1. BMayer@ChabotCollege.edu ENGR-25_Lec-28_Excel-1.ppt 1 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Engr/MATH/Physics 25 Sketch Fcn Graphs by MuPAD

2. BMayer@ChabotCollege.edu ENGR-25_Lec-28_Excel-1.ppt 2 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods ReCall from MTH1 Graph Sketching  Determine horizontal and vertical asymptotes of a graph  Use Algebra to find Axes InterCepts on a Function Graph  Use Derivatives to find Significant Points on the graph  Discuss and apply a general procedure for sketching graphs

3. BMayer@ChabotCollege.edu ENGR-25_Lec-28_Excel-1.ppt 3 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods T-Table Can Miss Features  Consider the Function  Make T-Table, Connect-Dots xY -5-6.00 -4-4.44 -3-3.06 -2-1.88 -0.86 00.00 10.74 21.39 31.95 42.45 52.89

4. BMayer@ChabotCollege.edu ENGR-25_Lec-28_Excel-1.ppt 4 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods MATLAB Code MATLAB Code % Bruce Mayer, PE % MTH-15 13Jul13 % XYfcnGraph6x6BlueGreenBkGndTemplate1306.m % ref: % % The Limits xmin = -35; xmax = 25; ymin = -15; ymax = 40; % The FUNCTION x = linspace(xmin,xmax,500); y = 10*x.*(x+8)./(x+10).^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) = = 10x(x+8)/(x+10)^2'),... title(['\fontsize{16}MTH15 GraphSketching',]),... annotation('textbox',[.51.05.0.1], 'FitBoxToText', 'on', 'EdgeColor', 'none', 'String', 'XYfcnGraph6x6BlueGreenBkGndTemplate1306.m','FontSize',7) hold on plot(zxv,zyv, 'k', zxh,zyh, 'k', 'LineWidth', 2) plot([-10 -10], [ymin, ymax], '-- m', [xmin xmax],[10 10], '-- m', 'LineWidth', 2) set(gca,'XTick',[xmin:5:xmax]); set(gca,'YTick',[ymin:5:ymax])

5. BMayer@ChabotCollege.edu ENGR-25_Lec-28_Excel-1.ppt 5 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods T-Table Can Miss Features  But Using Methods to be Discussed, Find

6. BMayer@ChabotCollege.edu ENGR-25_Lec-28_Excel-1.ppt 6 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods MATLAB Code MATLAB Code % Bruce Mayer, PE % MTH-15 23Jun13 % XYfcnGraph6x6BlueGreenBkGndTemplate1306.m % ref: % % The Limits xmin = -5; xmax = 5; ymin = -6; ymax = 3; % The FUNCTION x = [-5 -4 -3 -2 0 1 2 3 4 5]; y = [-6 -4.444444444 -3.06122449 -1.875 -0.864197531 0 0.743801653 1.388888889 1.952662722 2.448979592 2.888888889] % % 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) = 10x(x+8)/(x+10)^2'),... title(['\fontsize{16}MTH15 GraphSketching',]),... annotation('textbox',[.51.05.0.1], 'FitBoxToText', 'on', 'EdgeColor', 'none', 'String', 'XYfcnGraph6x6BlueGreenBkGndTemplate1306.m','FontSize',7) hold on plot(x,y, 'x m', 'MarkerSize', 15, 'LineWidth', 3) plot(zxv,zyv, 'k', zxh,zyh, 'k', 'LineWidth', 2) set(gca,'XTick',[xmin:1:xmax]); set(gca,'YTick',[ymin:1:ymax] hold off

7. BMayer@ChabotCollege.edu ENGR-25_Lec-28_Excel-1.ppt 7 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods T-Table Can Miss Features  In Order for T-Tables & ConnectDots to properly Characterize the Fcn Graph, the Domain (x) Column must Cover sufficiently Wide values Have sufficiently small increments  Unfortunately the Grapher does NOT know a-priori the x Span ∆x Increment Size x-Span InSufficent

8. BMayer@ChabotCollege.edu ENGR-25_Lec-28_Excel-1.ppt 8 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Better Graphing GamePlan 1.Find THE y-Intercept, if Any a.Set x = 0, find y b.Only TWO Functions do NOT have a y-intercepts –Of the form 1/x –x = const; x ≠ 0 2.Find x-Intercept(s), if Any a.Set y = 0, find x b.Many functions do NOT have x-intercepts

9. BMayer@ChabotCollege.edu ENGR-25_Lec-28_Excel-1.ppt 9 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Better Graphing GamePlan 3.Find VERTICAL (↨) Asymptotes, If Any a.Exist ONLY when fcn has a denom b.Set Denom = 0, solve for x –These Values of x are the Vertical Asymptote (VA) Locations 4.Find HORIZONTAL (↔) Asymptotes (HA), If Any a.HA’s Exist ONLY if the fcn has a finite limit-value when x→+∞, or when x→−∞

10. BMayer@ChabotCollege.edu ENGR-25_Lec-28_Excel-1.ppt 10 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Better Graphing GamePlan b.Find y-value for: –These Values of y are the HA Locations 5.Find the Extrema (Max/Min) Locations a.Set dy/dx = 0, solve for x E b.Find the corresponding y E = f(x E ) c.Determine by 2 nd Derivative, or ConCavity, test whether (x E, y E ) is a Max or a Min –See Table on Next Slide

11. BMayer@ChabotCollege.edu ENGR-25_Lec-28_Excel-1.ppt 11 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Better Graphing GamePlan –Determine Max/Min By Concavity 6.Find the Inflection Pt Locations a.Set d 2 y/dx 2 = 0, solve for x i b.Find the corresponding y i = f(x i ) c.Determine by 3 rd Derivative test The Inflection form: ↑-↓ or ↓-↑

12. BMayer@ChabotCollege.edu ENGR-25_Lec-28_Excel-1.ppt 12 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Better Graphing GamePlan 7.Find the Inflection Pt Locations a.Set d 2 y/dx 2 = 0, solve for x i b.Find the corresponding y i = f(x i ) c.Determine by 3 rd Derivative test The Inflection form: ↑-↓ or ↓- ↑ –Determine Inflection form by 3 rd Derivative

13. BMayer@ChabotCollege.edu ENGR-25_Lec-28_Excel-1.ppt 13 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Better Graphing GamePlan 8.Sign Charts for Max/Min and ↑-↓/↓-↑ a.To Find the “Flat Spot” behavior for dy/dx = 0, when d 2 y/dx 2 exists, but [d 2 y/dx 2 ] xE = 0 use the Direction-Diagram abc −−−−−−++++++−−−−−−++++++ x Slope df/dx Sign Critical (Break) Points MaxNO Max/Min Min

14. BMayer@ChabotCollege.edu ENGR-25_Lec-28_Excel-1.ppt 14 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Better Graphing GamePlan 9.Sign Charts for Max/Min and ↑-↓/↓-↑ a.To Find the ↑-↑ or ↓-↓ behavior for d 2 y/dx 2 = 0, when d 3 y/dx 3 exists, but [d 3 y/dx 3 ] xi = 0 use the Dome-Diagram abc −−−−−−++++++−−−−−−++++++ x ConCavity Form d 2 f/dx 2 Sign Critical (Break) Points InflectionNO Inflection Inflection

15. BMayer@ChabotCollege.edu ENGR-25_Lec-28_Excel-1.ppt 15 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Example  Sketch Rational Fcn  Sketch  Set x = 0 to Find y-intercept Thus y-intercept → (0, 4/3)  Set y = 0 to Find x-intercept(s), if any

16. BMayer@ChabotCollege.edu ENGR-25_Lec-28_Excel-1.ppt 16 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Example  Sketch Rational Fcn  y=0:  Solving for x:  Finding y(x):

17. BMayer@ChabotCollege.edu ENGR-25_Lec-28_Excel-1.ppt 17 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Example  Sketch Rational Fcn  The x-Intercepts (½,0); Multiplicity = 1 (LINE-Like) (−2,0); Multiplicity = 2 (PARABOLA Like)  The Horizontal Intercept(s)

18. BMayer@ChabotCollege.edu ENGR-25_Lec-28_Excel-1.ppt 18 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Example  Sketch Rational Fcn  Continuing with the Limit Thus have a HORIZONTAL asymptote at y = 0

19. BMayer@ChabotCollege.edu ENGR-25_Lec-28_Excel-1.ppt 19 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Example  Sketch Rational Fcn  To Find VERTICAL asymptote(s) set the DeNom/Divisor = 0 Using Zero Products Thus have VERTICAL Asymptotes at –x = −1 –x = 3

20. BMayer@ChabotCollege.edu ENGR-25_Lec-28_Excel-1.ppt 20 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Example  Sketch Rational Fcn  Use Computer Algebra System, MuPAD to find and Solve Derivatives  From the Derivatives Find Min at (−2,0) → ConCave UP Inflection Points –↓-to-↑ at (−2.63299, 0.16714) –↑-to-↓ at (0.63299, −0.29213)

21. BMayer@ChabotCollege.edu ENGR-25_Lec-28_Excel-1.ppt 21 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods The Graph

22. BMayer@ChabotCollege.edu ENGR-25_Lec-28_Excel-1.ppt 22 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods

23. BMayer@ChabotCollege.edu ENGR-25_Lec-28_Excel-1.ppt 23 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods

24. BMayer@ChabotCollege.edu ENGR-25_Lec-28_Excel-1.ppt 24 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods

25. BMayer@ChabotCollege.edu ENGR-25_Lec-28_Excel-1.ppt 25 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods

26. BMayer@ChabotCollege.edu ENGR-25_Lec-28_Excel-1.ppt 26 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods

27. BMayer@ChabotCollege.edu ENGR-25_Lec-28_Excel-1.ppt 27 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods All Done for Today A Graphic Scaling Machine