1. BMayer@ChabotCollege.edu MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics §5.3 Fundamental Theorem of Calc

2. BMayer@ChabotCollege.edu MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 2 Bruce Mayer, PE Chabot College Mathematics Review §  Any QUESTIONS About §5.2 → AntiDerivatives by Substitution  Any QUESTIONS About HomeWork §5.22 → HW-23 5.2

3. BMayer@ChabotCollege.edu MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 3 Bruce Mayer, PE Chabot College Mathematics §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. BMayer@ChabotCollege.edu MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 4 Bruce Mayer, PE Chabot College Mathematics Area Under the Curve (AUC)  The AUC has many Applications in Business, Science, and Engineering Calculation of Geographic Areas River Channel Cross Section Wind-Force Loading

5. BMayer@ChabotCollege.edu MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 5 Bruce Mayer, PE Chabot College Mathematics 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 pointsRight end pointsMidpoints Strip Width (n strips)

6. BMayer@ChabotCollege.edu MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 6 Bruce Mayer, PE Chabot College Mathematics Example: Strip Sum  Approximate the area under the graph of  Use n = 4 (4 strips) Strip MidPoints

7. BMayer@ChabotCollege.edu MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 7 Bruce Mayer, PE Chabot College Mathematics Example: Strip Sum GamePlan

8. BMayer@ChabotCollege.edu MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 8 Bruce Mayer, PE Chabot College Mathematics MATLAB Code MATLAB Code % 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 = [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 % Now make AREA Plot area(x,y, 'FaceColor', [1.8 1], 'LineWidth', 3), axis([xmin xmax ymin ymax]),... 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),

9. BMayer@ChabotCollege.edu MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 9 Bruce Mayer, PE Chabot College Mathematics MATLAB Code 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 = [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) = 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. BMayer@ChabotCollege.edu MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 10 Bruce Mayer, PE Chabot College Mathematics Example: Strip Sum  The Algebra midpoints

11. BMayer@ChabotCollege.edu MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 11 Bruce Mayer, PE Chabot College Mathematics 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. a b Width: (n strips)

12. BMayer@ChabotCollege.edu MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 12 Bruce Mayer, PE Chabot College Mathematics Area Under a Curve  Function, f(x), on interval [a,b] is: Continuous NonNegative  Then the Area Under the Curve, A  The x 1, x 2, …, x n-1,x n are arbitrary, n SubIntervals each with width (b − a)/n a b

13. BMayer@ChabotCollege.edu MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 13 Bruce Mayer, PE Chabot College Mathematics Riemann Sum ∑f(x k )·∆x  For a Continuous, NonNeg fcn 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. BMayer@ChabotCollege.edu MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 14 Bruce Mayer, PE Chabot College Mathematics Riemann ∑ → Definite Integral  For a Continuous, NonNeg fcn 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. BMayer@ChabotCollege.edu MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 15 Bruce Mayer, PE Chabot College Mathematics Riemann ∑ → Definite Integral  As the No. of Strips increase the AUC Calculation becomes more accurate  The Riemann-Sum to Definite-Integral Twenty StripsFifty Strips

16. BMayer@ChabotCollege.edu MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 16 Bruce Mayer, PE Chabot College Mathematics MATLAB Code 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 = [1/20:1/10:39/20]; y1 = 2*x1.^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) = 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)

17. BMayer@ChabotCollege.edu MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 17 Bruce Mayer, PE Chabot College Mathematics Integration Symbol lower limit of integration upper limit of integration integrand variable of integration (dummy variable) It is called a dummy variable because the answer does not depend on the symbol chosen; it depends only on a&b Definite Integral Symbology

18. BMayer@ChabotCollege.edu MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 18 Bruce Mayer, PE Chabot College Mathematics 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. BMayer@ChabotCollege.edu MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 19 Bruce Mayer, PE Chabot College Mathematics 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. BMayer@ChabotCollege.edu MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 20 Bruce Mayer, PE Chabot College Mathematics 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. BMayer@ChabotCollege.edu MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 21 Bruce Mayer, PE Chabot College Mathematics Example  Find AUC  Find the area under the graph of y = 2x 3  Then Gives the area since 2x 3 is nonnegative on [0, 2]. AntiderivativeFund. Thm. of Calculus

