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Chabot Mathematics
§3.1 Relative
Extrema
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
[email protected]
Chabot College Mathematics
1
Bruce Mayer, PE
[email protected] • MTH15_Lec-12_sec_3-1_Rel_Extrema_.pptx
Review §
2.6
Any QUESTIONS About
• §2.6 → Implicit Differentiation
Any QUESTIONS
About
HomeWork
• §2.6 →
HW-12
Chabot College Mathematics
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§3.1 Learning Goals
Discuss increasing and decreasing
functions
Define critical points and relative
extrema
Use the first
derivative test to
study relative
extrema and
sketch graphs
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Increasing & Decreasing Values
A function f is INcreasing if whenever
a<b, then:
f (a) < f (b)
• INcreasing is Moving UP from Left→Right
A function f is DEcreasing if whenever
a<b, then:
f ( a ) f (b )
• DEcreasing is Moving DOWN from Left→Right
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Inc & Dec Values Graphically
MTH15 • Bruce Mayer, PE
7
DEcreasing
6
5
4
y = f(x)
3
2
INcreasing
1
0
-1
-2
-3
-4
XY f cnGraph6x6BlueGreenBkGndTemplate1306.m
0
1
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2
3
4
5
x
6
7
8
9
10
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Inc & Dec with Derivative
df
If for every c on
f ' (c )
the interval [a,b]
dx
0
xc
• That is, the Slope is POSITIVE
Then f is INcreasing on [a,b]
If for every c on f ' ( c ) df
dx
the interval [a,b]
0
xc
• That is, the Slope is NEGATIVE
Then f is DEcreasing on [a,b]
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Example Inc & Dec
y f x x 3 4
2
The function, y = f(x),is decreasing on
[−2,3] and increasing on [3,8]
Chabot College Mathematics
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Example Inc & Dec Profit
The default list price of
p(x)
=10
a small bookstore’s
paperbacks Follows this Formula
• Where
– x ≡ The Estimated Sales Volume in No. Books
– p ≡ The Book Selling-Price in $/book
The bookstore buys paperbacks for $1
each, and has daily overhead of $50
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x
Example Inc & Dec Profit
For this Situation Find:
• Find the profit as a function of x
• intervals of increase and decrease for the
Profit Function
SOLUTION
Profit is the difference of revenue and
cost, so first determine the revenue as a
function of x:
R x x p x
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(
= x × 10 - x
)
10 x x
3/2
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Example Inc & Dec Profit
And now cost as a function of x:
C x variable
cost fixed cost 1 x 50
Then the Profit is the Revenue minus
the Costs: P x R x C x
= (10x - x 3/2 ) - ( x + 50)
= 9x - x 3/2 - 50.
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Example Inc & Dec Profit
Now we turn to determining the intervals
of increase and decrease.
3/2
The graph of the profit
y = 9x - x - 50
function is shown
next on the interval
[0,100] (where the
price and quantity
demanded are
both non-negative).
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Example Inc & Dec Profit
From the Plot Observe that The profit
function appears to be increasing until
some sales level below 40, and then
decreasing thereafter.
