1. BMayer@ChabotCollege.edu MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics §3.2 Concavity & Inflection

2. BMayer@ChabotCollege.edu MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx 2 Bruce Mayer, PE Chabot College Mathematics Review §  Any QUESTIONS About §3.1 → Relative Extrema  Any QUESTIONS About HomeWork §3.1 → HW-13 3.1

3. BMayer@ChabotCollege.edu MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx 3 Bruce Mayer, PE Chabot College Mathematics §3.2 Learning Goals  Introduce Concavity (a.k.a. Curvature)  Use the sign of the second derivative to find intervals of concavity  Locate and examine inflection points  Apply the second derivatives test for relative extrema

4. BMayer@ChabotCollege.edu MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx 4 Bruce Mayer, PE Chabot College Mathematics ConCavity Described  Concavity quantifies the Slope-Value Trend (Sign & Magnitude) of a fcn when moving Left→Right on the fcn Graph m≈+2.2 m≈0 m≈−1.4 m≈−4.4 m≈−1.4 m≈+2.2

5. BMayer@ChabotCollege.edu MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx 5 Bruce Mayer, PE Chabot College Mathematics MATLAB Code MATLAB Code % Bruce Mayer, PE % MTH-15 11Jul133 % XYfcnGraph6x6BlueGreenBkGndTemplate1306.m % % The data blue =[2.2 0 -1.4 -4.4] red = [-4.4 -1.4 0 2.2] % % the 6x6 Plot axes; set(gca,'FontSize',12); subplot(1,2,1) bar(blue, 'b'), grid, xlabel('\fontsize{14}Position, x'), ylabel('\fontsize{14}m = df/dx'),... title(['\fontsize{16}MTH15 BLUE',]), axis([0 5 - 5,3]) subplot(1,2,2) bar(red, 'r'), grid, xlabel('\fontsize{14}Position, x'), axis([0 5 -5,3]),... title(['\fontsize{16}MTH15 RED',]) set(get(gco,'BaseLine'),'LineWidth',4,'LineStyle',':')

6. BMayer@ChabotCollege.edu MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx 6 Bruce Mayer, PE Chabot College Mathematics ConCavity Defined  A differentiable function f on a < x < b is said to be: … concave DOWN (↓) if df/dx is DEcreasing on the interval …concave up if df/dx is INcreasing on the interval.

7. BMayer@ChabotCollege.edu MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx 7 Bruce Mayer, PE Chabot College Mathematics Example  Graphical Concavity  Consider the function f given in the graph and defined on the interval (−4,4).  Approximate all intervals on which the function is INcreasing, DEcreasing, concave up, or concave down

8. BMayer@ChabotCollege.edu MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx 8 Bruce Mayer, PE Chabot College Mathematics Example  Graphical Concavity  SOLUTION  Because we have NO equation for the function, we need to use our best judgment: around where the graph changes directions (increasing/decreasing) where the derivative of the graph changes directions (concave up or down).

9. BMayer@ChabotCollege.edu MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx 9 Bruce Mayer, PE Chabot College Mathematics Example  Graphical Concavity  To determine where the function is INcreasing, we look for the graph to “Rise to the Right (RR)” Rising

10. BMayer@ChabotCollege.edu MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx 10 Bruce Mayer, PE Chabot College Mathematics Example  Graphical Concavity  Similarly, the function is DEcreasing where the graph “Falls to the Right (FR)”: Falling

11. BMayer@ChabotCollege.edu MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx 11 Bruce Mayer, PE Chabot College Mathematics Example  Graphical Concavity  Conclude that f is increasing on the interval (0,4) and decreasing on the interval (−4,0)  Now Examine Concavity. Falling to Rt Rising to Rt

