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TS: Explicitly assessing information and drawing
conclusions
Increasing & Decreasing Functions
Objectives
To examine the relationship between the slope of tangent lines and the behavior of a curve.
To determine when a function is increasing, decreasing, or neither.
To find the critical points of a function.
To determine the intervals on which a function is increasing or decreasing.
The Derivative
The derivative is used to find: Instantaneous Rate of Change Slopes of Tangent Lines
Tangent Lines
The line tangent to the curve of a function emulates the behavior of the curve near the point of tangency.
Tangent Lines
Behavior of a Curve
Always “read” the graph from left to right.
Behavior of a Curve
The curve increases until it reaches a summit.
Behavior of a Curve
The curve decreases until it reaches a valley.
Behavior of a Curve
The curve increases again.
Behavior of a Curve
Question: How can you determine where the curve is increasing or decreasing?
Answer: Study the tangent lines.
On an interval, the sign of the derivative of a function indicates whether that function is increasing or decreasing.
Tangent Lines
Tangent line is positively sloped – function is increasing.
Tangent Lines
Tangent line is positively sloped – function is still increasing.
Tangent Lines
Tangent line levels off at the summit.
Tangent Lines
Tangent line is negatively sloped – function is decreasing.
Tangent Lines
Tangent line is negatively sloped – function is still decreasing.
Tangent Lines
Tangent line levels off at the valley..
Tangent Lines
Tangent line is positively sloped – function is increasing again.
Positive Derivative Function Increasing
Negative Derivative Function Decreasing
The Derivative
If f ’ (x) > 0 , then f (x) is increasing.
if f ’ (x) < 0 , then f (x) is decreasing.
Behavior of a Curve
Question: What if the derivative equals 0?
Answer: The function is neither increasing nor decreasing.
Values that make the derivative of a function equal zero are candidates for the location of maxima and minima of the function.
Behavior of a Curve
Tangent line has a slope of 0 at the summit.
Behavior of a Curve
Tangent line has a slope of 0 at the valley.
Max & Min
Behavior of a Curve
Consider: What is the function doing at x = 0
and at x = 10 ?
3 2( ) 10 1f x x x x
2'( ) 3 2 10f x x x 2'(0) 3(0) 2(0) 10f
'(0) 10f
The function is decreasing through x = 0.
Behavior of a Curve
2'(10) 3(10) 2(10) 10f
'(10) 300 20 10f
'(10) 310f
The function is increasing through x = 10.
2'( ) 3 2 10f x x x
Critical Points
cusp pointThe derivativeis not defined.
Neither a max nor a min.
x1 x2 x3 x5x4 x
y
Critical Pointsy
x1 x2 x3 x5x4 x
Critical Points
Critical points are the places on a function where the derivative equals zero or is undefined.
Interesting things happen at critical points.
Critical Points
Steps to find critical points:1. Take the derivative.2. Set the derivative equal to zero and solve.3. Find values where the derivative is
undefined. Set the denominator of the derivative equal
to zero to find points where the derivative could be undefined.
Critical Points
Find the critical points of: 2( ) 4 2 2f x x x
'( ) 8 2f x x
0 8 2x
8 2x 1
4x
Critical Points
Find the critical points of: 3 2( ) 3 9 1g x x x x 2'( ) 3 6 9g x x x
20 3( 2 3)x x
0 3( 1)( 3)x x
1 0x 3 0x 1x 3x
Critical Points
Find the critical points of:1
3( )h x x2
31'( )
3h x x
23
1'( )
3h x
x
3 20 3 x
0x
Increasing & Decreasing
Find the intervals on which the function is increasing or decreasing: 2( ) 4 2 2f x x x
'( ) 8 2f x x
0 8 2x
8 2x 1
4x
Increasing & Decreasing
14
'( ) 8 2f x x
'( 1) 8( 1) 2f
'( 1) 8 2f
'( 1) 6f
'(0) 8(0) 2f
'(0) 2f
0
1x
'( )f x
0x
Decreasing: Increasing:14( , ) 1
4( , )
Increasing & Decreasing
Find the intervals on which the function is increasing or decreasing:
3 2( ) 3 9 1g x x x x 2'( ) 3 6 9g x x x
20 3( 2 3)x x
0 3( 1)( 3)x x
1 0x 3 0x 1x 3x
Increasing & Decreasing
'( )g x1 3
0 0
2x '( 2) 0g
'( ) 3( 1)( 3)g x x x
0x '(0) 0g
4x '(4) 0g
Decreasing:
Increasing:
( 1, 3)
( , 1) (3, )
Increasing & Decreasing
Find the intervals on which the function is increasing or decreasing:
3 2
1'( )
3h x
x
3 20 3 x
0x
13( )h x x
Increasing & Decreasing
0
'( 1) 0h '(1) 0h
UND.
1x
'( )h x
1x
Decreasing: Increasing:Never ( , 0) (0, )
3 2
1'( )
3h x
x
Conclusion
The derivative is used to find the slope of the tangent line.
The line tangent to the curve of a function emulates the behavior of the curve near the point of tangency.
On an interval, the sign of the derivative of a function indicates whether that function is increasing or decreasing.
Conclusion
f (x) is increasing if f ’ (x) > 0.
f (x) is decreasing if f ’ (x) < 0.
Values that make the derivative of a function equal zero are candidates for the location of maxima and minima of the function.
Conclusion
Critical points are the places on a graph where the derivative equals zero or is undefined.
First derivative Positive Increasing
First derivative Negative Decreasing
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