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

Chabot Mathematics. §5.3 Fundamental Theorem of Calc. Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu. 5.2. Review §. Any QUESTIONS About §5.2 → AntiDerivatives by Substitution Any QUESTIONS About HomeWork §5.22 → HW-23. §5.3 Learning Goals.

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

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  1. Chabot Mathematics §5.3 FundamentalTheorem of Calc Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

  2. 5.2 Review § • Any QUESTIONS About • §5.2 → AntiDerivatives by Substitution • Any QUESTIONS About HomeWork • §5.22 → HW-23

  3. §5.3 Learning Goals • Show how area under a curve can be expressed as the limit of a sum • Define the definite integral and explore its properties • State the fundamental theorem of calculus, and use it to compute definite integrals • Use the fundamental theorem to solve applied problems involving net change • Provide a geometric justification of the fundamental theorem

  4. The AUC has many Applications in Business, Science, and Engineering Area Under the Curve (AUC) Calculation of Geographic Areas River ChannelCross Section Wind-ForceLoading

  5. Area Under Function Graph • For a Continuous Function, approximate the area between the Curve and the x-Axis by Summing Vertical Strips • Use Rectangles of Equal Width • Three Possible Forms Left end points Right end points Midpoints Strip Width (n strips)

  6. Example: Strip Sum • Approximate the area under the graph of • Use • n = 4 (4 strips) • StripMidPoints

  7. Example: Strip Sum GamePlan

  8. % Bruce Mayer, PE % MTH-15 • 24Jul13 % XY_Area_fcn_Graph_6x6_BlueGreen_BkGnd_Template_1306.m % % The FUNCTION xmin = 0; xmax = 2; ymin = 0; ymax = 8; % The FUNCTION x = linspace(xmin,xmax,20); y = 2*x.^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 % Now make AREA Plot area(x,y, 'FaceColor', [1 .8 1] , 'LineWidth', 3), axis([xminxmaxyminymax]),... grid, xlabel('\fontsize{14}x'), ylabel('\fontsize{14}y = f(x) = 2x^2'),... title(['\fontsize{16}MTH15 • Area by Strip Addition',]),... annotation('textbox',[.13 .82 .0 .1], 'FitBoxToText', 'on', 'EdgeColor', 'none', 'String', 'Bruce Mayer, PE • 24JUul13','FontSize',7) hold on set(gca,'Layer','top') plot(x,y, 'LineWidth', 3), MATLAB Code

  9. MATLAB Code % Bruce Mayer, PE % MTH-15 • 24Jul13 % % The Limits xmin = 0; xmax = 2; ymin = 0; ymax = 8; % The FUNCTION x = linspace(xmin,xmax,500); y = 2*x.^2; x1 = [0.25:.5:1.75]; y1 = 2*x1.^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) = 2x^2'),... title(['\fontsize{16}MTH15 • Area by Strip Addition',]),... annotation('textbox',[.13 .82 .0 .1], 'FitBoxToText', 'on', 'EdgeColor', 'none', 'String', 'Bruce Mayer • 24Jul13','FontSize',7) hold on area([(x1(1)-.25)*ones(1,100),(x1(1)+.25)*ones(1,100)],[y1(1)*ones(1,100),y1(1)*ones(1,100)],'FaceColor',[1 .8 1]) area([(x1(2)-.25)*ones(1,100),(x1(2)+.25)*ones(1,100)],[y1(2)*ones(1,100),y1(2)*ones(1,100)],'FaceColor',[1 .8 1]) area([(x1(3)-.25)*ones(1,100),(x1(3)+.25)*ones(1,100)],[y1(3)*ones(1,100),y1(3)*ones(1,100)],'FaceColor',[1 .8 1]) area([(x1(4)-.25)*ones(1,100),(x1(4)+.25)*ones(1,100)],[y1(4)*ones(1,100),y1(4)*ones(1,100)],'FaceColor',[1 .8 1]) plot(x,y, 'LineWidth', 4) set(gca,'Layer','top') plot(x1,y1,'g d', 'LineWidth', 4) plot([x1(1)-.25,x1(1)+.25],[y1(1),y1(1)], 'm', [x1(2)-.25,x1(2)+.25],[y1(2),y1(2)], 'm',... [x1(3)-.25,x1(3)+.25],[y1(3),y1(3)], 'm', [x1(4)-.25,x1(4)+.25],[y1(4),y1(4)], 'm','LineWidth',2) plot([x1(1)-.25,x1(1)-.25],[0,y1(1)], 'm',[x1(2)-.25,x1(2)-.25],[0,y1(2)], 'm',... [x1(3)-.25,x1(3)-.25],[0,y1(3)], 'm', [x1(4)-.25,x1(4)-.25],[0,y1(4)], 'm', 'LineWidth', 2) set(gca,'XTick',[xmin:0.5:xmax]); set(gca,'YTick',[ymin:1:ymax])