Although a graph is
informative, we turn
to calculus to
determine the
y = 9x - x 3/2 - 50
exact intervals
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Example Inc & Dec Profit
We know that if the derivative of a
function is POSITIVE on an open
interval, the function is INCREASING on
that interval. Similarly, if the derivative
is negative, the function is decreasing
So first compute the P ' x d 9 x x 3 / 2 50
dx
derivative, or Slope,
3 1/ 2
function:
9 x
2
Chabot College Mathematics
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Example Inc & Dec Profit
On Increasing intervals the Slope is
POSTIVE or NonNegative dP
3 1/ 2
9 x 0
so in this case need
dx
2
Solving
3 1/ 2
3 1/ 2
9 x 0
x
9
This
2
2
InEquality:
2
1/ 2
x 36
x 9
The profit
3
function is
DEcreasing on the interval [36,100]
Chabot College Mathematics
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Relative Extrema (Max & Min)
A relative maximum of a function f is
located at a value M such that
f(x) ≤ f(M) for all values of x on an
interval a<M<b
A relative minimum of a function f is
located at a value m such that
f(x) ≥ f(m) for all values of x on an
interval a<m<b
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Peaks & Valleys
Extrema is precise math terminology for
MTH15 • Bruce Mayer, PE
Both of
4
• The Bottom
of a Trough,
That is a
VALLEY
3
2
y = f(x)
• The TOP of a
Hill; that is,
a PEAK
PEAK
PEAK
VALLEY
1
0
-1
-2
-3
0
VALLEY
10
20
30
x
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40
50
MTH15 • Bruce Mayer, PE
4
3
y = f(x)
2
Rel&Abs Max& Min
Absolute
Max
Relative
Max
Relative
Min
1
0
-1
-2
-3
0
Absolute
Min
10
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20
30
x
40
50
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Critical Points
Let c be a value in the domain of f
Then c is a Critical Point If, and only if
df
dx
0
xc
HORIZONTAL
slope
at c
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OR
df
dx
xc
VERTICAL
slope
at c
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Critical Points GeoMetrically
Horizontal
Vertical
MTH15 • Zero Critical-Pt
MTH15 • Critical-Pt
1.3
20
1
0.9
16
14
y = f(x)
1.1
y=f(x)
18
(0.1695, 1.2597)
1.2
12
10
8
6
0.8
4
0.7
2
0.6
0.05
0.1
0.15
x
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0.2
0.25
0
0
0.5
1
1.5
2
x
Bruce Mayer, PE
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2.5
3
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MATLAB Code
% Bruce Mayer, PE
% MTH-15 • 07Jul13
% XYfcnGraph6x6BlueGreenBkGndTemplate1306.m
%
clear; clc;
% The Limits
xmin = 0; xmax = 0.27;
ymin =0; ymax = 1.3;
% The FUNCTION
x = linspace(xmin,xmax,1000); y1 = x.*(12-10*x-100*x.^2);
%
% The Max Condition
[yHi,I] = max(y1); xHi = x(I);
y2 = yHi*ones(1,length(x));
% The ZERO Lines
zxh = [xmin xmax]; zyh = [0 0]; zxv = [0 0]; zyv = [ymin ymax];
%
% the 6x6 Plot
axes; set(gca,'FontSize',12);
whitebg([0.8 1 1]); % Chg Plot BackGround to Blue-Green
plot(x,y1, 'LineWidth', 5),axis([.05 xmax .6 ymax]),...
grid, xlabel('\fontsize{14}x'),
ylabel('\fontsize{14}y=f(x)'),...
title(['\fontsize{16}MTH15 • Zero Critical-Pt',]),...
annotation('textbox',[.15 .05 .0 .1], 'FitBoxToText', 'on',
'EdgeColor', 'none', 'String', ' ','FontSize',7)
hold on
plot(x,y2, '-- m', xHi,yHi, 'd r', 'MarkerSize',
10,'MarkerFaceColor', 'r', 'LineWidth', 2)
set(gca,'XTick',[xmin:.05:xmax]); set(gca,'YTick',[ymin:.1:ymax])
Bruce Mayer, PE
Chabot
holdCollege
off Mathematics
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Bruce Mayer, PE
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MATLAB Code
% Bruce Mayer, PE
% MTH-15 • 23Jun13
% XYfcnGraph6x6BlueGreenBkGndTemplate1306.m
% ref:
%
% The Limits
xmin = 0; xmax = 3;
ymin = 0; ymax = 20;
% The FUNCTION
x = linspace(xmin,1.99,1000); y = -1./(x-2);
%
% The ZERO Lines
zxh = [xmin xmax]; zyh = [0 0]; zxv = [0 0]; zyv = [ymin ymax];
%
% the 6x6 Plot
axes; set(gca,'FontSize',12);
whitebg([0.8 1 1]); % Chg Plot BackGround to Blue-Green
plot(x,y, 'LineWidth', 4),axis([xmin xmax ymin ymax]),...
