1. Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu
Chabot Mathematics
§3.1 Relative Extrema
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer

2. 2.6 Review § Any QUESTIONS About Any QUESTIONS About HomeWork
§2.6 → Implicit Differentiation
Any QUESTIONS About HomeWork
§2.6 → HW-12

3. §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

4. Increasing & Decreasing Values
A function f is INcreasing if whenever a<b, then:
INcreasing is Moving UP from Left→Right
A function f is DEcreasing if whenever a<b, then:
DEcreasing is Moving DOWN from Left→Right

5. Inc & Dec Values Graphically
DEcreasing
INcreasing

6. Inc & Dec with Derivative
If for every c on the interval [a,b]
That is, the Slope is POSITIVE
Then f is INcreasing on [a,b]
If for every c on the interval [a,b]
That is, the Slope is NEGATIVE
Then f is DEcreasing on [a,b]

7. Example  Inc & Dec
The function, y = f(x),is decreasing on [−2,3] and increasing on [3,8]

8. Example  Inc & Dec Profit
The default list price of 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

9. Example  Inc & Dec Profit
For this Situation 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:

10. Example  Inc & Dec Profit
And now cost as a function of x:
Then the Profit is the Revenue minus the Costs:

11. Example  Inc & Dec Profit
Now we turn to determining the intervals of increase and decrease.
The graph of the profit function is shown next on the interval [0,100] (where the price and quantity demanded are both non-negative).
Profit Curve

12. 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 exact intervals
Profit Curve

13. 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 derivative, or Slope, function:

14. Example  Inc & Dec Profit
On Increasing intervals the Slope is POSTIVE or NonNegative so in this case need
Solving This InEquality:
The profit function is DEcreasing on the interval [36,100]

15. 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

16. Peaks & Valleys Extrema is precise math terminology for Both of
The TOP of a Hill; that is, a PEAK
The Bottom of a Trough, That is a VALLEY
PEAK
PEAK
VALLEY
VALLEY

17. AbsoluteMax
Relative Max
Relative Min
Rel&Abs Max& Min
Absolute Min

18. Critical Points Let c be a value in the domain of f
Then c is a Critical Point if, and only if
HORIZONTAL slope at c
VERTICAL slope at c

19. Critical Points GeoMetrically
Horizontal
Vertical
(0.1695, )

20. 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([ ]); % 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',[ ], '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])
hold off
MATLAB Code

21. 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([ ]); % 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',[ ], '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])
MATLAB Code

22. Discuss the next 7 slides, slides 23→29
14Mar17 CatchUp
Discuss the next 7 slides, slides 23→29

23. Direction Diagrams (Slope Chart)
Use “Direction Diagrams” to determine for Critical Points: MAX, MIN, Neither
MAX form
MIN form
“Neither” forms
Direction of Graph
++++++
−−−−−−
df/dx Sign
Direction of Graph
−−−−−−
++++++
df/dx Sign
Direction of Graph
−−−−−−
−−−−−−
++++++
++++++
df/dx Sign

24. Example  Critical Numbers
Find all critical numbers and classify them as a relative maximum, relative minimum, or neither for The Function:

25. 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
Think Division by Zero

26. Example  Critical Numbers
For the Zero Critical Point
Now need to consider critical points due to the derivative being undefined

27. 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

28. Example  Critical Numbers
Use Direction Diagram to Classify the Critical Point at x = 1
Calculating the derivative/slope at a test point to the left of 1 (e.g. x = 0.5) find
Similarly for x>1, say 2:
→ f is DEcreasing
→ f is INcreasing

29. 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.
Relative Minimum

30. Example  Evaluating Temperature
The average temperature, in degrees Fahrenheit, in an ice cave t hours after midnight is modeled by:
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?
Note When 𝑡 2 −𝑡+1=0, then 𝑇 𝑡 is Undefined
D.N.E → Does Not Exist

31. 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)

32. Example  Evaluating Temperature
Taking dT/dt:
Using the Quotient Rule

33. Example  Evaluating Temperature
Expanding and Simplifying
When dT/dt → ∞
The denominator being zero causes the derivative to be undefined
however,(t2−t +1)2 is zero exactly when t2−t is zero, so it results in NO critical values
That its, the FUNCTION and the DERIVATIVE both SIMULTANEOUSLY →∞

34. No Point → No Critical Point
Notice the last Statement
ReCall that since the original Function is Undefined at 𝑡 2 −𝑡−1=0 then any point where 𝑡 2 −𝑡−1=0 can NOT be a Critical Pt.
From the previous slide 𝑑𝑇 𝑑𝑡→∞ when 𝑡 2 −𝑡−1 2 =0, an Undefined Pt
Thus No CritPoint if 𝑡 2 −𝑡−1 =0
however,(t2−t +1)2 is zero exactly when t2−t is zero, so it results in NO critical values

35. No Point → No Critical Point
Solving 𝑡 2 −𝑡+1=0 yields Undefined Points for the ORIGINAL Function, which then can NOT produce critical points at these values
Undefined For
Orginal Fcn 10𝑇+1 𝑡 2 −𝑡+−1 → 𝑡 2 −𝑡+1=0 → NO valid pts at these values
Thus NO Critical Points Exist at these Point as denom = 𝑡 2 −𝑡+1 2 =0
Encircled value are the SAME which are NOT Real
Complex Solns!
←Same→

36. Example  Evaluating Temperature
When dT/dt = 0 Numerator = 0
Thus Find:
Using the quadratic formula (or a computer algebra system such as MuPAD), find
Critical Points

37. 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 ≈
See Previous Slide

38. Example  Evaluating Temperature
The Direction Diagram
We test the derivative function in each of the three regions to determine if T is increasing or decreasing. Testing t = −2
The negative Slope indicates that T is Decreasing left of −1.15

39. Example  Evaluating Temperature
The Direction Diagram
Now we test in the second region using t = 0:
The positive Slope indicates that T is Increasing to the Right of −1.15

40. Example  Evaluating Temperature
The Direction Diagram
Now we test in the second region using t = 1:
Again the negative Slope indicates that T is DEcreasing to the Left of 0.954

41. Example  Evaluating Temperature
Coldest
Warmest
The Completed Slope Direction-Diagram:
Can 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.

42. Example  Evaluating Temperature
Graphically
Relative Max (Stove OFF)
Relative Min (Stove On)
Note that there are NO
Vertical Asymtotes

43. MuPAD Plot Code

44. 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

45. Critical (Mach) Number
All Done for Today
Critical (Mach) Number
Ernst Mach

46. Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu
Chabot Mathematics
Appendix
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer –

54. P Hand Sketch

55. P MuPAD Graph

56. WhiteBd Graphic for P3.1-52

65. P MuPAD

67. Example  Evaluating Temperature
Find where 𝑑𝑓 𝑑𝑡→∞ when
Solve by Quadratic Eqn as 𝑡 2 −𝑡+1 does not Factor.
Using MuPAD For Quadratic Equation