Download Is it CRITICAL to go to EXTREMES?

Going to Extrema
Section 4.1
Is it CRITICAL to go
to EXTREMES?
•Should you do Minimum work or
Maximum work?
•Should you be Saddled with this
decision?
•Is homework ABSOLUTELY
necessary?
CRITICAL Numbers
If c is a critical number, then
•c must be an x-coordinate
•a function f must be defined at c
•f ’(c) = 0 or f is not differentiable at c
How
critical
am I?
Derivatives: Activity 1
You are given the graph of a function on a
grid. Assuming that the grid lines are
spaced 1 unit apart both horizontally and
vertically,
Sketch the graph of the derivative of each
function over the same interval.
4
m=-.8
m=-.7
m=-.5
m=-.3
-5
2
Derivatives:
Activity 1
m=.8
m=.7
m=.5
m=.3
m=-.2 m=.2
m=0
5
Graph 1
-2
Sketch the
graph of the
derivative of
each function
over the same
interval
2
-5
5
-2
6
Derivatives:
Activity 1
4
m=___
m=___
6
2
m=___
Graph 2
-5
4
5
m=___
m=___
-2
m=___
2
m=___
-4
m=___
m=___
-6
-5
5
m=___
-8
-2
m=___
-4
-6
6
4
m=___
m=___
2
6
m=___
-5
m=___
5
4
-2
m=___
2
-4
-5
5
-6
Derivatives:
Activity 1- Graph 3
-2
-4
-6
Derivatives and Graphs
6
6
On these graphs, where is the
derivative positive and how do
you know?
1st Derivative = slope of the
tangent line at a point! If the
slope is positive then the
derivative is positive.
4
4
2
2
2
-2
5
-4
-2
-6
What word could describe the
behavior of the function when
the derivative is positive?
Derivatives and Graphs
On these graphs, where is the
derivative negative and how do
you know?
1st Derivative = slope of the
tangent line at a point! If the
slope is negative then the
derivative is negative.
6
4
4
2
2
2
-2
-4
-2
What word could describe the
behavior of the function when
the derivative is negative?
Derivatives:
Activity 1 – Graph 4
8
6
6
4
m=___
4
m=___
m=___
2
2
m=___
-5
5
m=___
-5
5
-2
m=___
-2
m=___
-4
m=___
m=___
-4
-6
-6
m=___
-8
m=___
Derivatives: Activity 1 – Graph 5
6
6
4
4
2
2
-5
5
-5
5
-2
-2
-4
-4
-6
-6
6
Derivatives: Activity 1 – Graph 6
12
x
4
-5
slope
10
-4
2
8
-3
6
5
-2
4
-2
-1
2
-4
-6
0
5
Derivatives:
Activity 1 – Graph 7
8
6
6
4
4
2
2
-5
5
-5
x
slope
-5
-4
5
-2
-2
x
slope
1
-4
-3
2
-2
-1
0
3
4
5
-6
Are Derivatives telling you
more about graphs???
•
When the derivative is zero, describe
what could be happening on the graph
of the function at that point.
Are Derivatives telling you
more about graphs???
•
When the derivative is undefined (the
function is not differentiable at the
point), describe what could be
happening on the graph of the function
at that point.
Here is the graph of a
Derivative:
Local Max–
f ’changes from increasing to
decreasing and f ” is negative
4
Positive Derivatives =
2
Increasing Functions
-5
5
Negative Derivatives =
Decreasing Functions
-2
Local Min –
f ‘ changes from decreasing to
increasing and f ” is positive
Are Derivatives telling you
more about graphs???
When the derivative reaches a maximum or
minimum, describe what is happening to
the graph of the function at that point.
Here are the graphs of some
1
Derivative changes
from positive to
negative
Relative
Max!
-2
fx = x2-3ex
-1
-2
f'x = ex x2+2ex x+-3ex
-3
-4
-5
Relative
Min
2
Derivative
changes from
negative to
positive
Derivative
functions:
Where are the
extrema of the
original function
located? Label
each as a
maximum or
minimum.
Here are the graphs of some
 
