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

1. Chabot Mathematics §5.1Integration Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

2. 4.4 Review § • Any QUESTIONS About • §4.4 → Exp & Log Math Models • Any QUESTIONS About HomeWork • §4.4 → HW-21

3. §5.1 Learning Goals • Define AntiDerivative • Study and compute indefinite integrals • Explore differential equations and Initial/Boundary value problems • Set up and solve Variable-Separable differential equations

4. Fundamental Theorem of Calculus • 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

5. AntiDifferentiation • Using the 2nd Part of the Theorem • F(x) is called the AntiDerivative of f(x) • Example: Find f(x) when • ONE Answer is • As Verified by

6. Fundamental Property of Antiderivs • The Process of Finding an AntiDerivavite is Called: InDefinite Integration • The Fundamental Property of AntiDerivatives: • If F(x) is an AntiDerivative of the continuous fcnf(x), then any other AntiDerivative of f(x) has the formG(x) = F(x) + C, for some constant C

7. Fundamental Property of Antiderivs • Proof of G(x) = F(x) + C • Assertion: both G(x) & F(x)+C are AntiDerivatives of f(x); that is: • Using DerivativeRules Transitive Property Derivative of a Sum Derivative of a Const 

8. The Indefinite Integral • The family of ALL AntiDerivatives of f(x) is written • The result of ∫f(x)dx is called the indefinite integral of f(x) • Quick Example for: • u(x) has in INFINITE NUMBER of Results, Two Possibilities:

9. The Meaning of “C” • The Constant, C, is the y-axis “Anchor Point” for the “natural Response” fcnF(x) for which C = 0. • C is then the y-intercept of F(x)+C; i.e., • Adding C to F(x) creates a “family” of functions, or curves on the graph, with the SAME SHAPE, but Shifted VERTICALLY on the y-axis

10. The Meaning of “C” Graphically

11. % Bruce Mayer, PE % MTH-15 • 20Jul13 % XYfcnGraph6x6BlueGreenBkGndTemplate1306.m % % The Limits xmin = -4; xmax = 4; ymin = -10; ymax = 20; % The FUNCTION x = linspace(xmin,xmax,1000); y = 7*exp(-x/2.5) + 5*x -8; % % The ZERO Lines zxh = [xminxmax]; zyh = [0 0]; zxv = [0 0]; zyv = [yminymax]; % % the 6x6 Plot axes; set(gca,'FontSize',12); whitebg(['white']) % whitebg([0.8 1 1]); % Chg Plot BackGround to Blue-Green plot(x,y, x,y+9,x,y-pi,x,y+sqrt(13),x,y-7, 'LineWidth', 4),axis([xminxmaxyminymax]),... grid, xlabel('\fontsize{14}x'), ylabel('\fontsize{14}y = G(x) = F(x)+C = 7e^-^5^x^/^2 + 5x - 8 + C'),... title(['\fontsize{16}MTH15 • Familiy of AntiDerivatives',]),... annotation('textbox',[.71 .05 .0 .1], 'FitBoxToText', 'on', 'EdgeColor', 'none', 'String', 'B. Mayer • 20Jul13','FontSize',7) hold on plot(zxv,zyv, 'k', zxh,zyh, 'k', [-1.4995, -1.4995], [ymin,ymax], '--m', 'LineWidth', 2) set(gca,'XTick',[xmin:1:xmax]); set(gca,'YTick',[ymin:5:ymax]) MATLAB Code

12. MuPAD Code Bruce Mayer, PE MTH15 20Jul13 F(x) = 7*exp(-2*x/5) + 5*x -8 f(x) = int(G, x) G := 7*exp(-2*x/5) + 5*x -8 dgdx := diff(G, x) assume(x > -6): xmin := solve(dgdx, x) xminNo := float(xmin) Gmin := subs(G, x = xmin) GminNo := float(Gmin) plot(G, x=-4..4, GridVisible = TRUE,LineWidth = 0.04*unit::inch)

