Increasing and decreasing functions ap calc sec 3.3
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Increasing and Decreasing Increasing and Decreasing Functions and the First Functions and the First Derivative Test Derivative Test Objectives: 1.Find the intervals on which a function is increasing or decreasing. 2.Use the First Derivative Test to classify extrema as either a maximum or a minimum. AP Calculus – Section 3.3
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Transcript of Increasing and decreasing functions ap calc sec 3.3
Increasing and Decreasing Increasing and Decreasing Functions and the First Derivative Functions and the First Derivative TestTest
Objectives:
1.Find the intervals on which a function is increasing or decreasing.
2.Use the First Derivative Test to classify extrema as either a maximum or a minimum.
AP Calculus – Section 3.3
Increasing and Decreasing Increasing and Decreasing FunctionsFunctions
• The derivative is related to the slope of a function
Increasing and Decreasing Increasing and Decreasing FunctionsFunctions
On an interval in which a function f is continuous and differentiable, a function is…increasing if f ‘(x) is positive on that interval, ( f ‘ (x) > 0 )decreasing if f ‘(x) is negative on that interval, and ( f ‘ (x) < 0 )constant if f ‘(x) = 0 on that interval.
Visual ExampleVisual Example
f ‘(x) < 0 on (-5,-2)f(x) is decreasing on (-5,-2)
f ‘(x) = 0 on (-2,1)f(x) is constant on (-2,1)
f ‘(x) > 0 on (1,3)f(x) is increasing on (1,3)
Finding Finding Increasing/Decreasing Increasing/Decreasing Intervals for a FunctionIntervals for a FunctionTo find the intervals on which a
function is increasing/decreasing:1.Find critical numbers. - These determine the boundaries of your intervals.2.Pick a random x-value in each interval. 3.Determine the sign of the derivative on that interval.
ExampleExampleFind the intervals on which the function is increasing and decreasing.
Critical numbers:
23
23)( xxxf
xxxf 33)(' 2 033 2 xx0)1(3 xx
}1,0{x
ExampleExampleTest an x-value in each interval.
f(x) is increasing on and .f(x) is decreasing on .
Interval
Test Value
f ‘(x)
)0,( )1,0( ),1(
1 21
2
6)1(' f43
21'
f 6)2(' f
)0,( ),1( )1,0(
PracticePracticeFind the intervals on which the function is increasing and decreasing.
Critical numbers:
xxxxf 93)( 23
963)(' 2 xxxf0963 2 xx
0)1)(3(3 xx
}1,3{x
0)32(3 2 xx
PracticePracticeTest an x-value in each interval.
f(x) is increasing on and .f(x) is decreasing on .
Interval
Test Value
f ‘(x)
)3,( )1,3( ),1(
4 0 2
15)4(' f 90' f 15)2(' f
)3,( ),1( )1,3(
963)(' 2 xxxf
The First Derivative Test
AP Calculus – Section 3.3
The First Derivative TestSummary
The point where the first derivative changes sign is an extrema.
The First Derivative TestIf c is a critical number of a function f,
then:If f ‘(c) changes from negative to positive
at c, then f(c) is a relative minimum.If f ‘(c) changes from positive to negative
at c, then f(c) is a relative maximum.If f ‘(c) does not change sign at c, then
f(c) is neither a relative minimum or maximum.
GREAT picture on page 181!
Visual of First Derivative Visual of First Derivative TestTest
Find all intervals of increase/decrease and all relative extrema. 108)( 2 xxxf
82)(' xxf082 x
4xTest: )4,(
28)5(2)5(' fdecreasing is f
Test: ),4( 8)0(' f
increasing is f
CONCLUSION: f is decreasing before -4 and increasing after -4; so f(-4) is a MINIMUM.
Critical Points:
