Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu

1. Engr/MATH/Physics 25 Sketch FcnGraphs by MuPAD Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

2. 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 forsketching graphs

3. T-Table Can Miss Features • Consider the Function • Make T-Table,Connect-Dots

4. % 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 = [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) = = 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', [xminxmax],[10 10], '-- m', 'LineWidth', 2) set(gca,'XTick',[xmin:5:xmax]); set(gca,'YTick',[ymin:5:ymax]) MATLAB Code

5. T-Table Can Miss Features • But Using Methods to be Discussed, Find

6. 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 -1 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 = [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) = 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. 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-SpanInSufficent

8. Better Graphing GamePlan • Find THE y-Intercept, if Any • Set x = 0, find y • Only TWO Functions do NOT have a y-intercepts • Of the form 1/x • x = const; x ≠ 0 • Find x-Intercept(s), if Any • Set y = 0, find x • Many functions do NOT have x-intercepts

9. Better Graphing GamePlan • Find VERTICAL (↨) Asymptotes, If Any • Exist ONLY when fcn has a denom • Set Denom = 0, solve for x • These Values of x are the Vertical Asymptote (VA) Locations • Find HORIZONTAL (↔) Asymptotes (HA), If Any • HA’s Exist ONLY if the fcn has a finite limit-value when x→+∞, or when x→−∞

10. Better Graphing GamePlan • Find y-value for: • These Values of y are the HA Locations • Find the Extrema (Max/Min) Locations • Set dy/dx = 0, solve for xE • Find the corresponding yE = f(xE) • Determine by 2nd Derivative, or ConCavity, test whether (xE,yE) is a Max or a Min • See Table on Next Slide

11. Better Graphing GamePlan • Determine Max/Min By Concavity • Find the Inflection Pt Locations • Set d2y/dx2 = 0, solve for xi • Find the corresponding yi = f(xi) • Determine by 3rd Derivative test The Inflection form: ↑-↓ or ↓-↑

12. Better Graphing GamePlan • Find the Inflection Pt Locations • Set d2y/dx2 = 0, solve for xi • Find the corresponding yi = f(xi) • Determine by 3rd Derivative test The Inflection form: ↑-↓ or ↓- ↑ • Determine Inflection form by 3rd Derivative

13. Better Graphing GamePlan • Sign Charts for Max/Min and ↑-↓/↓-↑ • To Find the “Flat Spot” behavior for dy/dx = 0, when d2y/dx2 exists, but [d2y/dx2]xE = 0 use the Direction-Diagram Slope ++++++ −−−−−− −−−−−− ++++++ df/dx Sign x Critical (Break)Points a b c Max NOMax/Min Min

14. Better Graphing GamePlan • Sign Charts for Max/Min and ↑-↓/↓-↑ • To Find the ↑-↑ or ↓-↓ behavior for d2y/dx2 = 0, when d3y/dx3 exists, but [d3y/dx3]xi = 0 use the Dome-Diagram ConCavityForm ++++++ −−−−−− −−−−−− ++++++ d2f/dx2 Sign x Critical (Break)Points a b c Inflection NOInflection Inflection

15. 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. Example  Sketch Rational Fcn • y=0: • Solving for x: • Finding y(x):

17. Example  Sketch Rational Fcn • The x-Intercepts • (½,0); Multiplicity = 1 (LINE-Like) • (−2,0); Multiplicity = 2 (PARABOLA Like) • The Horizontal Intercept(s)

18. Example  Sketch Rational Fcn • Continuing with the Limit • Thus have a HORIZONTAL asymptote at y = 0

19. 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. 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. The Graph

27. All Done for Today A GraphicScalingMachine