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