12. BMayer@ChabotCollege.edu MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx 12 Bruce Mayer, PE Chabot College Mathematics Example  Graphical Concavity  A function is concave UP wherever its derivative is INcreasing. Visually, we look for where the graph is “curved upward”, or “Bowl-Shaped” Similarly, A function is concave DOWN wherever its derivative is DEcreasing. Visually, we look for where the graph is “curved downward”, or “Dome-Shaped”

13. BMayer@ChabotCollege.edu MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx 13 Bruce Mayer, PE Chabot College Mathematics Example  Graphical Concavity  The graph is “curved UPward” for values of x near zero, and might guess the curvature to be positive between −1 & 1 f is ConCave UP

14. BMayer@ChabotCollege.edu MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx 14 Bruce Mayer, PE Chabot College Mathematics Example  Graphical Concavity  The graph is “curved DOWNward” for values of x on the outer edges of the domain. f is ConCave DOWN

15. BMayer@ChabotCollege.edu MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx 15 Bruce Mayer, PE Chabot College Mathematics Example  Graphical Concavity  Thus the function is concave UP approximately on the interval (−1,1) and concave DOWN on the intervals (−4, −1) & (1,4) f is ConCave UP f is ConCave DOWN

16. BMayer@ChabotCollege.edu MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx 16 Bruce Mayer, PE Chabot College Mathematics Inflection Point Defined  A function has an inflection point at x=a if f is continuous and the CONCAVITY of f CHANGES at Pt-a ConCave DOWN ConCave UP Inflection Point

17. BMayer@ChabotCollege.edu MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx 17 Bruce Mayer, PE Chabot College Mathematics MATLAB Code MATLAB Code % Bruce Mayer, PE % MTH-15 10Jul13 % XYfcnGraph6x6BlueGreenBkGndTemplate1306.m % % The Limits xmin = -2; xmax = 9; ymin =-50; ymax = 50; % The FUNCTION x = linspace(xmin,xmax,1000); y =(x-4).^3/4 + (x+5).^2/7; yOf4 = (4-4).^3/4 + (4+5).^2/7 % % 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', 5),axis([xmin xmax ymin ymax]),... grid, xlabel('\fontsize{14}x'), ylabel('\fontsize{14}y = f(x)'),... title(['\fontsize{16}MTH15 Inflection Point',]) hold on plot(4, yOf4, 'd r', 'MarkerSize', 9,'MarkerFaceColor', 'r', 'LineWidth', 2) plot(zxv,zyv, 'k', zxh,zyh, 'k', 'LineWidth', 2) set(gca,'XTick',[xmin:1:xmax]); set(gca,'YTick',[ymin:10:ymax]) hold off

18. BMayer@ChabotCollege.edu MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx 18 Bruce Mayer, PE Chabot College Mathematics Example  Inflection Graphically  The function shown above has TWO inflection points. change from concave down to up change from concave up to down

19. BMayer@ChabotCollege.edu MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx 19 Bruce Mayer, PE Chabot College Mathematics 2nd Derivative Test

20. BMayer@ChabotCollege.edu MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx 20 Bruce Mayer, PE Chabot College Mathematics Example  Apply 2nd Deriv Test  Use the 2 nd Derivative Test to Find and classify all critical points for the Function  SOLUTION  Find the critical points by solving:

21. BMayer@ChabotCollege.edu MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx 21 Bruce Mayer, PE Chabot College Mathematics Example  Apply 2nd Deriv Test  By Zero-Products:  Also need to check for values of x that make the derivative undefined. ReCall the 1 st Derivative: Thus df/dx is UNdefined for x = −1, But the ORIGINAL function is ALSO Undefined at the this value –Thus there is NO Critical Point at x = −1

22. BMayer@ChabotCollege.edu MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx 22 Bruce Mayer, PE Chabot College Mathematics Example  Apply 2 nd Deriv Test  Thus the only critical points are at −2 & 0  Now use the second derivative test to determine whether each is a MAXimum or MINimum (or if the test is InConclusive):