  10. Example: Strip Sum • The Algebra midpoints

  11. Area under a Curve • GOAL: find the exact area under the graph of a function; i.e., the curve • PLAN: Use an infinite number of strips of equal width and compute their area with a limit. Width: (n strips) a b

  12. Area Under a Curve • Function, f(x), oninterval [a,b] is: • Continuous • NonNegative • Then the Area Under the Curve, A • The x1, x2, …,xn-1,xn are arbitrary, nSubIntervalseach with width (b − a)/n a b

  13. Riemann Sum ∑f(xk)·∆x • For a Continuous, NonNegfcn over [a,b] divided into n-intervals of Equal Width, ∆x = (b−a)/n, The AUC can be approximated by the sum the area of Vertical Strips Riemann ∑

  14. Riemann ∑ → Definite Integral • For a Continuous, NonNegfcn over [a,b] divided into n-intervals of Equal Width, ∆x = (b−a)/n, The AUC can be calculated EXACTLY by the Riemann sum as the number of strips becomes infinite. • This Process of finding an Infinite Sum is called “Integration”; • "to render (something) whole," from Latin integratus, past participle of integrare "make whole,"

  15. Riemann ∑ → Definite Integral • As the No. of Strips increase the AUC Calculation becomes more accurate • The Riemann-Sum to Definite-Integral Twenty Strips Fifty Strips

  16. % Bruce Mayer, PE % MTH-15 • 24Jul13 % % The Limits xmin = 0; xmax = 2; ymin = 0; ymax = 8; % The FUNCTION x = linspace(xmin,xmax,500); y = 2*x.^2; x1 = [1/20:1/10:39/20]; y1 = 2*x1.^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) = 2x^2'),... title(['\fontsize{16}MTH15 • Area by Strip Addition',]),... annotation('textbox',[.13 .82 .0 .1], 'FitBoxToText', 'on', 'EdgeColor', 'none', 'String', 'Bruce Mayer • 24Jul13','FontSize',7) hold on bar(x1,y1, 'BarWidth',1, 'FaceColor', [1 .8 1], 'EdgeColor','b', 'LineWidth', 2) set(gca,'XTick',[xmin:0.5:xmax]); set(gca,'YTick',[ymin:1:ymax]) set(gca,'Layer','top') plot(x,y, 'LineWidth', 3) MATLAB Code

  17. Definite Integral Symbology upper limit of integration Integration Symbol integrand variable of integration (dummy variable) lower limit of integration It is called a dummy variable because the answer does not depend on the symbol chosen; it depends only on a&b

  18. Recall Fundamental Theorem • The fundamental theorem* of calculus is a theorem that links the concept of the derivative of a function with the concept of the integral. • Part-1: Definite Integral (Area Under Curve) • Part-2: AntiDerivative * The Proof is Beyond the Scope of MTH15

  19. Fundamental Theorem – Part2 • Previously we stated that the AntiDerivative of f(x) is F(x), so then • Now consider the definite Integral (AUC) Relationship to the AntiDerivative

  20. DefiniteIntegral↔AntiDerivative • That is, The AUC for a continuous Function, f(x), spanning domain [a,b] is the AntiDerivative evaluated at b minus the AntiDerivative evaluated at a. • D. F. Riddle, Calculus and Analytical Geometry, Belmont, CA, Wadsworth, 1974, pp. 179-181, pg. 770

  21. Example  Find AUC • Find the area under the graph of y = 2x3 • Then Gives the area since 2x3 is nonnegative on [0, 2]. Antiderivative Fund. Thm. of Calculus

  22. Rules for Definite Integrals • Constant Rule: for any constant, k • Sum/DiffRule: • Zero WidthRule • DomainReversal Rule

