MTH15_Lec-21_sec_4-4_EXP-n-LOG_Applications.pptx 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer Chabot Mathematics §4.4 Exp & Log Applications

MTH15_Lec-21_sec_4-4_EXP-n-LOG_Applications.pptx 2 Bruce Mayer, PE Chabot College Mathematics Review §  Any QUESTIONS About §4.3 → Exp & Log Derivatives  Any QUESTIONS About HomeWork §4.3 → HW

MTH15_Lec-21_sec_4-4_EXP-n-LOG_Applications.pptx 3 Bruce Mayer, PE Chabot College Mathematics §4.4 Learning Goals  Use exponential and logarithmic derivatives in curve sketching  Examine applications involving exponential models

MTH15_Lec-21_sec_4-4_EXP-n-LOG_Applications.pptx 4 Bruce Mayer, PE Chabot College Mathematics 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 whole number

MTH15_Lec-21_sec_4-4_EXP-n-LOG_Applications.pptx 5 Bruce Mayer, PE Chabot College Mathematics Typical Log-Confusion  Beware  Beware that Logs do NOT behave Algebraically. In General:

MTH15_Lec-21_sec_4-4_EXP-n-LOG_Applications.pptx 6 Bruce Mayer, PE Chabot College Mathematics Exponent↔Logarithm Duality  Some Important Implications of the Properties of Logs & Exponents

MTH15_Lec-21_sec_4-4_EXP-n-LOG_Applications.pptx 7 Bruce Mayer, PE Chabot College Mathematics 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 = a x are interchanged Note that y = u x and y = log u x are Mirror images Mirror Line

MTH15_Lec-21_sec_4-4_EXP-n-LOG_Applications.pptx 8 Bruce Mayer, PE Chabot College Mathematics MATLAB Code MATLAB Code % 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 = [xmin xmax]; zyh = [0 0]; zxv = [0 0]; zyv = [ymin ymax]; % % the 6x6 Plot axes; set(gca,'FontSize',12); whitebg([ ]); % Chg Plot BackGround to Blue-Green plot(x1,y1, x2,y2, 'LineWidth', 4),axis([xmin xmax ymin ymax]),... 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',[ ], '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])

MTH15_Lec-21_sec_4-4_EXP-n-LOG_Applications.pptx 9 Bruce Mayer, PE Chabot College Mathematics Recall: Better Graphing GamePlan 1.Find THE y-Intercept, if Any a.Set x = 0, find y b.Only TWO Functions do NOT have a y-intercepts –Of the form 1/x –x = const; x ≠ 0 2.Find x-Intercept(s), if Any a.Set y = 0, find x b.Many functions do NOT have x-intercepts

MTH15_Lec-21_sec_4-4_EXP-n-LOG_Applications.pptx 10 Bruce Mayer, PE Chabot College Mathematics Better Graphing GamePlan 3.Find VERTICAL (↨) Asymptotes, If Any a.Exist ONLY when fcn has a denom b.Set Denom = 0, solve for x –These Values of x are the Vertical Asymptote (VA) Locations 4.Find HORIZONTAL (↔) Asymptotes (HA), If Any a.HA’s Exist ONLY if the fcn has a finite limit-value when x→+∞, or when x→−∞

MTH15_Lec-21_sec_4-4_EXP-n-LOG_Applications.pptx 11 Bruce Mayer, PE Chabot College Mathematics Better Graphing GamePlan b.Find y-value for: –These Values of y are the HA Locations 5.Find the Extrema (Max/Min) Locations a.Set dy/dx = 0, solve for x E b.Find the corresponding y E = f(x E ) c.Determine by 2 nd Derivative, or ConCavity, then test whether (x E, y E ) is a Max or a Min –See Table on Next Slide

