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

1. Chabot Mathematics §1.6 Limits& Continuity Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

2. 1.5 Review § • Any QUESTIONS About • §1.5 → Limits • Any QUESTIONS About HomeWork • §1.5 → HW-05

3. §1.6 Learning Goals • Compute and use one-sided limits • Explore the concept of continuity and examine the continuity of several functions • Investigate the intermediate value property

4. Limits • Limits are a very basic aspect of calculus which needs to be taught first, after reviewing old material. • The concept of limits is very important, since we will need to use limits to make new ideas and formulas in calculus. • In order to understand calculus, limits are very fundamental to know!

5. Continuous Functions • Generally Speaking A function is very likely to be “continuous” if: The graph has no holes or gaps and can be drawn on a piece of paper without lifting The Drawing Instrument(Pencil or Pen)

6. Smooth Functions • Generally Speaking A function is very likely to be “smooth” if: The graph of the function is a “flowing” curve. This means that the graph of the function does not contain any “sharp” corners • Smoothness Analysis will be covered after we learn how to evaluate the “Slope” of curved lines

7. Continuous vs. DisContinuous • CONTINUOUS Function Plot • DIScontinuousFunction Plot

8. Smooth vs. Kinked/Cornered • SMOOTH-Curved Function Plot • SHARP-Cornered Function Plot

9. ONEsided Limits - From LEFT • If f(x) Approaches L as x→c from the Left; i.e., x<c, write: • See Graph at Right

10. ONEsided Limits – From RIGHT • If f(x) Approaches L as x→c from the Left; i.e., x<c, write: • See Graph at Right

11. Example  PieceWiseFcn • Find the OneSidedLimits for Function: • Compute the one-sided limits of f(x) as xapproaches 1

12. Example  OneSided Limits • SOLUTION • Need to Determine: • Because the function is defined by the first expression for values of x ≤1, have • Also the fcn is defined by the second expression for values of x >1, have

13. Example  OneSided Limits • SOLUTION • ReCall the Requirement for Limit Existence • For the Given Fcn use the Transitive Property to Recognize that the Limit x→1 Does Not Exist as

14. % Bruce Mayer, PE % MTH-15 • 01Jul13 % XY_fcn_Graph_BlueGreenBkGnd_Solid_Marker_Template1306.m % % The Limits xmin = -3; xmax1 = 1; xmin2 = xmax1; xmax = 3; ymin = -4; ymax = 10; % The FUNCTION x1 = linspace(xmin,xmax1,500); y1 = 1-x1.^2; x2 = linspace(xmin2,xmax,500); y2 = 3*x2+1; % The Total Function by appending x = [x1, x2]; y = [y1, y2]; % % 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(x1,y1,'b', x2,y2,'b', zxv,zyv, 'k', zxh,zyh, 'k', x1(end),y1(end), 'ob', 'MarkerSize', 12, 'MarkerFaceColor', 'b',... 'LineWidth', 3),axis([xminxmaxyminymax]),... grid, xlabel('\fontsize{14}x'), ylabel('\fontsize{14}f(x) \rightarrowPieceWise'),... title(['\fontsize{14}MTH15 • Bruce Mayer, PE • 2-Sided Limit',]),... annotation('textbox',[.51 .05 .0 .1], 'FitBoxToText', 'on', 'EdgeColor', 'none', 'String', 'XYfcnGraphBlueGreenBkGndSolidMarkerTemplate1306.m','FontSize',7) hold on plot(x2(1),y2(1), 'ob', 'MarkerSize', 12, 'MarkerFaceColor', [0.8 1 1], 'LineWidth', 3) set(gca,'XTick',[xmin:1:xmax]); set(gca,'YTick',[ymin:1:ymax]) hold off MATLAB Code

15. Continuity Analysis • DEFININITION: A function, f(x) is continuous at a point c If and Only If The limit of f(x) is independent of the direction of Approach; that is the fcn is continuous if: • Note that this a Necessary AND Sufficient, Condition

16. Example  Continuity • Consider Function: • See Graph at Right • Determine if the Function is Continuous at • x = 4 • x = 5 • Use BiLateral Approach Limit Test

