Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics §3.1 Relative Extrema Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu

2.6 Review § Any QUESTIONS About Any QUESTIONS About HomeWork §2.6 → Implicit Differentiation Any QUESTIONS About HomeWork §2.6 → HW-12

§3.1 Learning Goals Discuss increasing and decreasing functions Define critical points and relative extrema Use the first derivative test to study relative extrema and sketch graphs

Increasing & Decreasing Values A function f is INcreasing if whenever a<b, then: INcreasing is Moving UP from Left→Right A function f is DEcreasing if whenever a<b, then: DEcreasing is Moving DOWN from Left→Right

Inc & Dec Values Graphically DEcreasing INcreasing

Inc & Dec with Derivative If for every c on the interval [a,b] That is, the Slope is POSITIVE Then f is INcreasing on [a,b] If for every c on the interval [a,b] That is, the Slope is NEGATIVE Then f is DEcreasing on [a,b]

Example  Inc & Dec The function, y = f(x),is decreasing on [−2,3] and increasing on [3,8]

Example  Inc & Dec Profit The default list price of a small bookstore’s paperbacks Follows this Formula Where x ≡ The Estimated Sales Volume in No. Books p ≡ The Book Selling-Price in $/book The bookstore buys paperbacks for $1 each, and has daily overhead of $50

Example  Inc & Dec Profit For this Situation Find the: Profit as a function of x Intervals of INcrease and DEcrease for the Profit Function SOLUTION Profit is the difference of revenue and cost, so first determine the revenue as a function of x:

Example  Inc & Dec Profit And now cost as a function of x: Then the Profit is the Revenue minus the Costs:

Example  Inc & Dec Profit Now we turn to determining the intervals of increase and decrease. The graph of the profit function is shown next on the interval [0,100] (where the price and quantity demanded are both non-negative). Profit Curve

Example  Inc & Dec Profit From the Plot Observe that The profit function appears to be increasing until some sales level below 40, and then decreasing thereafter. Although a graph is informative, we turn to calculus to determine the exact intervals Profit Curve

Example  Inc & Dec Profit We know that if the derivative of a function is POSITIVE on an open interval, the function is INCREASING on that interval. Similarly, if the derivative is negative, the function is decreasing So first compute the derivative, or Slope, function:

Example  Inc & Dec Profit On Increasing intervals the Slope is POSTIVE or NonNegative so in this case need Solving This InEquality: The profit function is DEcreasing on the interval [36,100]

Relative Extrema (Max & Min) A relative maximum of a function f is located at a value M such that f(x) ≤ f(M) for all values of x on an interval a<M<b A relative minimum of a function f is located at a value m such that f(x) ≥ f(m) for all values of x on an interval a<m<b

Peaks & Valleys Extrema is precise math terminology for Both of The TOP of a Hill; that is, a PEAK The Bottom of a Trough, That is a VALLEY PEAK PEAK VALLEY VALLEY

AbsoluteMax Relative Max Relative Min Rel&Abs Max& Min Absolute Min

Critical Points Let c be a value in the domain of f Then c is a Critical Point if, and only if HORIZONTAL slope at c VERTICAL slope at c

Critical Points GeoMetrically Horizontal Vertical (0.1695, 1.2597)

MATLAB Code % Bruce Mayer, PE % MTH-15 • 07Jul13 % XYfcnGraph6x6BlueGreenBkGndTemplate1306.m % clear; clc; % The Limits xmin = 0; xmax = 0.27; ymin =0; ymax = 1.3; % The FUNCTION x = linspace(xmin,xmax,1000); y1 = x.*(12-10*x-100*x.^2); % The Max Condition [yHi,I] = max(y1); xHi = x(I); y2 = yHi*ones(1,length(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([0.8 1 1]); % Chg Plot BackGround to Blue-Green plot(x,y1, 'LineWidth', 5),axis([.05 xmax .6 ymax]),... grid, xlabel('\fontsize{14}x'), ylabel('\fontsize{14}y=f(x)'),... title(['\fontsize{16}MTH15 • Zero Critical-Pt',]),... annotation('textbox',[.15 .05 .0 .1], 'FitBoxToText', 'on', 'EdgeColor', 'none', 'String', ' ','FontSize',7) hold on plot(x,y2, '-- m', xHi,yHi, 'd r', 'MarkerSize', 10,'MarkerFaceColor', 'r', 'LineWidth', 2) set(gca,'XTick',[xmin:.05:xmax]); set(gca,'YTick',[ymin:.1:ymax]) hold off MATLAB Code