MTH15_Lec-21_sec_4-4_EXP-n-LOG_Applications.pptx 12 Bruce Mayer, PE Chabot College Mathematics Better Graphing GamePlan –Determine Max/Min By Concavity 6.Find the Inflection Pt Locations a.Set d 2 y/dx 2 = 0, solve for x i b.Find the corresponding y i = f(x i ) c.Determine by 3 rd Derivative test The Inflection form: ↑-↓ or ↓-↑

MTH15_Lec-21_sec_4-4_EXP-n-LOG_Applications.pptx 13 Bruce Mayer, PE Chabot College Mathematics Better Graphing GamePlan 7.Find the Inflection Pt Locations a.Set d 2 y/dx 2 = 0, solve for x i b.Find the corresponding y i = f(x i ) c.Determine by 3 rd Derivative test The Inflection form: ↑-↓ or ↓- ↑ –Determine Inflection form by 3 rd Derivative

MTH15_Lec-21_sec_4-4_EXP-n-LOG_Applications.pptx 14 Bruce Mayer, PE Chabot College Mathematics Better Graphing GamePlan 8.Sign Charts for Max/Min and ↑-↓/↓-↑ a.To Find the “Flat Spot” behavior for dy/dx = 0, when d 2 y/dx 2 exists, but [d 2 y/dx 2 ] xE = 0 use the Direction-Diagram abc −−−−−−++++++−−−−−− x Slope df/dx Sign Critical (Break) Points MaxNO Max/Min Min

MTH15_Lec-21_sec_4-4_EXP-n-LOG_Applications.pptx 15 Bruce Mayer, PE Chabot College Mathematics Better Graphing GamePlan 9.Sign Charts for Max/Min and ↑-↓/↓-↑ a.To Find the ↑-↑ or ↓-↓ behavior for d 2 y/dx 2 = 0, when d 3 y/dx 3 exists, but [d 3 y/dx 3 ] xi = 0 use the Dome-Diagram abc −−−−−−++++++−−−−−− x ConCavity Form d 2 f/dx 2 Sign Critical (Break) Points InflectionNO Inflection Inflection

MTH15_Lec-21_sec_4-4_EXP-n-LOG_Applications.pptx 16 Bruce Mayer, PE Chabot College Mathematics 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 a.If there are 5000 thousand susceptible individuals in the population, then find the values of constants a and b.

MTH15_Lec-21_sec_4-4_EXP-n-LOG_Applications.pptx 17 Bruce Mayer, PE Chabot College Mathematics Example  Exp Inoculation b.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.

MTH15_Lec-21_sec_4-4_EXP-n-LOG_Applications.pptx 18 Bruce Mayer, PE Chabot College Mathematics Example  Exp Inoculation  Using (x,I) = (0,5)  Now Use (5,0)  But From Before  Substituting

MTH15_Lec-21_sec_4-4_EXP-n-LOG_Applications.pptx 19 Bruce Mayer, PE Chabot College Mathematics Example  Exp Inoculation  Doing the algebra

MTH15_Lec-21_sec_4-4_EXP-n-LOG_Applications.pptx 20 Bruce Mayer, PE Chabot College Mathematics 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

MTH15_Lec-21_sec_4-4_EXP-n-LOG_Applications.pptx 21 Bruce Mayer, PE Chabot College Mathematics 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

MTH15_Lec-21_sec_4-4_EXP-n-LOG_Applications.pptx 22 Bruce Mayer, PE Chabot College Mathematics 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.

MTH15_Lec-21_sec_4-4_EXP-n-LOG_Applications.pptx 23 Bruce Mayer, PE Chabot College Mathematics Example  Logistic Curve  Taking the Second Derivative

MTH15_Lec-21_sec_4-4_EXP-n-LOG_Applications.pptx 24 Bruce Mayer, PE Chabot College Mathematics Example  Logistic Curve  Concavity changes at Inflection-Points when the 2 nd Derivative equals Zero  Because the first two factors are always NonZero, the equation reduces to  Now chk the sign of the 2 nd derivative on either side of 0, at x = −1 & x = 1

