Transcript Student Version - Parkway C-2

Objectives:
1. Be able to apply the first derivative test to find
relative extrema of a function
Critical Vocabulary:
Increasing, Decreasing, Constant
Warm Up: Determine (using calculus) where each function is
increasing, decreasing, and constant.
1. f(x) = 2x3 - 3x2 - 36x + 14
2. f(x) = (x2 - 4)2/3
x4 1
3. f ( x) 
x2
The previous lesson discussed the behavior of an interval
(increasing, decreasing, and constant).
This lesson is going to discuss the behavior of a critical
number.
If an interval changes from increasing to decreasing
(or decreasing to increasing), then what is happening at
the critical number that separates the intervals?
Interval
-∞ < x < 0
0<x<1
1<x<∞
Test Value
x = -5
x=½
x = 25
Sign of f’(x) f’(x) = (+)
Conclusion
Increasing
f’(x) = (-)
Decreasing
f’(x) = (+)
Increasing
Let c be a critical number of a function f that is continuous
on the open interval I containing c. If f is differentiable on the
interval, except possibly at c, then f(c) can be classified as
follows:
1. If f’(x) changes from negative to positive at c, then f(c) is
a _____________________ of f.
2. If f’(x) changes from positive to negative at c, then f(c) is
a _____________________ of f.
3. If f’(x) does not change signs at c, then f(c) is neither a
______________________ of f.
Summary:
1. Use increasing/decreasing test to find the intervals.
2. Use the First Derivative Test to determine if a
critical number is a relative max or min.
Directions: For the following exercises, find the critical numbers of f
(if any). Find the open intervals on which the function is increasing or
decreasing. Locate all the relative extrema. Use a graphing calculator
to confirm your results.
Example 1: f(x) = 2x3 - 3x2 - 36x + 14
Interval
Test Value
Increasing: __________
Sign f’(x)
Decreasing: __________
Conclusion
Constant: ___________
Relative Max: _____
Relative Min: _____
Directions: For the following exercises, find the critical numbers of f
(if any). Find the open intervals on which the function is increasing or
decreasing. Locate all the relative extrema. Use a graphing calculator
to confirm your results.
Example 2: f(x) = (x2 - 4)2/3
Interval
Test Value
Sign f’(x)
Conclusion
Increasing: __________
Decreasing: __________
Constant: ___________
Relative Max: _____
Relative Min: _____
Directions: For the following exercises, find the critical numbers of f
(if any). Find the open intervals on which the function is increasing or
decreasing. Locate all the relative extrema. Use a graphing calculator
to confirm your results.
x4 1
Example 3: f ( x) 
x2
Interval
Test Value
Sign f’(x)
Conclusion
Increasing: __________
Decreasing: __________
Constant: ___________
Relative Max: _____
Relative Min: _____
Page 334-335 #11-33 odd (skip 27), 55,
(MUST USE CALCULUS!!!!)