MTH15_Lec-12_sec_3-1_Rel_Extrema_.pptx 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer Chabot Mathematics §3.1 Relative Extrema

MTH15_Lec-12_sec_3-1_Rel_Extrema_.pptx 2 Bruce Mayer, PE Chabot College Mathematics Review §  Any QUESTIONS About §2.6 → Implicit Differentiation  Any QUESTIONS About HomeWork §2.6 → HW

MTH15_Lec-12_sec_3-1_Rel_Extrema_.pptx 3 Bruce Mayer, PE Chabot College Mathematics §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

MTH15_Lec-12_sec_3-1_Rel_Extrema_.pptx 4 Bruce Mayer, PE Chabot College Mathematics 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

MTH15_Lec-12_sec_3-1_Rel_Extrema_.pptx 5 Bruce Mayer, PE Chabot College Mathematics Inc & Dec Values Graphically INcreasing DEcreasing

MTH15_Lec-12_sec_3-1_Rel_Extrema_.pptx 6 Bruce Mayer, PE Chabot College Mathematics 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]

MTH15_Lec-12_sec_3-1_Rel_Extrema_.pptx 7 Bruce Mayer, PE Chabot College Mathematics Example Inc & Dec  The function, y = f(x),is decreasing on [−2,3] and increasing on [3,8]

MTH15_Lec-12_sec_3-1_Rel_Extrema_.pptx 8 Bruce Mayer, PE Chabot College Mathematics 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

MTH15_Lec-12_sec_3-1_Rel_Extrema_.pptx 9 Bruce Mayer, PE Chabot College Mathematics Example  Inc & Dec Profit  For this Situation Find: 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:

MTH15_Lec-12_sec_3-1_Rel_Extrema_.pptx 10 Bruce Mayer, PE Chabot College Mathematics Example Inc & Dec Profit  And now cost as a function of x:  Then the Profit is the Revenue minus the Costs:

MTH15_Lec-12_sec_3-1_Rel_Extrema_.pptx 11 Bruce Mayer, PE Chabot College Mathematics 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).

MTH15_Lec-12_sec_3-1_Rel_Extrema_.pptx 12 Bruce Mayer, PE Chabot College Mathematics 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

MTH15_Lec-12_sec_3-1_Rel_Extrema_.pptx 13 Bruce Mayer, PE Chabot College Mathematics 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:

MTH15_Lec-12_sec_3-1_Rel_Extrema_.pptx 14 Bruce Mayer, PE Chabot College Mathematics 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]

MTH15_Lec-12_sec_3-1_Rel_Extrema_.pptx 15 Bruce Mayer, PE Chabot College Mathematics 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

MTH15_Lec-12_sec_3-1_Rel_Extrema_.pptx 16 Bruce Mayer, PE Chabot College Mathematics 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 VALLEY

MTH15_Lec-12_sec_3-1_Rel_Extrema_.pptx 17 Bruce Mayer, PE Chabot College Mathematics Rel&Abs Max& Min Rel&Abs Max& Min Relative Max Absolute Max Relative Min Absolute Min

MTH15_Lec-12_sec_3-1_Rel_Extrema_.pptx 18 Bruce Mayer, PE Chabot College Mathematics 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

MTH15_Lec-12_sec_3-1_Rel_Extrema_.pptx 19 Bruce Mayer, PE Chabot College Mathematics Critical Points GeoMetrically  Horizontal  Vertical (0.1695, )

MTH15_Lec-12_sec_3-1_Rel_Extrema_.pptx 20 Bruce Mayer, PE Chabot College Mathematics MATLAB Code 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([ ]); % 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',[ ], '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

MTH15_Lec-12_sec_3-1_Rel_Extrema_.pptx 21 Bruce Mayer, PE Chabot College Mathematics 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([ ]); % 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',[ ], '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])

MTH15_Lec-12_sec_3-1_Rel_Extrema_.pptx 22 Bruce Mayer, PE Chabot College Mathematics Example  Critical Numbers  Find all critical numbers and classify them as a relative maximum, relative minimum, or neither for The Function:

MTH15_Lec-12_sec_3-1_Rel_Extrema_.pptx 23 Bruce Mayer, PE Chabot College Mathematics 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

MTH15_Lec-12_sec_3-1_Rel_Extrema_.pptx 24 Bruce Mayer, PE Chabot College Mathematics Example Critical Numbers  For the Zero Critical Point  Now need to consider critical points due to the derivative being undefined

MTH15_Lec-12_sec_3-1_Rel_Extrema_.pptx 25 Bruce Mayer, PE Chabot College Mathematics Example  Critical Numbers  The Derivative Fcn, f’ = 4 − 4/x 3 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

MTH15_Lec-12_sec_3-1_Rel_Extrema_.pptx 26 Bruce Mayer, PE Chabot College Mathematics 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

MTH15_Lec-12_sec_3-1_Rel_Extrema_.pptx 27 Bruce Mayer, PE Chabot College Mathematics 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

MTH15_Lec-12_sec_3-1_Rel_Extrema_.pptx 28 Bruce Mayer, PE Chabot College Mathematics 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?

