section 3.3 – increasing and decreasing functions and the first derivative test
TRANSCRIPT
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Section 3.3 – Increasing and Decreasing Functions and the
First Derivative Test
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How Derivatives Affect the Shape of the Graph
Many of the applications of calculus depend on our ability to deduce facts about a function f from in information concerning its derivatives. Since the derivative of f represents the slope of tangents lines, it tells us the direction in which the curve proceeds at each point. Thus, it should seem reasonable that the derivative of a function can reveal characteristics of the graph of the function.
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Increasing and Decreasing Functions
The function f is strictly increasing on an interval I if f (x1) < f (x2) whenever x1 < x2.
The function f is strictly Decreasing on an interval I if f (x1) > f (x2) whenever x1 < x2.
f(x)
x
D
AC
B IncreasingDecreasing
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How the Derivative is connected to Increasing/Decreasing Functions
When the function is increasing, what is the sign (+ or –) of the slopes of the tangent lines?
When the function is decreasing, what is the sign (+ or –) of the slopes of the tangent lines?
f(x)
x
D
AC
B
POSITIVE Slope
NEGATIVE Slope
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Test for Increasing and Decreasing Functions
Let f be differentiable on the open interval (a,b)
If f '(x) > 0 on (a,b), then f is strictly increasing on (a,b).
If f '(x) < 0 on (a,b), then f is strictly decreasing on (a,b).
If f '(x) = 0 on (a,b), then f is constant on (a,b).
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Procedure for Finding Intervals on which a Function is increasing or Decreasing
If f is a continuous function on an open interval (a,b). To find the open intervals on which f is increasing or decreasing:
1. Find the critical numbers of f in (a,b) AND all values (a,b) of x in that make the derivative undefined.
2. Make a sign chart: The critical numbers and x-values that make the derivative undefined divide the x-axis into intervals. Test the sign (+ or –) of the derivative inside each of these intervals.
3. If f '(x) > 0 in an interval, then f is increasing in that same interval. If f '(x) < 0 in an interval, then f is decreasing in that same interval.
4. State your conclusion(s) with a “because” statement using the sign chart.A sign chart does NOT stand on its own.
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Example 1Use the graph of f '(x) below to determine when f is
increasing and decreasing.
(-∞,-1)Increasing: Decreasing: (-1,3)U (3,∞)
f is increasing when the
derivative is positive.
f ' (x)
x
f is decreasing when the
derivative is negative.
f is increasing when the
derivative is positive.
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White Board Challenge
The graph of f is shown below. Sketch a graph of the derivative of f.
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White Board Challenge
Find the critical numbers of:
3 22 9 12f x x x x
2 1x or x
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Example 2Find where the function is
increasing and where it is decreasing. 4 3 23 4 12 5f x x x x
4 3 2' 3 4 12 5ddxf x x x x
Find the derivative.
3 2' 12 12 24f x x x x
Find the critical numbers/where the derivative is undefined
3 20 12 12 24x x x Find where the derivative is 0 or undefined
20 12 2x x x
0 12 2 1x x x 0, 2, 1x
-1 20
Find the sign of the derivative on each interval.
2x 0.5x 1x 3x ' 2 96f
' 0.5 7.5f ' 1 24f
' 3 144f
Answer the question
Domain of f:All Reals
'f x
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Example 2: Answer
The function is increasing on (-1,0)U(2,∞) because the first
derivative is positive on this interval.
The function is decreasing on (-∞,-1)U(0,2) because the first
derivative is negative on this interval.
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Example 3Use the graph of f (x) below to determine when f is
increasing and decreasing.
(-∞,-3)Increasing: Decreasing: (-3,∞)
f is increasing when the
function’s outputs are getting larger
as the input increases.
f (x)
x
f is decreasing when the
function’s outputs are getting
smaller as the input increases.
Notice how the function changes from increasing to decreasing at x=-3. But since -3 is not
in the domain of the function, it is not a critical point. Thus, critical points are
not the only points to include in sign
charts.
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Example 4Find where the function is increasing and where
it is decreasing. 1
1xf x
2
1 1 1 1
1'
d ddx dxx x
xf x
Find the derivative.
21
1'
xf x
Find the critical numbers/where the derivative is undefinedFind where the derivative is 0 or undefined
-1
Find the sign of the derivative on each interval.
2x 0x ' 2 1f ' 0 1f
Answer the question
Domain of f : All Reals except -1
The function does not have any critical points: the derivative is never equal to 0 and the
derivative is only undefined at a point not in the functions domain (x=-1).