  23. Rules for Definite Integrals • SubDivision Rule, for (a<b<c)

  24. Example  Eval Definite Integral • Find a Value for • The Reduction using the Term-by-Term rule

  25. Example  Def Int by Substitution • Find: • Let: • Then find dx(du) and u(x=0), and u(x=1) ClarifyLimits

  26. Example  Def Int by Substitution • SubOutx2+3x, and the Limits • Dividing out the 2x+3 • Then • Thus Ansx

  27. The Average Value of a Function • At y = yavg there at EQUAL AREAS above & below the Avg-Line Avg Line

  28. % Bruce Mayer, PE % MTH-15 • 24Jul13 % Area_Between_fcn_Graph_BlueGreen_BkGnd_Template_1306.m % Ref: E. B. Magrab, S. Azarm, B. Balachandran, J. H. Duncan, K. E. % Herhold, G. C. Gregory, "An Engineer's Guide to MATLAB", ISBN % 978-0-13-199110-1, Pearson Higher Ed, 2011, pp294-295 % clc; clear % The Function xmin = 0; xmax = 16; ymin = 0; ymax = 350; xct = 1000 x = linspace(xmin,xmax,xct); y1 = .5*x.^3-9*x.^2+11*x+330; yavg = mean(y1) y2 = yavg*ones(1,xct) % % % Find Zeros plot(x,y1, x,y2, 'k','LineWidth', 2), axis([0 xmaxyminymax]),... grid, xlabel('\fontsize{14}x'), ylabel('\fontsize{14}y = f(x)'),... title(['\fontsize{16}MTH15 • Meaning of Avg',]),... annotation('textbox',[.13 .05 .0 .1], 'FitBoxToText', 'on', 'EdgeColor', 'none', 'String', 'Bruce Mayer, PE • 24Jul13','FontSize',7) display('Showing 2Fcn Plot; hit ANY KEY to Continue') % "hold" = Retain current graph when adding new graphs hold on % nct = 500 xn = linspace(xmin, xmax, nct); fill([xn,fliplr(xn)],[.5*xn.^3-9*xn.^2+11*xn+330, fliplr(yavg*ones(1,nct))],'m'),grid plot(x,y1), grid MATLAB Code

  29. The Average Value of a Function • Mathematically - If f is integrable on [a, b], then the average value of fover [a, b] is • Example  Find the Avg Value: • Use Average Definition:

  30. Net Change • If the Rate of Change (RoC), dQ/dx = Q’(x) is continuous over the interval [a,b], then the NET CHANGE in Q(x) is Given by

  31. Example  Find Net Change • A small importer of Gladiator merchandise has modeled her monthly profits since the company was created on January 1, 1997 by the formula • Where • P ≡ $-Profit in 100’s of Dollars ($c or c-Notes) • t ≡ year of operation for the company

  32. Example  Find Net Change • What is the importer’s net change in profit between the beginning of the years 2000 and 2003? • SOLUTION: • Recall t is in years after 1997, Thus • Year 2000 corresponds to t = 3 • Year 2003 corresponds to t = 6 • Then in this case the Net Change in Profit over [3,6] →

  33. Example  Find Net Change • Thus Her monthly profits increased by about $1,354.50 between 2000 & 2003

  34. WhiteBoard Work • Problems From §5.3 • P74 → Water Consumption • P80 → Distance Traveled

  35. All Done for Today StudentsShould Calc

  36. All Done for Today FundamentalTheoremPart-1

  37. Chabot Mathematics Appendix Do On Wht/BlkBorad Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu –

  38. Let area under the curve from a to x. (“a” is a constant) Fundamental Theorem Proof Then:

  39. min f max f h The area of a rectangle drawn under the curve would be less than the actual area under the curve. The area of a rectangle drawn above the curve would be more than the actual area under the curve.

  40. As h gets smaller, min f and max f get closer together. This is the definition of derivative! initial value Take the anti-derivative of both sides to find an explicit formula for area.

  41. As h gets smaller, min f and max f get closer together. Area under curve from a to x = antiderivative at x minus antiderivative at a.

  42. ConCavity Sign Chart ConCavityForm ++++++ −−−−−− −−−−−− ++++++ d2f/dx2 Sign x Critical (Break)Points a b c Inflection NOInflection Inflection

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