17. Example  Continuity • Find for x = 4 The BiLateral Limits • At x = 3.9999 • At x = 4.0001 • By the PolyNomial Limit Rule • The Left Approach (3.9999) and the Right Approach (4.0001) Both Lead to 235, thus the fcn IS Continuous at x = 4

18. Example  Continuity • Now Check Continuity at x = 5 • Use Approach Tables • From Approach Tables Note:

19. PieceWise Continuity • A NONontinuousPieceWise-Defined Function can be made continuous thru the process of Break-Point Matching. • BreakPoint Matching • One Fcn Left Unchanged • At Least ONE Variable-Term in the other Fcn is multiplied by a CONSTANT • The two Fcns are then equated at the BreakPoint Value

20. Example  Make Continuous • Consider the Fcn: • This Fcn isNONcontinuous asshown in the Plot • Make this Plot Continuous for Constants P & Q:

21. Example  Continuous at 8 • The FineTunedFcn • ThePlot

22. Example  Continuous at −13 • The FineTunedFcn • ThePlot

23. % Bruce Mayer, PE % MTH-15 • 01Jul13 % XY_fcn_Graph_BlueGreenBkGnd_Solid_Marker_Template1306.m % % The Limits xmin = -2; xmax1 = 1; xmin2 = xmax1; xmax = 3; ymin = -15; ymax = 15; % The FUNCTION x1 = linspace(xmin,xmax1,500); y1 = 24*x1.^2 - 5*x1 - 11 ; x2 = linspace(xmin2,xmax,500); y2 = sqrt(x2) + 7; % The Total Function by appending x = [x1, x2]; y = [y1, y2]; % % 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(x1,y1,'b', x2,y2,'b', zxv,zyv, 'k', zxh,zyh, 'k',... 'LineWidth', 3),axis([xminxmaxyminymax]),... grid, xlabel('\fontsize{14}x'), ylabel('\fontsize{14}f(x) \rightarrowPieceWise'),... title(['\fontsize{14}MTH15 • Bruce Mayer, PE • PcWise Continuous',]),... annotation('textbox',[.41 .05 .0 .1], 'FitBoxToText', 'on', 'EdgeColor', 'none', 'String', 'XYfcnGraphBlueGreenBkGndSolidMarkerTemplate1306.m','FontSize',7) hold on set(gca,'XTick',[xmin:1:xmax]); set(gca,'YTick',[ymin:5:ymax]) hold off P MATLAB Code

24. % Bruce Mayer, PE % MTH-15 • 01Jul13 % XY_fcn_Graph_BlueGreenBkGnd_Solid_Marker_Template1306.m % % The Limits xmin = -2; xmax1 = 1; xmin2 = xmax1; xmax = 3; ymin = -20; ymax = 10; % The FUNCTION x1 = linspace(xmin,xmax1,500); y1 = 3*x1.^2 - 5*x1 - 11 ; x2 = linspace(xmin2,xmax,500); y2 = (-13/8)*(sqrt(x2) + 7); % The Total Function by appending x = [x1, x2]; y = [y1, y2]; % % 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(x1,y1,'b', x2,y2,'b', zxv,zyv, 'k', zxh,zyh, 'k',... 'LineWidth', 3),axis([xminxmaxyminymax]),... grid, xlabel('\fontsize{14}x'), ylabel('\fontsize{14}f(x) \rightarrowPieceWise'),... title(['\fontsize{14}MTH15 • Bruce Mayer, PE • PcWise Continuous',]),... annotation('textbox',[.41 .05 .0 .1], 'FitBoxToText', 'on', 'EdgeColor', 'none', 'String', 'XYfcnGraphBlueGreenBkGndSolidMarkerTemplate1306.m','FontSize',7) hold on set(gca,'XTick',[xmin:1:xmax]); set(gca,'YTick',[ymin:5:ymax]) hold off Q MATLAB Code

25. Intermediate Value Theorem • If f(x) is a continuous function on a closed interval [a, b] and L is any number between f(a) and f(b), then there is at least one number c in [a, b] such that f(c) = L f(b) f(c) = L f(a) a c b