Even though -1 is not a critical point, it can still be a point where a function changes from increasing to decreasing. ALWAYS include every x value that
makes the derivative undefined on a sign chart (even if its not a critical point).
'f x
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Example 4: Answer
The function is decreasing on (-∞,-1)U(-1,∞) because the
first derivative is negative on this interval.
Make sure not to include -1 in the interval because it is not in the domain of the function.
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White Board Challenge
Find the maximum and minimum values attained by the given function on the indicated closed interval:
4 ; 1,4xf x x
max : 5 @ 1,4
min : 4 @ 2
x
x
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Critical Values and Relative Extrema
Remember that if a function has a relative minimum or a maximum at c, then c must be a critical number of the function. Unfortunately not every critical number results in a relative extrema.
A new calculus method is needed to determine whether relative extrema exist at a critical point and if it is a maximum or minimum.
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How the Derivative is connected to Relative Minimum and Maximum
When a critical point is a relative maximum, what are the characteristics of the function?
When a critical point is a relative minimum, what are the characteristics of the function?
f(x)
x
D
AC
B
The function changes from increasing to decreasing
The function changes from decreasing to increasing
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The First Derivative TestSuppose that c is a critical number of a continuous function
f(x).
(a) If f '(x) changes from positive to negative at c, then f(x) has a relative maximum at c.
f(x)
xc
f '(x) < 0 f '(x) > 0
Relative Maximum
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The First Derivative TestSuppose that c is a critical number of a continuous function
f(x).
(b) If f '(x) changes from negative to positive at c, then f(x) has a relative minimum at c.
f(x)
xc
f '(x) > 0 f '(x) < 0
Relative Minimum
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The First Derivative TestSuppose that c is a critical number of a continuous function
f(x).
(c) If f '(x) does not change sign at c (that is f '(x) is positive on both sides of c or negative on both sides), then f(x) has no relative maximum or minimum at c.
f(x)
x
f '(x) < 0 f '(x) > 0 No Relative
Maximum or
Minimumf '(x) > 0
f(x)
xc c
f '(x) < 0
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Example 1Use the graph of f '(x) below to determine where f has a
relative minimum or maximum.
@ -1Relative Maximum: Relative Minimum: @ 3
Find the Critical Numbersf ' (x)
x
-1 32x 0x 4x
Make a sign chart and Find the sign of the derivative on each
interval.
Apply the First Derivative Test.
'f x
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Example 2Find where the function on 0≤x≤2π has
relative extrema. 2sinf x x x
' 2sinddxf x x x
Find the derivative.
' 1 2cosf x x
Find the critical numbers
0 1 2cos x Find where the derivative is 0 or undefined
1 2cos x 12 cos x
2 43 3,x
2π/3 4π/3
Find the sign of the derivative on each interval.
1x 3x 5x ' 1 2.08f
' 3 0.98f ' 5 1.57f
Answer the question
Domain of f:0≤x≤2π
0 2π 'f x 2 2
3 33f Find the value of the function:
4 43 3 3f
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Example 2: Answer
The function has a relative maximum of 3.826 at x = 2π/3 because the first derivative changes from positive to
negative values at this point.
The function has a relative minimum 2.457 at x = 4π/3 because the first
derivative changes from negative to positive values at this point.
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Example 3
Find the relative extrema values of . 2 31 3 3f x x x
2 31 3' 3ddxf x x x
Find the derivative.
2 3 1 32 3 1 31 23 3' 3 3f x x x x x
Find the critical numbers
2 31 3
2 3 1 3
3 23 3 3
0 x xx x
Find where the derivative is 0 or undefined
2 31 3
2 3 1 3
3 23 3 3
x xx x
3 3 6x x
1x
Find the sign of the derivative on each interval.
2 31 3
2 3 1 3
3 23 3 3
' x xx x
f x
3 9 6x x 9 9x
The derivative
is undefined at x=-3,0
-3 0-14x 2x 0.5x 1x ' 4 1.19f
' 2 0.63f ' 0.5 0.58f
' 1 1.26f
NOTE: 0 is not a relative extrema since the derivative
does not change sign.
Domain of f:
All Reals
Answer the question
'f x
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Example 3: Answer
The function has a relative maximum of 0 at x = -3 because the first derivative changes
from positive to negative values at this point.
The function has a relative minimum of -1.587 at x = -1 because the first
derivative changes from negative to positive values at this point.
3 0f First find the value of the function:
2 31 2f Now answer the question:
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White Board Challenge
Find the intervals on which the function below is increasing or decreasing.
31f x x x
14
14
increasing :
decreasing :
x
x
BONUS: How many critical numbers are
there?