23. BMayer@ChabotCollege.edu MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx 23 Bruce Mayer, PE Chabot College Mathematics Example  Apply 2 nd Deriv Test  Before expanding the BiNomials, note that the numerator and denominator can be simplified by removing a common factor of (x+1) from all terms:

24. BMayer@ChabotCollege.edu MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx 24 Bruce Mayer, PE Chabot College Mathematics Example  Apply 2nd Deriv Test  Now expand BiNomials:  Now Check Value of f’’’(0) & f’’’(−2)

25. BMayer@ChabotCollege.edu MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx 25 Bruce Mayer, PE Chabot College Mathematics Example  Apply 2nd Deriv Test  The 2 nd Derivative is NEGATIVE at x = −2 Thus the orginal fcn is ConCave DOWN at x = −2, and a Relative MAX exists at this Pt  Conversely, 2 nd Derivative is POSITIVE at x = 0 Thus the orginal fcn is ConCave UP at x = 0 and a Relative MIN exists at this Pt

26. BMayer@ChabotCollege.edu MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx 26 Bruce Mayer, PE Chabot College Mathematics Example  Apply 2 nd Deriv Test  Confirm by Plot →  Note the relative MINimum at 0, relative MAXimum at −2, and a vertical asymptote where the function is undefined at x=−1 (although the vertical line is not part of the graph of the function)

27. BMayer@ChabotCollege.edu MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx 27 Bruce Mayer, PE Chabot College Mathematics ConCavity Sign Chart  A form of the df/dx (Slope) Sign Chart (Direction-Diagram) Analysis Can be Applied to d 2 f/dx 2 (ConCavity)  Call the ConCavity Sign-Charts “Dome- Diagrams” for INFLECTION Analysis abc −−−−−−++++++−−−−−−++++++ x ConCavity Form d 2 f/dx 2 Sign Critical (Break) Points InflectionNO Inflection Inflection

28. BMayer@ChabotCollege.edu MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx 28 Bruce Mayer, PE Chabot College Mathematics Example  Dome-Diagram  Find All Inflection Points for Notes on this (and all other) PolyNomial Function exists for ALL x  Use the ENGR25 Computer Algebra System, MuPAD, to find Derivatives Critical Points

29. BMayer@ChabotCollege.edu MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx 29 Bruce Mayer, PE Chabot College Mathematics Example  Dome-Diagram  The Derivatives  The Critical Points  The ConCavity Values Between Break Pts At x = −1 At x = ½

30. BMayer@ChabotCollege.edu MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx 30 Bruce Mayer, PE Chabot College Mathematics MyPAD Code

31. BMayer@ChabotCollege.edu MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx 31 Bruce Mayer, PE Chabot College Mathematics Example  Dome-Diagram  Draw Dome-Diagram  The ConCavity Does NOT change at 0, but it DOES at 1 Since Inflection requires Change, the only Inflection-Pt occurs at x = 1 01 −−−−−− ++++++ x ConCavity Form d 2 f/dx 2 Sign Critical (Break) Points NO Inflection Inflection

32. BMayer@ChabotCollege.edu MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx 32 Bruce Mayer, PE Chabot College Mathematics Example  Dome-Diagram  The Fcn Plot Showing Inflection Point at (1,y(1)) = (1,−3) (1,−3)

33. BMayer@ChabotCollege.edu MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx 33 Bruce Mayer, PE Chabot College Mathematics MATLAB Code MATLAB Code % Bruce Mayer, PE % MTH-15 11Jul13 % XYfcnGraph6x6BlueGreenBkGndTemplate1306.m % % The Limits xmin = -1.5; xmax = 2.5; ymin =-15; ymax = 15; % The FUNCTION x = linspace(xmin,xmax,1000); y =3*x.^5 - 5*x.^4 - 1; % % 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', 5),axis([xmin xmax ymin ymax]),... grid, xlabel('\fontsize{14}x'), ylabel('\fontsize{14}y = f(x) = 3x^5 - 5x^4 - 1'),... title(['\fontsize{16}MTH15 Dome-Diagram',]) hold on plot(1,-3, 'd r', 'MarkerSize', 10,'MarkerFaceColor', 'r', 'LineWidth', 2) plot(zxv,zyv, 'k', zxh,zyh, 'k', 'LineWidth', 2) set(gca,'XTick',[xmin:0.5:xmax]); set(gca,'YTick',[ymin:5:ymax]) hold off