MATLAB Code % Bruce Mayer, PE % MTH-15 • 23Jun13 % XYfcnGraph6x6BlueGreenBkGndTemplate1306.m % ref: % % The Limits xmin = 0; xmax = 3; ymin = 0; ymax = 20; % The FUNCTION x = linspace(xmin,1.99,1000); y = -1./(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 plot(x,y, 'LineWidth', 4),axis([xmin xmax ymin ymax]),... grid, xlabel('\fontsize{14}x'), ylabel('\fontsize{14}y = f(x)'),... title(['\fontsize{16}MTH15 • \infty Critical-Pt',]),... annotation('textbox',[.51 .05 .0 .1], 'FitBoxToText', 'on', 'EdgeColor', 'none', 'String', ' ','FontSize',7) hold on plot([2 2], [ymin,ymax], '--m', 'LineWidth', 3) set(gca,'XTick',[xmin:0.5:xmax]); set(gca,'YTick',[ymin:2:ymax]) MATLAB Code

Discuss the next 7 slides, slides 23→29 14Mar17 CatchUp Discuss the next 7 slides, slides 23→29

Direction Diagrams (Slope Chart) Use “Direction Diagrams” to determine for Critical Points: MAX, MIN, Neither MAX form MIN form “Neither” forms Direction of Graph ++++++ −−−−−− df/dx Sign Direction of Graph −−−−−− ++++++ df/dx Sign Direction of Graph −−−−−− −−−−−− ++++++ ++++++ df/dx Sign

Example  Critical Numbers Find all critical numbers and classify them as a relative maximum, relative minimum, or neither for The Function:

Example  Critical Numbers SOLUTION Relative extrema can only take place at critical points (but not necessarily all critical points end up being extrema!) Thus we need to find the critical points of f. In other words, values of x so that Think Division by Zero

Example  Critical Numbers For the Zero Critical Point Now need to consider critical points due to the derivative being undefined

Example  Critical Numbers The Derivative Fcn, 𝑓 ′ = 4 − 4/x3 is undefined when x = 0. However, it is very important to note that 0 cannot be the location of a critical point, because 𝑓 is also undefined at 0 In other words, no critical point of a function can exist at c if no point on 𝒇 exists at c

Example  Critical Numbers Use Direction Diagram to Classify the Critical Point at x = 1 Calculating the derivative/slope at a test point to the left of 1 (e.g. x = 0.5) find Similarly for x>1, say 2: → f is DEcreasing → f is INcreasing

Example  Critical Numbers From our Direction Diagram it appears that f has a relative minimum at x = 1. A graph of the function corroborates this assessment. Relative Minimum

Example  Evaluating Temperature The average temperature, in degrees Fahrenheit, in an ice cave t hours after midnight is modeled by: Use the Model to Answer Questions: At what times was the temperature INcreasing? DEcreasing? The cave occupants light a camp stove in order to raise the temperature. At what times is the stove turned on and then off? Note When 𝑡 2 −𝑡+1=0, then 𝑇 𝑡 is Undefined D.N.E → Does Not Exist

Example  Evaluating Temperature SOLUTION: The Temperature “Changes Direction” before & after a Max or Min (Extrema) Thus need to find the Critical Points which give the Location of relative Extrema To find critical points of T, determine values of t such that one these occurs dT/dt = 0 or dT/dt → ±∞ (undefined)