MTH15_Lec-21_sec_4-4_EXP-n-LOG_Applications.pptx 25 Bruce Mayer, PE Chabot College Mathematics Example  Logistic Curve  The Sign Tests  The Sign Chart (Dome-Diagram 101 −−−−−− x ConCavity Form d 2 f/dx 2 Sign Critical Point Inflection

MTH15_Lec-21_sec_4-4_EXP-n-LOG_Applications.pptx 26 Bruce Mayer, PE Chabot College Mathematics Example  Logistic Curve  The 2 nd Derivative function is Concave UP for all real no.s less than 0 Concave DOWN for all real no.s greater than 0.  Because the graph changes concavity at x = 0, an inflection point exists at his location.  Next investigate asymptotes.

MTH15_Lec-21_sec_4-4_EXP-n-LOG_Applications.pptx 27 Bruce Mayer, PE Chabot College Mathematics 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

MTH15_Lec-21_sec_4-4_EXP-n-LOG_Applications.pptx 28 Bruce Mayer, PE Chabot College Mathematics 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 NO x-intercepts exist

MTH15_Lec-21_sec_4-4_EXP-n-LOG_Applications.pptx 29 Bruce Mayer, PE Chabot College Mathematics Example  Logistic Curve

MTH15_Lec-21_sec_4-4_EXP-n-LOG_Applications.pptx 30 Bruce Mayer, PE Chabot College Mathematics Example  Logistic Curve  Graphically Horizontal Asymptotes Inflection Point

MTH15_Lec-21_sec_4-4_EXP-n-LOG_Applications.pptx 31 Bruce Mayer, PE Chabot College Mathematics 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

MTH15_Lec-21_sec_4-4_EXP-n-LOG_Applications.pptx 32 Bruce Mayer, PE Chabot College Mathematics 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:

MTH15_Lec-21_sec_4-4_EXP-n-LOG_Applications.pptx 33 Bruce Mayer, PE Chabot College Mathematics Example  Marginal Inoculation

MTH15_Lec-21_sec_4-4_EXP-n-LOG_Applications.pptx 34 Bruce Mayer, PE Chabot College Mathematics Example  Marginal Inoculation  Now du/dx  BackSub e u = 0.9 x & du/dx = ln(0.9)

MTH15_Lec-21_sec_4-4_EXP-n-LOG_Applications.pptx 35 Bruce Mayer, PE Chabot College Mathematics Example  Marginal Inoculation

MTH15_Lec-21_sec_4-4_EXP-n-LOG_Applications.pptx 36 Bruce Mayer, PE Chabot College Mathematics WhiteBoard Work  Problems From §4.4 P36 → Marginal Analysis Special Prob → Sketch Log Fcn

MTH15_Lec-21_sec_4-4_EXP-n-LOG_Applications.pptx 37 Bruce Mayer, PE Chabot College Mathematics All Done for Today Finding Pwr Fcn by Log-Log

MTH15_Lec-21_sec_4-4_EXP-n-LOG_Applications.pptx 38 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer Chabot Mathematics Appendix –

MTH15_Lec-21_sec_4-4_EXP-n-LOG_Applications.pptx 39 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

MTH15_Lec-21_sec_4-4_EXP-n-LOG_Applications.pptx 40 Bruce Mayer, PE Chabot College Mathematics

MTH15_Lec-21_sec_4-4_EXP-n-LOG_Applications.pptx 41 Bruce Mayer, PE Chabot College Mathematics

MTH15_Lec-21_sec_4-4_EXP-n-LOG_Applications.pptx 42 Bruce Mayer, PE Chabot College Mathematics

MTH15_Lec-21_sec_4-4_EXP-n-LOG_Applications.pptx 43 Bruce Mayer, PE Chabot College Mathematics

MTH15_Lec-21_sec_4-4_EXP-n-LOG_Applications.pptx 44 Bruce Mayer, PE Chabot College Mathematics

MTH15_Lec-21_sec_4-4_EXP-n-LOG_Applications.pptx 45 Bruce Mayer, PE Chabot College Mathematics

