MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer Chabot Mathematics §3.3 Curve Sketching

MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx 2 Bruce Mayer, PE Chabot College Mathematics Review §  Any QUESTIONS About §3.2 → ConCavity & InflectionPoints  Any QUESTIONS About HomeWork §3.2 → HW

MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx 3 Bruce Mayer, PE Chabot College Mathematics §3.3 Learning Goals  Determine horizontal and vertical asymptotes of a graph  Use Algebra to find Axes InterCepts on a Funciton Graph  Use Derivatives to find Significant Points on the graph  Discuss and apply a general procedure for sketching graphs

MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx 4 Bruce Mayer, PE Chabot College Mathematics T-Table Can Miss Features  Consider the Function  Make T-Table, Connect-Dots xY

MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx 5 Bruce Mayer, PE Chabot College Mathematics 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([ ]); % 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',[ ], 'FitBoxToText', 'on', 'EdgeColor', 'none', 'String', 'XYfcnGraph6x6BlueGreenBkGndTemplate1306.m','FontSize',7) hold on plot(zxv,zyv, 'k', zxh,zyh, 'k', 'LineWidth', 2) plot([ ], [ymin, ymax], '-- m', [xmin xmax],[10 10], '-- m', 'LineWidth', 2) set(gca,'XTick',[xmin:5:xmax]); set(gca,'YTick',[ymin:5:ymax])

MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx 6 Bruce Mayer, PE Chabot College Mathematics T-Table Can Miss Features  But Using Methods to be Discussed, Find

MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx 7 Bruce Mayer, PE Chabot College Mathematics 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 = [ ]; y = [ ] % % 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) = 10x(x+8)/(x+10)^2'),... title(['\fontsize{16}MTH15 GraphSketching',]),... annotation('textbox',[ ], '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

MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx 8 Bruce Mayer, PE Chabot College Mathematics 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

MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx 9 Bruce Mayer, PE Chabot College Mathematics 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

MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx 10 Bruce Mayer, PE Chabot College Mathematics 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→−∞

MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx 11 Bruce Mayer, PE Chabot College Mathematics 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, to test whether (x E, y E ) is a Max or a Min –See Table on Next Slide

MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx 12 Bruce Mayer, PE Chabot College Mathematics 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-concavity form: ↑-↓ or ↓- ↑

MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx 13 Bruce Mayer, PE Chabot College Mathematics 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

MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx 14 Bruce Mayer, PE Chabot College Mathematics 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

MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx 15 Bruce Mayer, PE Chabot College Mathematics 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

MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx 16 Bruce Mayer, PE Chabot College Mathematics 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

MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx 17 Bruce Mayer, PE Chabot College Mathematics Example  Sketch Rational Fcn  y=0:  Solving for x:  Finding y(x):

MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx 18 Bruce Mayer, PE Chabot College Mathematics Example  Sketch Rational Fcn  The x-Intercepts (½,0); Multiplicity = 1 (Linear Form) (−2,0); Multiplicity = 2 (Parabolic Form)  The Horizontal Intercept(s)

MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx 19 Bruce Mayer, PE Chabot College Mathematics Example  Sketch Rational Fcn  Continuing with the Limit Thus have a HORIZONTAL asymptote at y = 2

MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx 20 Bruce Mayer, PE Chabot College Mathematics 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

MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx 21 Bruce Mayer, PE Chabot College Mathematics 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 (− , ) –↑-to-↓ at ( , − )

MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx 22 Bruce Mayer, PE Chabot College Mathematics The Graph

MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx 23 Bruce Mayer, PE Chabot College Mathematics WhiteBoard Work  Problems From §3.3 P46 → Inventory Cost P60 → Immunization

MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx 24 Bruce Mayer, PE Chabot College Mathematics All Done for Today A Graphic Scaling Machine

MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx 25 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer Chabot Mathematics Appendix –

MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx 26 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

MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx 27 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

MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx 28 Bruce Mayer, PE Chabot College Mathematics

MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx 29 Bruce Mayer, PE Chabot College Mathematics

MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx 30 Bruce Mayer, PE Chabot College Mathematics

MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx 31 Bruce Mayer, PE Chabot College Mathematics

MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx 32 Bruce Mayer, PE Chabot College Mathematics

MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx 33 Bruce Mayer, PE Chabot College Mathematics

MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx 34 Bruce Mayer, PE Chabot College Mathematics

MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx 35 Bruce Mayer, PE Chabot College Mathematics

MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx 36 Bruce Mayer, PE Chabot College Mathematics

MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx 37 Bruce Mayer, PE Chabot College Mathematics

MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx 38 Bruce Mayer, PE Chabot College Mathematics P33-46

MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx 39 Bruce Mayer, PE Chabot College Mathematics

MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx 40 Bruce Mayer, PE Chabot College Mathematics

MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx 41 Bruce Mayer, PE Chabot College Mathematics

MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx 42 Bruce Mayer, PE Chabot College Mathematics

MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx 43 Bruce Mayer, PE Chabot College Mathematics P3.3-56

MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx 44 Bruce Mayer, PE Chabot College Mathematics