grid, xlabel('\fontsize{14}x'), ylabel('\fontsize{14}y =
f(x)'),...
title(['\fontsize{16}MTH15 • \infty Critical-Pt',]),...
annotation('textbox',[.51 .05 .0 .1], 'FitBoxToText', 'on',
'EdgeColor', 'none', 'String', ' ','FontSize',7)
hold on
plot([2 2], [ymin,ymax], '--m', 'LineWidth', 3)
set(gca,'XTick',[xmin:0.5:xmax]); set(gca,'YTick',[ymin:2:ymax])
Example Critical Numbers
Find all critical numbers and classify
them as a relative maximum, relative
minimum, or neither for The Function:
MTH15 • Bruce Mayer, PE
f x 4 x
40
2
x
30
2
20
y = f(x)
10
0
-10
-20
-30
-40
-10
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XY f cnGraph6x6BlueGreenBkGndTemplate1306.m
-8
-6
-4
-2
0
x
2
4
6
8
Bruce Mayer, PE
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Example Critical Numbers
SOLUTION
Relative extrema can only take place at
critical points (but not necessarily all
critical points end up being extrema!)
Thus we need to find the critical points
of f. In other words, values of x so that
df
dx
0
OR
df
UnDefined
dx
Think Division by Zero
Chabot College Mathematics
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Example Critical Numbers
For
the
Zero
Critical
Point
d
2
d
2
0 f '( x)
4
x
4
x
2
x
2
dx
x dx
0 = 4 + 2 × -2x -3
-4 = -4x -3
-3
1= x
x 1
Now need to consider critical points due
to the derivative being undefined
Chabot College Mathematics
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Example Critical Numbers
The Derivative Fcn, f’ = 4 − 4/x3 is
undefined when x = 0.
However, it is very important to note
that 0 cannot be the location of a critical
point, because f is also undefined at 0
In other words, no critical point of a
function can exist at c if no point on f
exists at c
Chabot College Mathematics
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Bruce Mayer, PE
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Example Critical Numbers
Use Direction Diagram to Classify the
Critical Point at x = 1
|
1
Calculating the derivative/slope at a test
point to the left of 1 (e.g. x = 0.5) find
3
f ' ( 0 . 5 ) 4 4 ( 0 . 5 ) 28 → f is DEcreasing
Similarly for x>1, say 2:
f ' (2) 4 4(2)
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3
3 .5
→ f is INcreasing
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Example Critical Numbers
From our Direction Diagram it appears
that f has a relative minimum at x = 1.
A graph of the function corroborates this
assessment.
f (x) = 4x +
2
x2
Relative Minimum
Chabot College Mathematics
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Example Evaluating Temperature
The average temperature, in degrees
Fahrenheit, in an ice cave t hours after
10 t 1
midnight is
T t 2
modeled by:
t t 1
Use the Model to Answer Questions:
• At what times was the temperature
INcreasing? DEcreasing?
• The cave occupants light a camp stove in
order to raise the temperature. At what
times is the stove turned on and then off?