   
Derivative
functions:
8
fx = ln
x
-2x2sgn
1+x2
6
f'x =
x
sgn
x2+1
+
x4+2x2+1
x
x2+1
x2+1
x
x2+1
4
Derivative changes
from positive to
negative
-5
2
Where are the
extrema of the
original function
Derivative changes
located? Label
from positive to
each as a
negative
maximum or
What’s happening
minimum.
at x = 0?
Relative
Max!
5
Relative
Max!
-2
-4
-6
-8
So, how do you find the
extrema
of a function?
1. Find the critical numbers of the function.
2. Find the absolute extrema if you have a
closed interval.
3. Find the relative/local extrema.
Find the critical numbers of the function.
a) Find the 1st derivative of the function
b) Find any values where the derivative does
not exist to find some critical numbers
c) Set the derivative equal to 0 and solve to
find the rest of the critical numbers
Extrema on a closed interval
Find the absolute extrema if its on a
closed interval.
a) Evaluate the function at each endpoint and
each critical number to identify absolute
extrema.
b) The point with the lowest y-coordinate is the
absolute minimum.
c) The point with the highest y-coordinate is the
absolute maximum.
Find the relative/local extrema.
a) Find the 2nd derivative of the function.
b) Evaluate the 2nd derivative at each critical number to
identify and label the relative/local extrema
c) If the 2nd derivative at a critical number is positive then
it is a relative minimum.
d) If the 2nd derivative at a critical number is negative, then
it is a relative maximum.
e) If the 2nd derivative at a critical number is zero, then it is
not relative extrema.
Find the extrema of
f(x) = 2x – 3x2/3 on [-1,3]
1. Find the 1st derivative of the function
2. Find any values where the derivative does not
exist and then set the derivative equal to 0 and
solve to find all of the critical numbers
3. Evaluate the function at each endpoint and each
critical number to identify absolute extrema
4. Find the 2nd derivative of the function.
5. Evaluate the 2nd derivative at each critical number
to identify and label the relative/local extrema
6. Graph the function and check your work!!!
Find the extrema of
f(x) = 2x – 3x2/3 on [-1,3]
Work Here…
4
2
fx = 2x-3x 3
2
-5
relative and
absolute max
at (0,0)
5
-2
relative min
 1,-1 
-1
f'x = -2x 3 +2
-4
absolute min at
(-1,-5)
-4
f''x =
-6
-8
2x 3
3
Find the extrema of
f(x) = 2sin(x) – cos(2x) on [0,2]
1. Find the 1st derivative of the function
2. Find any values where the derivative does not
exist and then set the derivative equal to 0 and
solve to find all of the critical numbers
3. Evaluate the function at each endpoint and each
critical number to identify absolute extrema
4. Find the 2nd derivative of the function.
5. Evaluate the 2nd derivative at each critical number
to identify and label the relative/local extrema
6. Graph the function and check your work!!!
Work Here…
Find the extrema of
f(x) = 2sin(x) – cos(2x) on [0,2]
6
4
relative and
absolute max

at
,3
2
fx = 2sinx-cos2x
f'x = 2cosx+2sin2x
 
f''x = -2sinx+4cos2x
2
5
 2,-1
(0,-1)
-2
-4
-6
relative and
absolute mins at
7
3
,and
6
2
11
3
,6
2




10
Homework- Due Wed.
• Re-read section 4.1 and finish
taking notes on it
• Do p. 209 # 1, 3, 6, 9, 12, 15, 18,
21, 24, 33, 42
Homework- Due Thurs.
Do p. 210 # 45, 51, 54, 61, 62, 67 - 74
So, is it CRITICAL to go
to EXTREMES?
• Should you do Maximum work or Minimum work?
Maximizing your Study Skills will often Minimize your work
and Minimize your mistakes!
• Should you be Saddled with this decision?
Yes – you are becoming an adult!
• Is homework ABSOLUTELY necessary?
Yes, ABSOLUTELY!