MTH15_Lec-21_sec_4-4_EXP-n-LOG_Applications.pptx 46 Bruce Mayer, PE Chabot College Mathematics

MTH15_Lec-21_sec_4-4_EXP-n-LOG_Applications.pptx 47 Bruce Mayer, PE Chabot College Mathematics

MTH15_Lec-21_sec_4-4_EXP-n-LOG_Applications.pptx 48 Bruce Mayer, PE Chabot College Mathematics

MTH15_Lec-21_sec_4-4_EXP-n-LOG_Applications.pptx 49 Bruce Mayer, PE Chabot College Mathematics

MTH15_Lec-21_sec_4-4_EXP-n-LOG_Applications.pptx 50 Bruce Mayer, PE Chabot College Mathematics P Graph

MTH15_Lec-21_sec_4-4_EXP-n-LOG_Applications.pptx 51 Bruce Mayer, PE Chabot College Mathematics MATLAB Code MATLAB Code % Bruce Mayer, PE % MTH-15 19Jul13 % XYfcnGraph6x6BlueGreenBkGndTemplate1306.m % % The Limits xmin = 0; xmax = 500; ymin = 0; ymax = 1600; % The FUNCTION x = linspace(xmin,xmax,1000); y = 1000*exp(-x/50).*(x-125); % % The ZERO Lines zxh = [xmin xmax]; zyh = [0 0]; zxv = [0 0]; zyv = [ymin ymax]; % % the 6x6 Plot axes; set(gca,'FontSize',12); whitebg([ ]); % 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 = 1000e^-^x^/^5^0(x-125)'),... title(['\fontsize{16}MTH15 P4.4-36',]),... annotation('textbox',[ ], 'FitBoxToText', 'on', 'EdgeColor', 'none', 'String', 'Bruce Mayer, PE 19JUl13','FontSize',7) hold on set(gca,'XTick',[xmin:50:xmax]); set(gca,'YTick',[ymin:200:ymax])

MTH15_Lec-21_sec_4-4_EXP-n-LOG_Applications.pptx 52 Bruce Mayer, PE Chabot College Mathematics Example  Graph  Use Graphing GamePlane 1.Find y-intercept if it exists 2.Find any x-intercept(s) 3.Use Denom→0 to Check for Vertical Asymptote(s) 4.Use Denom→∞ to Check for Horizontal Asymptote(s) 5.Find max/min pts by dy/dx = 0

MTH15_Lec-21_sec_4-4_EXP-n-LOG_Applications.pptx 53 Bruce Mayer, PE Chabot College Mathematics Example  Graph  Use Graphing GamePlane 6.Find Inflection Points by [d 2 y/(dx) 2 ] = 0 7.Check form of inflection points using 3 rd Derivative Test [d 3 y/(dx) 3 ] InflPts

MTH15_Lec-21_sec_4-4_EXP-n-LOG_Applications.pptx 54 Bruce Mayer, PE Chabot College Mathematics

MTH15_Lec-21_sec_4-4_EXP-n-LOG_Applications.pptx 55 Bruce Mayer, PE Chabot College Mathematics MATLAB Code MATLAB Code % Bruce Mayer, PE % MTH-15 18Jul13 % XYfcnGraph6x6BlueGreenBkGndTemplate1306.m % % The Limits xmin = 0; xmax = 20; ymin = 0; ymax = 4; % The ZERO Lines zxh = [xmin xmax]; zyh = [0 0]; zxv = [.05.05]; zyv = [ymin ymax]; % % the 6x6 Plot axes; set(gca,'FontSize',12); plot(zxv,zyv, 'k', zxh,zyh, 'k', 'LineWidth', 2),axis([xmin xmax ymin ymax]),... grid, annotation('textbox',[ ], 'FitBoxToText', 'on', 'EdgeColor', 'none', 'String', 'Bruce Mayer, PE 18Jul13','FontSize',7) set(gca,'XTick',[xmin:2:xmax]); set(gca,'YTick',[ymin:0.5:ymax])