MTH15_Lec-12_sec_3-1_Rel_Extrema_.pptx 29 Bruce Mayer, PE Chabot College Mathematics 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)

MTH15_Lec-12_sec_3-1_Rel_Extrema_.pptx 30 Bruce Mayer, PE Chabot College Mathematics Example Evaluating Temperature  Taking dT/dt:  Using the Quotient Rule

MTH15_Lec-12_sec_3-1_Rel_Extrema_.pptx 31 Bruce Mayer, PE Chabot College Mathematics Example Evaluating Temperature  Expanding and Simplifying  When dT/dt → ∞ The denominator being zero causes the derivative to be undefined –however,(t 2 −t +1) 2 is zero exactly when t 2 −t + 1 is zero, so it results in NO critical values

MTH15_Lec-12_sec_3-1_Rel_Extrema_.pptx 32 Bruce Mayer, PE Chabot College Mathematics Example Evaluating Temperature  When dT/dt = 0  Thus Find:  Using the quadratic formula (or a computer algebra system such as MuPAD), find

MTH15_Lec-12_sec_3-1_Rel_Extrema_.pptx 33 Bruce Mayer, PE Chabot College Mathematics Example  Evaluating Temperature  For dT/dt = 0 find: t ≈ −1.15 or t ≈  Because T is always continuous (check that the DeNom fcn, (t 2 −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 ≈

MTH15_Lec-12_sec_3-1_Rel_Extrema_.pptx 34 Bruce Mayer, PE Chabot College Mathematics 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

MTH15_Lec-12_sec_3-1_Rel_Extrema_.pptx 35 Bruce Mayer, PE Chabot College Mathematics Example Evaluating Temperature  The Direction Diagram  Now we test in the second region using t = 0:  The positive Slope indicates that T is INcreasing

MTH15_Lec-12_sec_3-1_Rel_Extrema_.pptx 36 Bruce Mayer, PE Chabot College Mathematics 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

MTH15_Lec-12_sec_3-1_Rel_Extrema_.pptx 37 Bruce Mayer, PE Chabot College Mathematics Example Evaluating Temperature  The Completed Slope Direction-Diagram:  We 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.

MTH15_Lec-12_sec_3-1_Rel_Extrema_.pptx 38 Bruce Mayer, PE Chabot College Mathematics Example  Evaluating Temperature  Graphically Relative Max (Stove OFF) Relative Min (Stove On)

MTH15_Lec-12_sec_3-1_Rel_Extrema_.pptx 39 Bruce Mayer, PE Chabot College Mathematics MuPAD Plot Code

MTH15_Lec-12_sec_3-1_Rel_Extrema_.pptx 40 Bruce Mayer, PE Chabot College Mathematics 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

MTH15_Lec-12_sec_3-1_Rel_Extrema_.pptx 41 Bruce Mayer, PE Chabot College Mathematics All Done for Today Critical (Mach) Number Ernst Mach

MTH15_Lec-12_sec_3-1_Rel_Extrema_.pptx 42 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer Chabot Mathematics Appendix

MTH15_Lec-12_sec_3-1_Rel_Extrema_.pptx 43 Bruce Mayer, PE Chabot College Mathematics

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MTH15_Lec-12_sec_3-1_Rel_Extrema_.pptx 50 Bruce Mayer, PE Chabot College Mathematics P Hand Sketch

MTH15_Lec-12_sec_3-1_Rel_Extrema_.pptx 51 Bruce Mayer, PE Chabot College Mathematics P MuPAD Graph

MTH15_Lec-12_sec_3-1_Rel_Extrema_.pptx 52 Bruce Mayer, PE Chabot College Mathematics WhiteBd Graphic for P3.1-52

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MTH15_Lec-12_sec_3-1_Rel_Extrema_.pptx 61 Bruce Mayer, PE Chabot College Mathematics P MuPAD

MTH15_Lec-12_sec_3-1_Rel_Extrema_.pptx 62 Bruce Mayer, PE Chabot College Mathematics