34. BMayer@ChabotCollege.edu MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx 34 Bruce Mayer, PE Chabot College Mathematics Example  Population Growth  A population model finds that the number of people, P, living in a city, in kPeople, t years after the beginning of 2010 will be:  Questions In what year will the population be decreasing most rapidly? What will be the population at that time?

35. BMayer@ChabotCollege.edu MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx 35 Bruce Mayer, PE Chabot College Mathematics Example  Population Growth  SOLUTION:  “Decreasing most rapidly” is a phrase that requires some examination. “Decreasing” suggests a negative derivative.  “Decreasing most rapidly” means a value for which the negative derivative is as negative as possible. In other words, where the derivative is a MIN

36. BMayer@ChabotCollege.edu MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx 36 Bruce Mayer, PE Chabot College Mathematics Example  Population Growth  Need to find relative minima of functions (derivative functions are no exception) where the rate of change is equal to 0.  “Rate of change in the population derivative, set equal to zero” TRANSLATES mathematically to

37. BMayer@ChabotCollege.edu MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx 37 Bruce Mayer, PE Chabot College Mathematics Example  Population Growth  The only time at which the second derivative of P is equal to zero is the beginning of 2013. Need to verify that the derivative is, in fact, negative at that point: 

38. BMayer@ChabotCollege.edu MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx 38 Bruce Mayer, PE Chabot College Mathematics Example  Population Growth  Thus the function is decreasing most rapidly at the inflection point at the beginning of 2013:  The Model Predicts 2013 Population: xx

39. BMayer@ChabotCollege.edu MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx 39 Bruce Mayer, PE Chabot College Mathematics WhiteBoard Work  Problems From §3.2 P45 → Sketch Graph using General Description P66 → Spreading a Rumor

40. BMayer@ChabotCollege.edu MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx 40 Bruce Mayer, PE Chabot College Mathematics All Done for Today Rememgering ConCavity: cUP & frOWN

41. BMayer@ChabotCollege.edu MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx 41 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics Appendix –

42. BMayer@ChabotCollege.edu MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx 42 Bruce Mayer, PE Chabot College Mathematics ConCavity Sign Chart abc −−−−−−++++++−−−−−−++++++ x ConCavity Form d 2 f/dx 2 Sign Critical (Break) Points InflectionNO Inflection Inflection

43. BMayer@ChabotCollege.edu MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx 43 Bruce Mayer, PE Chabot College Mathematics Max/Min Sign Chart abc −−−−−−++++++−−−−−−++++++ x Slope df/dx Sign Critical (Break) Points MaxNO Max/Min Min

44. BMayer@ChabotCollege.edu MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx 44 Bruce Mayer, PE Chabot College Mathematics

45. BMayer@ChabotCollege.edu MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx 45 Bruce Mayer, PE Chabot College Mathematics

46. BMayer@ChabotCollege.edu MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx 46 Bruce Mayer, PE Chabot College Mathematics

47. BMayer@ChabotCollege.edu MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx 47 Bruce Mayer, PE Chabot College Mathematics

48. BMayer@ChabotCollege.edu MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx 48 Bruce Mayer, PE Chabot College Mathematics

49. BMayer@ChabotCollege.edu MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx 49 Bruce Mayer, PE Chabot College Mathematics

50. BMayer@ChabotCollege.edu MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx 50 Bruce Mayer, PE Chabot College Mathematics