Graphs of the slope function

3
Features of +x Graphs
The original function is…
f(x) is…
6
y is…
y
5
Stationary
4
3
2
Increasing
Decreasing
1
x
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
-1
-2
Increasing
-3
Stationary
-4
-5
-6
1
Investigate the tangents of +x3 Graphs
The slope function is…
f’(x) is…
6
dy/dx is…
y
5
4
3
Slope values
are decreasing
2
1
x
-6
-5
-4
-3
-2
-1
Point of
Inflection -1
=
slopes stop
-2
decreasing
and start-3
increasing
1
2
3
4
5
6
Slope values
are increasing
-4
-5
-6
2
Features of the Slope Function Graph
Reading the features of the graph of the slope function from the
original function
6
5
y
slope function = 0 (cuts x-axis)
dy/dx= 0 4
3
2
-6
-5
Slope values
are decreasing
→slope function
-4
-3
decreasing
1
-2
-1
Turning Point
of the slope
function:
where
slopes turn
from
decreasing
to increasing
= min
1
2
Slope values
are increasing
→slope function
3 increasing
4
x
5
6
-1
-2
-3
-4
dy/dx= 0
slope function = 0 (cuts x-axis)
-5
-6
Slope Function: U shaped (positive cubic graph will have positive derivative graph)
Minimum point at same x value as the point of inflection
3
Cuts x-axis at the x values of the turning points
The slope function is…
f’(x) is…
6
dy/dx is…
y
ORIGINAL FUNCTION
5
dy/dx= 0; slope function
=0
y = f(x)
4
Turning Point:
Decreasing to
increasing
= min pt
3
Slope values
are decreasing
2
1
-6
-5
-4
-3
-2
-1
1
2
3
4
5
x
6
Slope values
are increasing
-1
-2
-3
dy/dx= 0; slope function = 0
-4
-5
-6
6
y
SLOPE FUNCTION
5
y = f’(x)
4
3
Slope values
are decreasing
Slope values
are increasing
2
1
-6
-5
-4
-3
-2
dy/dx= 0; slope function = 0
-1
1
-1
2
3
4
dy/dx= 0; slope function = 0
5
x
6
x
-2
Turning Point:
-3
Decreasing
to
increasing
-4
= min pt
-5
-6
4
Also, we can read where the slope function is above and below the
+ + + + from
+ + + 0the
- - -original
- - - - - - -function
0+++++++
x-axis
6
y
5
4
3
Slopes are
positive
-6
-5
-4
-3
-2
2
1
Slopes are
negative
-1
1
Slopes are
positive
2
3
4
5
x
6
5
x
6
-1
-2
-3
-4
-5
-6
6
y
5
4
Slope
Function
above x-axis
-6
-5
-4
-3
-2
Slope
Function
above x-axis
3
2
1
-1
1
2
3
4
-1
-2
Slope
-3
Function
-4
-5
below
x-axis
-6
5
At what rate is the slope function changing? f’’(x) is… d2y/dx2 is...
6
How fast is
the rate of
decrease of
the slopes?
y
5
4
3
2
1
x
-6
-5
-4
-3
-2
-1
1
-1
-2
-3
-4
2
3
4
5
6
How fast is
the rate of
increase of
the slopes?
-5
-6
Finding the rate of change of the rate of change…. Finding the second derivative
6
A step further to investigate the tangents of the slope function.
Second Derivative Function is…
f’’(x) is… d2y/dx2 is...
6
y
ORIGNAL FUNCTION
5
dy/dx= 0; slope function
=0
y = f(x)
4
Turning Point:
Decreasing to
increasing
= min pt
3
Slope values
are decreasing
2
1
-6
-5
-4
-3
-2
-1
1
2
3
4
5
x
6
Slope values
are increasing
-1
-2
-3
dy/dx= 0; slope function = 0
-4
-5
-6
6
y
SLOPE FUNCTION
5
y = f’(x)
4
3
Slope values
are decreasing
Slope values
are increasing
2
1
-6
-5
-4
-3
-2
dy/dx= 0; slope function = 0
-1
1
-1
2
3
4
dy/dx= 0; slope function = 0
5
x
6
-2
Turning
Point:
-3
Decreasing to
-4
increasing
-5= min pt
-6
7
6
y
SLOPE FUNCTION
5
y = f’(x)
4
3
Slope values
are decreasing
Slope values
are increasing
2
1
-6
-5
-4
-3
-2
dy/dx= 0; slope function = 0
-1
1
-1
-2
Turning
Point:
-3
Decreasing to
-4
increasing
-5= min pt
-6
2
3
4
dy/dx= 0; slope function = 0
5
x
6
6
y
SLOPE FUNCTION
5
y = f’(x)
4
-6
Slope values
are increasing
→Second Derivative
Function
is -4
increasing
-5
-3
Slope values
are increasing
→Second Derivative
Function
3
4is increasing
5
3
2
1
-2
-1
1
2
x
6
-1
-2
-3 (d2y/dx2 = 0)
Slope=0
Second Derivative
Function =0
-4
(cuts
x-axis)
-5
-6
6
y
5
SECOND DERIVATIVE
FUNCTION
4
3
y = f’’(x)
2
1
-6
-5
-4
-3
-2
-1
1
-1
-2
-3
-4
-5
-6
2
3
4
5
x
6
Original Function, First Derivative Function, Second Derivative Function
6 y
5
4
3
2
1
y = f(x)
-6
-5
-4
-3
-2
-1
-1
-2
-3
-4
-5
-6
1
2
1
2
3
4
5
6 y
5
4
3
2
1
y = f’(x)
-6
-5
-4
-3
-2
-1
y = f’’(x)
-6
-5
-4

=
<

-3
-2
-1
-1
-2
-3
-4
-5
-6
6 y
5
4
3
2
1
-1
-2
-3
-4
-5
-6
x
6
x
3
4

5  6

=

=
>

1
2
3
4
5
x
6
10