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

Chabot Mathematics. §4.4 Exp & Log Applications. Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu. 4.3. Review §. Any QUESTIONS About §4.3 → Exp & Log Derivatives Any QUESTIONS About HomeWork §4.3 → HW-20. §4.4 Learning Goals.

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

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  1. Chabot Mathematics §4.4 Exp & Log Applications Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

  2. 4.3 Review § • Any QUESTIONS About • §4.3 → Exp & Log Derivatives • Any QUESTIONS About HomeWork • §4.3 → HW-20

  3. §4.4 Learning Goals • Use exponential and logarithmic derivatives in curve sketching • Examine applications involving exponential models

  4. Summary of Log Rules • Solving Logarithmic Equations Often Requires the Use of Logarithms Laws • For any positive numbers M, N, and a with a≠ 1,p a wholenumber

  5. Typical Log-Confusion • Beware that Logs do NOT behave Algebraically. In General:

  6. Exponent↔Logarithm Duality • Some Important Implications of the Properties of Logs & Exponents

  7. Alternative Graph: Swap x & y • It will be helpful in later work to be able to graph an equation in which the x and y in y = ax are interchanged • Note that y = ux and y = logux are Mirror images Mirror Line

  8. % Bruce Mayer, PE % MTH-15 • 18Jul13 % XYfcnGraph6x6BlueGreenBkGndTemplate1306.m % ref: % % The Limits xmin = -6; xmax = 6; ymin = -6; ymax = 6; % The FUNCTION x = linspace(xmin,xmax,1000); x1=x; y1=2.3.^x; x2=y1; y2=x; x3=x; y3=x; % % 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, x2,y2, 'LineWidth', 4),axis([xminxmaxyminymax]),... grid, xlabel('\fontsize{14}x, x = 2.3^y'), ylabel('\fontsize{14}y = 2.3^x, y '),... title(['\fontsize{16}MTH15 • y=2.3^x & x = 2.3^y ',]),... annotation('textbox',[.51 .05 .0 .1], 'FitBoxToText', 'on', 'EdgeColor', 'none', 'String', 'Bruce Mayer, PE • 18Jul13','FontSize',7) hold on plot(x3,y3, '--m', zxv,zyv, 'k', zxh,zyh, 'k', 'LineWidth', 2) set(gca,'XTick',[xmin:1:xmax]); set(gca,'YTick',[ymin:1:ymax]) MATLAB Code

  9. Recall: Better Graphing GamePlan • Find THE y-Intercept, if Any • Set x = 0, find y • Only TWO Functions do NOT have a y-intercepts • Of the form 1/x • x = const; x ≠ 0 • Find x-Intercept(s), if Any • Set y = 0, find x • Many functions do NOT have x-intercepts

  10. Better Graphing GamePlan • Find VERTICAL (↨) Asymptotes, If Any • Exist ONLY when fcn has a denom • Set Denom = 0, solve for x • These Values of x are the Vertical Asymptote (VA) Locations • Find HORIZONTAL (↔) Asymptotes (HA), If Any • HA’s Exist ONLY if the fcn has a finite limit-value when x→+∞, or when x→−∞

  11. Better Graphing GamePlan • Find y-value for: • These Values of y are the HA Locations • Find the Extrema (Max/Min) Locations • Set dy/dx = 0, solve for xE • Find the corresponding yE = f(xE) • Determine by 2nd Derivative, or ConCavity, then test whether (xE,yE) is a Max or a Min • See Table on Next Slide

  12. Better Graphing GamePlan • Determine Max/Min By Concavity • Find the Inflection Pt Locations • Set d2y/dx2 = 0, solve for xi • Find the corresponding yi = f(xi) • Determine by 3rd Derivative test The Inflection form: ↑-↓ or ↓-↑

  13. Better Graphing GamePlan • Find the Inflection Pt Locations • Set d2y/dx2 = 0, solve for xi • Find the corresponding yi = f(xi) • Determine by 3rd Derivative test The Inflection form: ↑-↓ or ↓- ↑ • Determine Inflection form by 3rd Derivative

  14. Better Graphing GamePlan • Sign Charts for Max/Min and ↑-↓/↓-↑ • To Find the “Flat Spot” behavior for dy/dx = 0, when d2y/dx2 exists, but [d2y/dx2]xE = 0 use the Direction-Diagram Slope ++++++ −−−−−− −−−−−− ++++++ df/dx Sign x Critical (Break)Points a b c Max NOMax/Min Min

  15. Better Graphing GamePlan • Sign Charts for Max/Min and ↑-↓/↓-↑ • To Find the ↑-↑ or ↓-↓ behavior for d2y/dx2 = 0, when d3y/dx3 exists, but [d3y/dx3]xi = 0 use the Dome-Diagram ConCavityForm ++++++ −−−−−− −−−−−− ++++++ d2f/dx2 Sign x Critical (Break)Points a b c Inflection NOInflection Inflection

  16. Example  Exp Inoculation • In a researcher’s model, inoculating x individuals to a virus suggests kPeople will become infected as • Where a & b are Constants • Find • If there are 5000 thousand susceptible individuals in the population, then find the values of constants a and b.