Example  Evaluating Temperature Taking dT/dt: Using the Quotient Rule

Example  Evaluating Temperature Expanding and Simplifying When dT/dt → ∞ The denominator being zero causes the derivative to be undefined however,(t2−t +1)2 is zero exactly when t2−t + 1 is zero, so it results in NO critical values That its, the FUNCTION and the DERIVATIVE both SIMULTANEOUSLY →∞

No Point → No Critical Point Notice the last Statement ReCall that since the original Function is Undefined at 𝑡 2 −𝑡−1=0 then any point where 𝑡 2 −𝑡−1=0 can NOT be a Critical Pt. From the previous slide 𝑑𝑇 𝑑𝑡→∞ when 𝑡 2 −𝑡−1 2 =0, an Undefined Pt Thus No CritPoint if 𝑡 2 −𝑡−1 =0 however,(t2−t +1)2 is zero exactly when t2−t + 1 is zero, so it results in NO critical values

No Point → No Critical Point Solving 𝑡 2 −𝑡+1=0 yields Undefined Points for the ORIGINAL Function, which then can NOT produce critical points at these values Undefined For Orginal Fcn 10𝑇+1 𝑡 2 −𝑡+−1 → 𝑡 2 −𝑡+1=0 → NO valid pts at these values Thus NO Critical Points Exist at these Point as denom = 𝑡 2 −𝑡+1 2 =0 Encircled value are the SAME which are NOT Real Complex Solns! ←Same→

Example  Evaluating Temperature When dT/dt = 0 Numerator = 0 Thus Find: Using the quadratic formula (or a computer algebra system such as MuPAD), find Critical Points

Example  Evaluating Temperature For dT/dt = 0 find: t ≈ −1.15 or t ≈ 0.954 Because T is always continuous (check that the DeNom fcn, (t2−t +1)2 has no real solutions) these are the only two values at which T can change direction Thus Construct a Direction Diagram with Two BreakPoints: t ≈ −1.15 t ≈ +0.954 See Previous Slide

Example  Evaluating Temperature The Direction Diagram We test the derivative function in each of the three regions to determine if T is increasing or decreasing. Testing t = −2 The negative Slope indicates that T is Decreasing left of −1.15

Example  Evaluating Temperature The Direction Diagram Now we test in the second region using t = 0: The positive Slope indicates that T is Increasing to the Right of −1.15

Example  Evaluating Temperature The Direction Diagram Now we test in the second region using t = 1: Again the negative Slope indicates that T is DEcreasing to the Left of 0.954

Example  Evaluating Temperature Coldest Warmest The Completed Slope Direction-Diagram: Can conclude that the function is increasing on the approximate interval (−1.15, 0.954) and decreasing on the intervals (−∞, −1.15) & (0.954, +∞) It appears that the stove was lit around 10:51pm (1.15 hours before midnight) and turned off around 12:57am (0.95 hours after midnight), since these are the relative extrema of the graph.

Example  Evaluating Temperature Graphically Relative Max (Stove OFF) Relative Min (Stove On) Note that there are NO Vertical Asymtotes

MuPAD Plot Code

WhiteBoard Work Problems From §3.1 P40 → Use Calculus to Sketch Graph Similar to P52 → Sketch df/dx for f(x) Graph at right P60 → Machine Tool Depreciation

Critical (Mach) Number All Done for Today Critical (Mach) Number http://www.webelements.com/webelements/elements/text/Ag/xtal.html Ernst Mach

Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics Appendix Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu –

P3.1-40 Hand Sketch

P3.1-40 MuPAD Graph

WhiteBd Graphic for P3.1-52

P3.1-60 MuPAD

Example  Evaluating Temperature Find where 𝑑𝑓 𝑑𝑡→∞ when Solve by Quadratic Eqn as 𝑡 2 −𝑡+1 does not Factor. Using MuPAD For Quadratic Equation