MTH15_Lec-21_sec_4-4_EXP-n-LOG_Applications.pptx 56 Bruce Mayer, PE Chabot College Mathematics

MTH15_Lec-21_sec_4-4_EXP-n-LOG_Applications.pptx 57 Bruce Mayer, PE Chabot College Mathematics

MTH15_Lec-21_sec_4-4_EXP-n-LOG_Applications.pptx 58 Bruce Mayer, PE Chabot College Mathematics

MTH15_Lec-21_sec_4-4_EXP-n-LOG_Applications.pptx 59 Bruce Mayer, PE Chabot College Mathematics

MTH15_Lec-21_sec_4-4_EXP-n-LOG_Applications.pptx 60 Bruce Mayer, PE Chabot College Mathematics

MTH15_Lec-21_sec_4-4_EXP-n-LOG_Applications.pptx 61 Bruce Mayer, PE Chabot College Mathematics

MTH15_Lec-21_sec_4-4_EXP-n-LOG_Applications.pptx 62 Bruce Mayer, PE Chabot College Mathematics

MTH15_Lec-21_sec_4-4_EXP-n-LOG_Applications.pptx 63 Bruce Mayer, PE Chabot College Mathematics

MTH15_Lec-21_sec_4-4_EXP-n-LOG_Applications.pptx 64 Bruce Mayer, PE Chabot College Mathematics

MTH15_Lec-21_sec_4-4_EXP-n-LOG_Applications.pptx 65 Bruce Mayer, PE Chabot College Mathematics

MTH15_Lec-21_sec_4-4_EXP-n-LOG_Applications.pptx 66 Bruce Mayer, PE Chabot College Mathematics

MTH15_Lec-21_sec_4-4_EXP-n-LOG_Applications.pptx 67 Bruce Mayer, PE Chabot College Mathematics MATLAB Code MATLAB Code % Bruce Mayer, PE % MTH-15 18Jul13 % XYfcnGraph6x6BlueGreenBkGndTemplate1306.m % % The Limits xmin = 0; xmax =16; ymin = 0; ymax = 3; % The FUNCTION x = linspace(xmin,xmax,1000); y = 4*(log(x).^2)./x; % % The ZERO Lines zxh = [xmin xmax]; zyh = [0 0]; zxv = [0 0]; zyv = [ymin ymax]; % % the 6x6 Plot axes; set(gca,'FontSize',12); whitebg([ ]); % 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 = 4ln^2x/x'),... title(['\fontsize{16}MTH15 Sketch ln',]),... annotation('textbox',[ ], 'FitBoxToText', 'on', 'EdgeColor', 'none', 'String', 'BMayer 18Jul13','FontSize',7) hold on set(gca,'XTick',[xmin:2:xmax]); set(gca,'YTick',[ymin:.5:ymax])

MTH15_Lec-21_sec_4-4_EXP-n-LOG_Applications.pptx 68 Bruce Mayer, PE Chabot College Mathematics Example  Exp Inoculation  Using Pts (0,5) & (5,0) in the Model  Simplifying, we can solve the first equation for a and then substitute into the second equation.

MTH15_Lec-21_sec_4-4_EXP-n-LOG_Applications.pptx 69 Bruce Mayer, PE Chabot College Mathematics Example  Exp Inoculation  Running the Numbers  Now Back SubStitute to find a:  Sub the Values of a & b into Model:

MTH15_Lec-21_sec_4-4_EXP-n-LOG_Applications.pptx 70 Bruce Mayer, PE Chabot College Mathematics Example  Exp Inoculation  Then the target level of infection is 2000 People, which translatesto solving the equation I(x) = 2  State: when 2,676 individuals are inoculated, only 2000 will get sick This suggests that even Partial inoculation reduces disease transmission

MTH15_Lec-21_sec_4-4_EXP-n-LOG_Applications.pptx 71 Bruce Mayer, PE Chabot College Mathematics