  17. Example  Exp Inoculation • How many individuals become infected when 2000 are inoculated? • SOLUTION a. • 5000 susceptible individuals could imply that the point (0,5) should be on the graph of the function (no individuals inoculated means all get sick). It also means that if everyone is inoculated, nobody should get sick. In other words, (5,0) is on the graph.

  18. Example  Exp Inoculation • Using (x,I) = (0,5) • Now Use (5,0) • But From Before • Substituting

  19. Example  Exp Inoculation • But From Before • Thus ans a) • SOLUTION b) • Using above to findwhen =2k • Doing the algebra

  20. Example  Logistic Curve • A version of the “Logistic Function” → • Determine where the fcn is increasing & decreasing and where its graph is concave Up & concave Down. • Sketch the graph of the function. Show as many key features as possible • high and low points, points of inflection, vertical/horizontal asymptotes, intercepts, cusps, vertical tangents

  21. Example  Logistic Curve • SOLUTION: • Finding intervals of increase and decrease (along with any relative extrema) can be accomplished using the derivative. • First, rewrite the function in a form avoids the quotient rule • Then

  22. Example  Logistic Curve • Note that df/dx is always positive (each factor is always positive), so the original function is increasing on its entire domain. • This also implies that the function has NO relative extrema. • Now find intervals on which the function is concave up or concave down. • This requires the use of the second derivative.

  23. Example  Logistic Curve • Taking the Second Derivative

  24. Example  Logistic Curve • Concavity changes at Inflection-Points when the 2nd Derivative equals Zero • Because the first two factors are always NonZero, the equation reduces to • Now chk the sign of the 2nd derivative on either side of 0, at x = −1 & x = 1

  25. Example  Logistic Curve • TheSignTests • The Sign Chart (Dome-Diagram ConCavityForm ++++++ −−−−−− d2f/dx2 Sign Inflection x 1 0 1 Critical Point

  26. Example  Logistic Curve • The 2nd Derivative function is • Concave UP for all real no.sless than 0 • Concave DOWN for all real no.sgreater than 0. • Because the graph changes concavity at x = 0, an inflection point exists at his location. • Next investigate asymptotes.

  27. Example  Logistic Curve • Because the function has no errors (Div-by-Zero) in its domain, conclude that there are NO vertical asymptotes • Letting x→±∞ reveals TWO horizontal Asymptotes • Thus Have Horizontal Asymptotes at • y = 0 • y = 5

  28. Example  Logistic Curve • Check for y-intercept at x = 0 • Have y-intercept at (0, 2.5) • Check for x-intercept at y = 0 • This CONTRADICTION (5=0) means that there is NO soln to the eqn, and thus NOx-intercepts exist

  29. Example  Logistic Curve • Finally, to find any cusps or vertical tangents, look for those values of x where the derivative function is undefined. Recall df/dx • UnDefinition Occurs when the Divisor Equals Zero, or: • But Since and are ALWAYS Positive this eqn has NO Solutions, so the fcn has no Cusps or Vertical Tangents

  30. Example  Logistic Curve • Graphically Horizontal Asymptotes Inflection Point

  31. Example  Marginal Inoculation • Consider the inoculation function from the Previous Example • Use marginal/incremental analysis to estimate the change in the number of infected individuals when increasing the number of inoculated person from 1000 to 1010

  32. Example  Marginal Inoculation • SOLUTION: • ReCall Marginal analysis is the process of using the derivative to predict change in a function in the short run. Recall that for a function f(x), value a, and small number ∆x; to Whit: • In this case with x in kPeople, estimate:

  33. Example  Marginal Inoculation • Calc : • Next Find : • Now Let0.9x = eu

  34. Example  Marginal Inoculation • Now du/dx • BackSubeu= 0.9x & du/dx = ln(0.9)

  35. Example  Marginal Inoculation • Find at x = 1.01 • By marginal analysis the Estimated value at an inoculation level of 1010 • The estimated number of infected is 3,767 using marginal analysis.

  36. WhiteBoard Work • Problems From §4.4 • P36 → Marginal Analysis • Special Prob → Sketch Log Fcn

  37. All Done for Today FindingPwrFcn by Log-Log

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

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

  40. P4.4-36 Graph

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