Understanding Increasing and Decreasing Functions for Optimal Profits

1. Chabot Mathematics §3.1 RelativeExtrema Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

2. 2.6 Review § • Any QUESTIONS About • §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: • 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).

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

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 thederivative, or Slope,function:

14. Example  Inc & Dec Profit • On Increasing intervals the Slope is POSTIVE or NonNegativeso in this case need • SolvingThisInEquality: • 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. Rel&Abs Max& Min AbsoluteMax RelativeMax RelativeMin AbsoluteMin

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

19. Critical Points GeoMetrically • Horizontal • Vertical (0.1695, 1.2597)

20. % 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 = [xminxmax]; zyh = [0 0]; zxv = [0 0]; zyv = [yminymax]; % % 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]) 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 = [xminxmax]; zyh = [0 0]; zxv = [0 0]; zyv = [yminymax]; % % 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([xminxmaxyminymax]),... 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])

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

23. Example  Critical Numbers • SOLUTION • Relative extremacan 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

24. Example  Critical Numbers • For theZeroCriticalPoint • Now need to consider critical points due to the derivative being undefined

25. 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 fexists at c

26. 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

27. 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

28. 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?

29. 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)

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

31. 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 + 1 is zero, so it results in NO critical values

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

33. Example  Evaluating Temperature • For dT/dt = 0 find: t ≈ −1.15 ort ≈ 0.954 • Because T is always continuous (check that the DeNomfcn, (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

34. Example  Evaluating Temperature • The DirectionDiagram • 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

35. Example  Evaluating Temperature • The DirectionDiagram • Now we test in the second region using t = 0: • The positive Slope indicates that T is INcreasing

36. Example  Evaluating Temperature • The DirectionDiagram • Now we test in the second region using t = 1: • Again the negative Slope indicates that T is DEcreasing

37. Example  Evaluating Temperature • The Completed SlopeDirection-Diagram: • 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.

38. Example  Evaluating Temperature • Graphically Relative Max (Stove OFF) Relative Min (Stove On)

39. MuPAD Plot Code

40. 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 → MachineTool Depreciation

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

42. Chabot Mathematics Appendix Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

50. P3.1-40 Hand Sketch