13. Evaluating C by Initial/Boundary • A number can be found for C if the situation provides a value for a SINGLE known point for G(x) → (x, G(x)); e.g., (xn, G(xn)) = (73.2, 4.58) • For Temporal (Time-Based) problems the known point is called the INITIAL Value • Called Initial Value Problems • For Spatial (Distance-Based) problems the known point is called the BOUDARY Value • Called Boundary Value Problems

14. Common Fcn Integration Rules • Constant Rule: for any constant, k • Power Rule:for any n ≠ −1 • Logarithmic Rule:for any x ≠ 0 • Exponential Rule:for any constant, k

15. Integration Algebra Rules • Constant Multiple Rule: For any constant, a • The Sum or Difference Rule: • This often called the Term-by-Term Rule

16. Example  Use the Rules • Find the family of AntiDerivativescorresponding to • SOLUTION: • First Term-by-Term →break up each term over addition and subtraction:

17. Example  Use the Rules • Move out the constant in the 2nd integral (2), and state sqrt as fractional power • Using the Power Rule • CleaningUp →

18. Example  Propensity to Consume • The propensity to consume (PC) is the fraction of income dedicated to spending (as opposed to saving). • A Math Model for the marginal propensity to consume (MPC) for a certain population: • Where • MPC is the rate of change in PC • x is the fraction of income that is disposable.

19. Example  Propensity to Consume • If the propensity to consume is 0.8 when disposable income is 0.92 of total income, find a formula for PC(x) • SOLUTION: • From the Problem Statement that the MPC is a marginal function discern that • Thus the PCfcn is the AntiDerivative of MPC(x)

20. Example  Propensity to Consume • Find PC byIntegrating • This is satisfactory for a general solution, but need the particular solution so that PC(0.92) = 0.8

21. Example  Propensity to Consume • Use the (x,PC) = (0.92,0.8) Boundary Value to Find a NUMBER for the Constant of Integration, C • With C ≈ 1.4, state the particular solution to this Boundary Value Problem

22. Differential Equations (DE’s) • A Differential Equation is an equation that involves differentials or derivatives, and a function that satisfies such an equation is called a solution • A Simple Differential Equation is an equation which includestwo differentials in the formof a derivative

23. Differential Equations (DE’s) • For some function f. Such a Simple Differential Equation can be solved by integrating: • In summary the Solution, y, to a Simple DE can be found by the integration

24. Example  Simple DE • From the Previous Example • As previously solved for the general solution by Integration: • Then used the Boundary Value, (0.92, 0.8), to find the Particular Solution

25. Variable-Separable DE’s • A Variable Separable Differential equation is a differential equation of the form • For some integrable functions f and g • Such a differential equation can be solved by separating the single-variable functions and integrating:

26. Example  Fluid Dynamics • The rate of change in volume (in cubic centimeters) of water in a draining container is proportional to the square root of the depth (in cm) of the water after t seconds, with constant of proportionality 0.044. • Find a model for the volume of water after t seconds, given that initially the container holds 400 cubic centimeters.

27. Example  Fluid Dynamics • SOLUTION: • First, TRANSLATE the written description into an equation: • “rate of change in volume” • “is proportional to thesquare root of volume” • “with constant of proportionality equal to 0.044”

28. Example  Fluid Dynamics • So the (Differential)Equation • Note that the right side does not explicitly depend on t, so we can’t simply integrate with respect to t. • Instead move the expression containing V to the left side: • The Variables are now Separated, allowing simple integration

29. Example  Fluid Dynamics • Integrating • Where • SquaringBoth SidesFind:

30. Example  Fluid Dynamics • For The particular solution find the a number for C using the Initial Value: when t = 0, V = 400 cc: • Sub (0,400) into DE Solution • Thus the volume of water in the Draining Container as a fcn of time:

31. WhiteBoard Work • Problems From §5.1 • P58 → Oil Production(not a Gusher…) • P73 → Car StoppingDistance

32. All Done for Today LOTS moreon DE’sin MTH25

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

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