22. BMayer@ChabotCollege.edu MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 22 Bruce Mayer, PE Chabot College Mathematics Rules for Definite Integrals 1.Constant Rule: for any constant, k 2.Sum/Diff Rule: 3.Zero Width Rule 4.Domain Reversal Rule

23. BMayer@ChabotCollege.edu MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 23 Bruce Mayer, PE Chabot College Mathematics Rules for Definite Integrals 5.SubDivision Rule, for (a<b<c)

24. BMayer@ChabotCollege.edu MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 24 Bruce Mayer, PE Chabot College Mathematics Example  Eval Definite Integral  Find a Value for  The Reduction using the Term-by-Term rule

25. BMayer@ChabotCollege.edu MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 25 Bruce Mayer, PE Chabot College Mathematics Example  Def Int by Substitution  Find:  Let:  Then find dx(du) and u(x=0), and u(x=1) Clarify Limits

26. BMayer@ChabotCollege.edu MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 26 Bruce Mayer, PE Chabot College Mathematics Example  Def Int by Substitution  SubOut x 2 +3x, and the Limits  Dividing out the 2x+3  Then  Thus Ans x

27. BMayer@ChabotCollege.edu MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 27 Bruce Mayer, PE Chabot College Mathematics The Average Value of a Function  At y = y avg there at EQUAL AREAS above & below the Avg-Line Avg Line

28. BMayer@ChabotCollege.edu MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 28 Bruce Mayer, PE Chabot College Mathematics MATLAB Code MATLAB Code % 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 xmax ymin ymax]),... 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

29. BMayer@ChabotCollege.edu MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 29 Bruce Mayer, PE Chabot College Mathematics The Average Value of a Function  Mathematically - If f is integrable on [a, b], then the average value of f over [a, b] is  Example  Find the Avg Value:  Use Average Definition:

30. BMayer@ChabotCollege.edu MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 30 Bruce Mayer, PE Chabot College Mathematics 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. BMayer@ChabotCollege.edu MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 31 Bruce Mayer, PE Chabot College Mathematics 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. BMayer@ChabotCollege.edu MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 32 Bruce Mayer, PE Chabot College Mathematics 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. BMayer@ChabotCollege.edu MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 33 Bruce Mayer, PE Chabot College Mathematics Example  Find Net Change  Thus Her monthly profits increased by about $1,354.50 between 2000 & 2003

34. BMayer@ChabotCollege.edu MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 34 Bruce Mayer, PE Chabot College Mathematics WhiteBoard Work  Problems From §5.3 P74 → Water Consumption P80 → Distance Traveled

35. BMayer@ChabotCollege.edu MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 35 Bruce Mayer, PE Chabot College Mathematics All Done for Today Students Should Calc

36. BMayer@ChabotCollege.edu MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 36 Bruce Mayer, PE Chabot College Mathematics All Done for Today Fundamental Theorem Part-1

37. BMayer@ChabotCollege.edu MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 37 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics Appendix –

38. BMayer@ChabotCollege.edu MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 38 Bruce Mayer, PE Chabot College Mathematics Let area under the curve from a to x. (“ a ” is a constant) Then: Fundamental Theorem Proof

39. BMayer@ChabotCollege.edu MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 39 Bruce Mayer, PE Chabot College Mathematics min f max f 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. h

40. BMayer@ChabotCollege.edu MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 40 Bruce Mayer, PE Chabot College Mathematics As h gets smaller, min f and max f get closer together. This is the definition of derivative! Take the anti-derivative of both sides to find an explicit formula for area. initial value

41. BMayer@ChabotCollege.edu MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 41 Bruce Mayer, PE Chabot College Mathematics 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. BMayer@ChabotCollege.edu MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 42 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

43. BMayer@ChabotCollege.edu MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 43 Bruce Mayer, PE Chabot College Mathematics

44. BMayer@ChabotCollege.edu MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 44 Bruce Mayer, PE Chabot College Mathematics

45. BMayer@ChabotCollege.edu MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 45 Bruce Mayer, PE Chabot College Mathematics

46. BMayer@ChabotCollege.edu MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 46 Bruce Mayer, PE Chabot College Mathematics

47. BMayer@ChabotCollege.edu MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 47 Bruce Mayer, PE Chabot College Mathematics