26. Example  IVT • Given Fcn → • Show That f(x)=0 has a solution on [1,2] • SOLUTION • Since the Function is a PolyNomial the Fcn IS Continuous for all x • Check Interval EndPoints

27. Example  IVT • STATE: f(x) is continuous (polynomial) and since f(1) < 0 and f(2) > 0, by the Intermediate Value Theorem there exists c on [1, 2] such that f(c) = 0. (c,0)

28. % Bruce Mayer, PE % MTH-15 • 01Jul13 % XY_fcn_Graph_BlueGreenBkGnd_Solid_Marker_Template1306.m % % The Limits xmin = 0; xmax1 = 3; xmin2 = xmax1; xmax = 3; ymin = -10; ymax = 15; % The FUNCTION x1 = linspace(xmin,xmax1,500); y1 = 3*x1.^2 - 2*x1 - 5 ; x2 = linspace(xmin2,xmax,500); y2 = 3*x2.^2 - 2*x2 - 5; % The Total Function by appending x = [x1, x2]; y = [y1, y2]; % % 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(x1,y1,'b', x2,y2,'b',zxh,zyh, 'k',... 'LineWidth', 3),axis([xminxmaxyminymax]),... grid, xlabel('\fontsize{14}x'), ylabel('\fontsize{14}y = f(x)=3x^2 - 2x - 5'),... title(['\fontsize{14}MTH15 • Bruce Mayer, PE • IVT',]),... annotation('textbox',[.41 .05 .0 .1], 'FitBoxToText', 'on', 'EdgeColor', 'none', 'String', 'XYfcnGraphBlueGreenBkGndSolidMarkerTemplate1306.m','FontSize',7) hold on set(gca,'XTick',[xmin:1:xmax]); set(gca,'YTick',[ymin:5:ymax]) hold off MATLAB Code

29. WhiteBoard Work • Problems From §1.6 • P13 → Find Limit Using Algebra • P52 → Electrically Charged Sphere • P56 → Create Continuity

30. All Done for Today KnowYourLimits

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

32. Make Continuous - P

33. Make Continuous - Q

36. Charge Hollow Sphere E-fld

37. % Bruce Mayer, PE % MTH-15 • 01Jul13 % XY_fcn_Graph_BlueGreenBkGnd_Solid_Marker_Template1306.m % clear; clc; % InDepVar = x/R % The Limits xmin = 0; xmax1 = 1; xmin2 = xmax1; xmax = 3; ymin = -.1; ymax = 1.1; % The FUNCTION x1 = linspace(xmin,xmax1,500); y1 = 0*x1 ; x2 = linspace(xmin2,xmax,500); y2 = 1./x2.^2; x3 = 1; y3 = 1/(2*1^2) % The Total Function by appending x = [x1, x2]; y = [y1, y2]; % % 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(x1,y1,'b', x2,y2,'b',... 'LineWidth', 3),axis([xminxmaxyminymax]),... grid, xlabel('\fontsize{14}x/R'), ylabel('\fontsize{14}y = E(x) (Volt/meter)'),... title(['\fontsize{14}MTH15 • Bruce Mayer, PE • P1.6-52 Charged Sphere',]),... annotation('textbox',[.41 .05 .0 .1], 'FitBoxToText', 'on', 'EdgeColor', 'none', 'String', 'XYfcnGraphBlueGreenBkGndSolidMarkerTemplate1306.m','FontSize',7) hold on plot(x3,y3, 'ob', 'MarkerSize', 6, 'MarkerFaceColor', 'b', 'LineWidth', 3) plot(x2(1),y2(1), 'ob', 'MarkerSize', 6, 'MarkerFaceColor', [0.8 1 1], 'LineWidth', 3) plot(x1(end),y1(end), 'ob', 'MarkerSize', 6, 'MarkerFaceColor', [0.8 1 1], 'LineWidth', 3) set(gca,'XTick',[xmin:1:xmax]); set(gca,'YTick',[ymin:.1:ymax]) hold off MATLAB Code

39. P1.6-52(B)

40. P1.6-56 Continuous Plot