Chabot College Mathematics
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Bruce Mayer, PE
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Example Evaluating Temperature
SOLUTION:
The Temperature “Changes Direction”
before & after a Max or Min (Extrema)
• Thus need to find the Critical Points which
give the Location of relative Extrema
• To find critical points of T, determine
values of t such that one these occurs
– dT/dt = 0 or
– dT/dt → ±∞ (undefined)
Chabot College Mathematics
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Bruce Mayer, PE
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Example Evaluating Temperature
Taking dT/dt:
d
10 t 1
dT
d
2
T
t
10
t
1
t
t
1
2
dt
t t 1
dt
dt
Using the Quotient Rule
dT
t
2
t 1
d
dt
dt
dT
t
2
t
2
t 1
dt
t
2
t 1
2
t 1 10 10 t 1 2 t 1
dt
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10 t 1 10 t 1
d
t
2
t 1
2
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2
Example Evaluating Temperature
Expanding and Simplifying
dT
10 t 10 t 10 20 t 8 t 1
2
2
t
dt
2
t 1
2
10 t 2 t 11
2
t
2
t 1
2
When dT/dt → ∞
dT
10 t 2 t 11
2
dt
t
2
t 1
2
t t 1 0
2
2
• The denominator being zero causes the
derivative to be undefined
– however,(t2−t +1)2 is zero exactly when t2−t + 1
is zero, so it results in NO critical values
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Example Evaluating Temperature
When dT/dt = 0
2
10 t 2 t 11 2
t t 1
0
2
2
t t 1
2
0 t t 1
2
Thus Find: 10 t 2 2 t 11 0
Using the quadratic
formula (or a
computer algebra
system such as
MuPAD), find
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2
10 t 2 t 11
2
Bruce Mayer, PE
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Example Evaluating Temperature
For dT/dt = 0 find: t ≈ −1.15 or t ≈ 0.954
Because T is always continuous (check
that the DeNom fcn, (t2−t +1)2 has no
real solutions) these are the only two
values at which T can change direction
Thus Construct a Direction Diagram
with Two BreakPoints:
• t ≈ −1.15
• t ≈ +0.954
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Bruce Mayer, PE
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Example Evaluating Temperature
The Direction
|
|
Diagram
1 . 15
0 . 954
We test the derivative function in each
of the three regions to determine if T is
increasing or decreasing. Testing t = −2
2
dT
10 2 2 ( 2 ) 11
25
2
2
dt t 2
49
2 ( 2 ) 1
The negative Slope indicates that T is
DEcreasing
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Bruce Mayer, PE
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Example Evaluating Temperature
The Direction
|
|
Diagram
1 . 15
0 . 954
Now we test in the second region using
2
t = 0: dT
10 0 2 ( 0 ) 11
dt
t0
0
2
(0) 1
2
11
The positive Slope indicates that T is
INcreasing
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Bruce Mayer, PE
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Example Evaluating Temperature
The Direction
|
|
Diagram
1 . 15
0 . 954
Now we test in the second region using
t = 1: dT
2
10 1 2 (1) 11
dt
t 1
1
2
(1) 1
2
1
Again the negative Slope indicates that
T is DEcreasing
Chabot College Mathematics
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Example Evaluating Temperature
The Completed Slope
|
|
Direction-Diagram:
1 . 15
0 . 954
We conclude that the function is
increasing on the approximate interval
(−1.15, 0.954) and decreasing on the
intervals (−∞, −1.15) & (0.954, +∞)
• It appears that the stove was lit around
10:51pm (1.15 hours before midnight) and
turned off around 12:57am (0.95 hours
after midnight), since these are the relative
extrema of the graph.
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•
Bruce Mayer, PE
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Example Evaluating Temperature
Graphically
Relative Max
(Stove OFF)
Relative Min
(Stove On)
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T t
10 t 1
t t 1
2
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MuPAD Plot Code
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WhiteBoard Work
Problems From §3.1
• P40 → Use Calculus to Sketch Graph
• Similar to P52 →
Sketch df/dx for
f(x) Graph at right
• P60 → Machine
Tool Depreciation
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All Done for Today
Critical
(Mach)
Number
Ernst Mach
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Chabot Mathematics
Appendix
r s r s r s
2
2
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
[email protected]
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Bruce Mayer, PE
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Bruce Mayer, PE
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Bruce Mayer, PE
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P3.1-40 Hand Sketch
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P3.1-40 MuPAD Graph
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WhiteBd Graphic
for P3.1-52
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Bruce Mayer, PE
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P3.1-60 MuPAD
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Bruce Mayer, PE
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Bruce Mayer, PE
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