Youth Involvement in an Innovative Coconut Value Chain by Mwalimu Menza
Increasing and decreasing functions ap calc sec 3.3
1. Increasing and Decreasing Functions and
the First Derivative Test
AP Calculus – Section 3.3
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.
3. Increasing and Decreasing Functions
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.
4. Visual 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)
5. Finding Increasing/Decreasing
Intervals for a Function
To 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.
6. Example
Find the intervals on which the function
3
f ( x) = x − x is increasing and decreasing.
2
3
2
Critical numbers:
f ' ( x) = 3x 2 − 3 x
3x 2 − 3x = 0
3 x( x − 1) = 0
x = {0,1}
7. Example
Test an x-value in each interval.
Interval
Test Value
f ‘(x)
(−∞,0)
(0,1)
(1, ∞)
−1
1
2
2
f ' (−1) = 6
3
1
f ' = −
4
2
f ' ( 2) = 6
f(x) is increasing on (−∞,0) and (1, ∞)
.
f(x) is decreasing on (0,1).
8. Practice
Find the intervals on which the function
f ( x) = x 3 + 3 x 2 − 9 x is increasing and decreasing.
Critical numbers:
f ' ( x) = 3x 2 + 6 x − 9
3x 2 + 6 x − 9 = 0
3( x 2 + 2 x − 3) = 0
3( x + 3)( x − 1) = 0
x = {−3,1}
9. f ' ( x) = 3x 2 + 6 x − 9
Practice
Test an x-value in each interval.
Interval
(−∞,−3)
(−3,1)
(1, ∞)
Test Value
−4
0
2
f ‘(x)
f ' (−4) = 15 f ' ( 0) = −9
f ' (2) = 15
f(x) is increasing on (−∞ ,− 3) and (1, ∞)
.
f(x) is decreasing on (−3,1)
.
11. The First Derivative Test
Summary
The
point where the first derivative
changes sign is an extrema.
12. The First Derivative Test
If 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!
14. Find all intervals of increase/decrease and
all relative extrema.
f ( x) = x 2 + 8 x + 10
Critical Points:
Test:
(−∞,−4)
f ' ( x) = 2 x + 8
2x + 8 = 0
x = −4
f ' (−5) = 2(−5) + 8 = −2
f is decreasing
CONCLUSION:
Test:
(−4, ∞)
f ' ( 0) = 8
f is increasing
f is decreasing before -4 and
increasing after -4; so f(-4) is a MINIMUM.