify Extrema of a Function
Estimate and classify the extrema to the nearest
0.5 unit for the graph of f (x). Support the
answers numerically.
29
Example 2
Estimate and Identify Extrema of a Function
30
Example 2
Estimate and Identify Extrema of a Function
Support Numerically Choose x-values in half unit
intervals on either side of the estimated x-value
for each extremum, as well as one very large and
one very small value for x.
Because f (1.5) gt f (1) and f (0.5) gt f (1),
there is a relative minimum in the interval
(1.5, 0.5) near 1. The approximate value of
this relative minimum is f (1) or 7.0.
31
Example 2
Estimate and Identify Extrema of a Function
Likewise, because f (1.5) lt f (2) and f (2.5) lt f
(2), there is a relative maximum in the interval
(1.5, 2.5) near 2. The approximate value of this
relative maximum is f (2) or 14.
f (100) lt f (2) and f (100) gt f (1), which
supports our conjecture that f has no absolute
extrema.
Answer
32
Example 2
Estimate and Identify Extrema of a Function
Likewise, because f (1.5) lt f (2) and f (2.5) lt f
(2), there is a relative maximum in the interval
(1.5, 2.5) near 2. The approximate value of this
relative maximum is f (2) or 14.
f (100) lt f (2) and f (100) gt f (1), which
supports our conjecture that f has no absolute
extrema.
Answer To the nearest 0.5 unit, there is a
relative minimum at x 1 and a relative maximum
at x 2. There are no absolute extrema.
33
Example 2
Estimate and classify the extrema to the nearest
0.5 unit for the graph of f (x). Support the
answers numerically.
A. There is a relative minimum of 2 at x 1 and
a relative maximum of 1 at x 0. There are no
absolute extrema. B. There is a relative maximum
of 2 at x 1 and a relative minimum of 1 at x
0. There are no absolute extrema. C. There is a
relative maximum of 2 at x 1 and no relative
minimum. There are no absolute extrema. D. There
is no relative maximum and there is a relative
minimum of 1 at x 0. There are no absolute
extrema.
34
Example 2
Estimate and classify the extrema to the nearest
0.5 unit for the graph of f (x). Support the
answers numerically.
A. There is a relative minimum of 2 at x 1 and
a relative maximum of 1 at x 0. There are no
absolute extrema. B. There is a relative maximum
of 2 at x 1 and a relative minimum of 1 at x
0. There are no absolute extrema. C. There is a
relative maximum of 2 at x 1 and no relative
minimum. There are no absolute extrema. D. There
is no relative maximum and there is a relative
minimum of 1 at x 0. There are no absolute
extrema.
35
Example 3
Use a Graphing Calculator to Approximate Extrema
GRAPHING CALCULATOR Approximate to the nearest
hundredth the relative or absolute extrema of f
(x) x 4 5x 2 2x 4. State the x-value(s)
where they occur.
f (x) x 4 5x 2 2x 4 Graph the function
and adjust the window as needed so that all of
the graphs behavior is visible.
36
Example 3
Use a Graphing Calculator to Approximate Extrema
From the graph of f, it appears that the function
has one relative minimum in the interval (2,
1), an absolute minimum in the interval (1, 2),
and one relative maximum in the interval (1, 0)
of the domain. The end behavior of the graph
suggests that this function has no absolute
extrema.
37
Example 3
Use a Graphing Calculator to Approximate Extrema
Using the minimum and maximum selection from the
CALC menu of your graphing calculator, you can
estimate that f(x) has a relative minimum of 0.80
at x 1.47, an absolute minimum of 5.51 at x
1.67, and a relative maximum of 4.20 at x
0.20.
38
Example 3
Use a Graphing Calculator to Approximate Extrema
Answer
39
Example 3
Use a Graphing Calculator to Approximate Extrema
Answer relative minimum (1.47, 0.80)
relative maximum (0.20, 4.20)absolute
minimum (1.67, 5.51)
40
Example 3
GRAPHING CALCULATOR Approximate to the nearest
hundredth the relative or absolute extrema of f
(x) x 3 2x 2 x 1. State the x-value(s)
where they occur.
A. relative minimum (0.22, 1.11)relative
maximum (1.55, 1.63) B. relative minimum
(1.55, 1.63) relative maximum (0.22,
1.11) C. relative minimum (0.22,
1.11)relative maximum none D. relative
minimum (0.22, 0) relative minimum (0.55,
0)relative maximum (1.55, 1.63)
41
Example 3
GRAPHING CALCULATOR Approximate to the nearest
hundredth the relative or absolute extrema of f
(x) x 3 2x 2 x 1. State the x-value(s)
where they occur.
A. relative minimum (0.22, 1.11)relative
maximum (1.55, 1.63) B. relative minimum
(1.55, 1.63) relative maximum (0.22,
1.11) C. relative minimum (0.22,
1.11)relative maximum none D. relative
minimum (0.22, 0) relative minimum (0.55,
0)relative maximum (1.55, 1.63)
42
Example 4
Use Extrema for Optimization
FUEL ECONOMY Advertisements for a new car claim
that a tank of gas will take a driver and three
passengers about 360 miles. After researching on
the Internet, you find the function for miles per
tank of gas for the car is f (x) ?0.025x 2
3.5x 240, where x is the speed in miles per
hour . What speed optimizes the distance the car
can travel on a tank of gas? How far will the car
travel at that optimum speed?
43
Example 4
Use Extrema for Optimization
We want to maximize the distance a car can travel
on a tank of gas. Graph the function f (x)
0.025x 2 3.5x 240 using a graphing
calculator. Then use the maximum selection from
the CALC menu to approximate the x-value that
will produce the greatest value for f (x).
44
Example 4
Use Extrema for Optimization
The graph has a maximum of 362.5 for x 70. So
the speed that optimizes the distance the car can
travel on a tank of gas is 70 miles per hour. The
distance the car travels at that speed is 362.5
miles.
Answer
45
Example 4
Use Extrema for Optimization
The graph has a maximum of 362.5 for x 70. So
the speed that optimizes the distance the car can
travel on a tank of gas is 70 miles per hour. The
distance the car travels at that speed is 362.5
miles.
Answer The optimal speed is about 70 miles per
hour. The car will travel 362.5 miles when
traveling at the optimum speed.
46
Example 4
VOLUME A square with side length x is cut from
each corner of a rectangle with dimensions 8
inches by 12 inches. Then the figure is folded to
form an open box, as shown in the diagram.
Determine the length and width of the box that
will allow the maximum volume.
A. 6.43 in. by 10.43 in. B. 4.86 in. by 8.86
in. C. 3 in. by 7 in. D. 1.57 in. by 67.6 in.
47
Example 4
VOLUME A square with side length x is cut from
each corner of a rectangle with dimensions 8
inches by 12 inches. Then the figure is folded to
form an open box, as shown in the diagram.
Determine the length and width of the box that
will allow the maximum volume.
A. 6.43 in. by 10.43 in. B. 4.86 in. by 8.86
in. C. 3 in. by 7 in. D. 1.57 in. by 67.6 in.
48
Key Concept3
49
Example 5
Find Average Rates of Change
A. Find the average rate of change of f (x)
2x 2 4x 6 on the interval 3, 1.
Use the Slope Formula to find the average rate of
change of f on the interval 3, 1.
Substitute 3 for x1 and 1 for x2.
Evaluate f(1) and f(3).
50
Example 5
Find Average Rates of Change
Simplify.
The average rate of change on the interval 3,
1 is 12. The graph of the secant line supports
this conclusion.
Answer
51
Example 5
Find Average Rates of Change
Simplify.
The average rate of change on the interval 3,
1 is 12. The graph of the secant line supports
this conclusion.
Answer 12
52
Example 5
Find Average Rates of Change
B. Find the average rate of change of f (x)
2x 2 4x 6 on the interval 2, 5.
Use the Slope Formula to find the average rate of
change of f on the interval 2, 5.
Substitute 2 for x1 and 5 for x2.
Evaluate f(5) and f(2).
53
Example 5
Find Average Rates of Change
Simplify.
The average rate of change on the interval 2, 5
is 10. The graph of the secant line supports
this conclusion.
Answer
54
Example 5
Find Average Rates of Change
Simplify.
The average rate of change on the interval 2, 5
is 10. The graph of the secant line supports
this conclusion.
Answer 10
55
Example 5
Find the average rate of change of f (x) 3x
3 2x 3 on the interval 2, 1.
56
Example 5
Find the average rate of change of f (x) 3x
3 2x 3 on the interval 2, 1.
57
Example 6
Find Average Speed
A. GRAVITY The formula for the distance traveled
by falling objects on the Moon is d (t) 2.7t 2,
where d (t) is the distance in feet and t is the
time in seconds. Find and interpret the average
speed of the object for the time interval of 1 to
2 seconds.
Substitute 1 for t1 and 2 for t2.
Evaluate d(2) and d(1).
Simplify.
58
Example 6
Find Average Speed
The average rate of change on the interval is 8.1
feet per second. Therefore, the average speed of
the object in this interval is 8.1 feet per
second.
Answer
59
Example 6
Find Average Speed
The average rate of change on the interval is 8.1
feet per second. Therefore, the average speed of
the object in this interval is 8.1 feet per
second.
Answer 8.1 feet per second
60
Example 6
Find Average Speed
B. GRAVITY The formula for the distance traveled
by falling objects on the Moon is d (t) 2.7t 2,
where d (t) is the distance in feet and t is the
time in seconds. Find and interpret the average
speed of the object for the time interval of 2 to
3 seconds.
Substitute 2 for t1 and 3 for t2.
Evaluate d(3) and d(2).
Simplify.
61
Example 6
Find Average Speed
The average rate of change on the interval is
13.5 feet per second. Therefore, the average
speed of the object in this interval is 13.5 feet
per second.
Answer
62
Example 6
Find Average Speed
The average rate of change on the interval is
13.5 feet per second. Therefore, the average
speed of the object in this interval is 13.5 feet
per second.
Answer 13.5 feet per second
63
Example 6
PHYSICS Suppose the height of an object dropped
from the roof of a 50 foot building is given by h
(t) 16t 2 50, where t is the time in seconds
after the object is thrown. Find and interpret
the average speed of the object for the time
interval 0.5 to 1 second.
A. 8 feet per second B. 12 feet per second C. 24
feet per second D. 132 feet per second
64
Example 6
PHYSICS Suppose the height of an object dropped
from the roof of a 50 foot building is given by h
(t) 16t 2 50, where t is the time in seconds
after the object is thrown. Find and interpret
the average speed of the object for the time
interval 0.5 to 1 second.
A. 8 feet per second B. 12 feet per second C. 24
feet per second D. 132 feet per second
65
End of the Lesson