MTH15_Lec-08_sec_2-3_Higher_Order_Derivatives_.pptx 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical.

BMayer@ChabotCollege.edu MTH15_Lec-08_sec_2-3_Higher_Order_Derivatives_.pptx 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics §2.3 Higher Order Derivatives

BMayer@ChabotCollege.edu MTH15_Lec-08_sec_2-3_Higher_Order_Derivatives_.pptx 2 Bruce Mayer, PE Chabot College Mathematics Review §  Any QUESTIONS About §2.2 → Techniques of Differentiation  Any QUESTIONS About HomeWork §2.2 → HW-8 2.2

BMayer@ChabotCollege.edu MTH15_Lec-08_sec_2-3_Higher_Order_Derivatives_.pptx 3 Bruce Mayer, PE Chabot College Mathematics §2.3 Learning Goals  Use the product and quotient rules to find derivatives  Define and study the second derivative and higher-order derivatives

BMayer@ChabotCollege.edu MTH15_Lec-08_sec_2-3_Higher_Order_Derivatives_.pptx 4 Bruce Mayer, PE Chabot College Mathematics Product INequality  The Derivative Defintion (at right) is NONLinear Such That:  In other words, the derivative of a product of functions does NOT EQUAL the Product of the individual Derivatives

BMayer@ChabotCollege.edu MTH15_Lec-08_sec_2-3_Higher_Order_Derivatives_.pptx 5 Bruce Mayer, PE Chabot College Mathematics Example  Product INequality  Compute Similar-Looking Derivatives &  Notice that the two expressions, 5x 4 & 6x 3, are NOT EQUAL

BMayer@ChabotCollege.edu MTH15_Lec-08_sec_2-3_Higher_Order_Derivatives_.pptx 6 Bruce Mayer, PE Chabot College Mathematics Rule Roster – Product Rule  If f(x) and g(x) are differentiable at x, then so is their product, f(x)·g(x), and  Or in LaGrange Notation  The Summary Statement: The 1 st times the Derivative of the 2 nd Plus the 2 nd times the Derivative of the 1 st

BMayer@ChabotCollege.edu MTH15_Lec-08_sec_2-3_Higher_Order_Derivatives_.pptx 7 Bruce Mayer, PE Chabot College Mathematics ProductRule Proof Do On White Board

BMayer@ChabotCollege.edu MTH15_Lec-08_sec_2-3_Higher_Order_Derivatives_.pptx 8 Bruce Mayer, PE Chabot College Mathematics Example  Product Rule on  Compute the Derivative of the Product:  SOLUTION  Let: f(x) = x 2 & g(x) = x 3 in the Product Rule so that:

BMayer@ChabotCollege.edu MTH15_Lec-08_sec_2-3_Higher_Order_Derivatives_.pptx 9 Bruce Mayer, PE Chabot College Mathematics Example  Product Rule on  Or:  This is the SAME as the correct answer in the Previous Example

BMayer@ChabotCollege.edu MTH15_Lec-08_sec_2-3_Higher_Order_Derivatives_.pptx 10 Bruce Mayer, PE Chabot College Mathematics Example  CellPhone Revenue  A Smart Industrial Engineer at Apple © Develops a Model Math Function for the Demand for SmartPhones: Where –D ≡ Phone-Demand in k-Phones –p ≡ Phone-Price in $k

BMayer@ChabotCollege.edu MTH15_Lec-08_sec_2-3_Higher_Order_Derivatives_.pptx 11 Bruce Mayer, PE Chabot College Mathematics Example  CellPhone Revenue  Use the IE’s Demand Model to Find At what rate is revenue changing With Respect To (W.R.T.) price when Selling phones at 0.2 $k ($200 per phone)?  SOLUTION  First construct a revenue function as the product of the price per phone and number of phones sold:

BMayer@ChabotCollege.edu MTH15_Lec-08_sec_2-3_Higher_Order_Derivatives_.pptx 12 Bruce Mayer, PE Chabot College Mathematics Example  CellPhone Revenue  Subbing for D(p) find for R(p): Note that R has units of ($k/Ph)·(kPh) = $M –i.e.; R has units of MegaBucks  Recall the RoC is simply the Derivative Find dR/dp using the product Rule

BMayer@ChabotCollege.edu MTH15_Lec-08_sec_2-3_Higher_Order_Derivatives_.pptx 13 Bruce Mayer, PE Chabot College Mathematics Example  CellPhone Revenue  Engaging the Product Rule

BMayer@ChabotCollege.edu MTH15_Lec-08_sec_2-3_Higher_Order_Derivatives_.pptx 14 Bruce Mayer, PE Chabot College Mathematics Example  CellPhone Revenue  Next determine the rate of change in revenue at a unit price of $200.  In other words need to find dR/dp at a price of $0.2k

BMayer@ChabotCollege.edu MTH15_Lec-08_sec_2-3_Higher_Order_Derivatives_.pptx 15 Bruce Mayer, PE Chabot College Mathematics Example  CellPhone Revenue

BMayer@ChabotCollege.edu MTH15_Lec-08_sec_2-3_Higher_Order_Derivatives_.pptx 16 Bruce Mayer, PE Chabot College Mathematics Example  CellPhone Revenue  The Calculation Shows  Thus we can say that at a Selling Price of $0.2k per phone Revenue will DEcrease $4,000 for every $1 INcrease in the Phone Price

BMayer@ChabotCollege.edu MTH15_Lec-08_sec_2-3_Higher_Order_Derivatives_.pptx 17 Bruce Mayer, PE Chabot College Mathematics Example  CellPhone Revenue RoC (Sensitivity) is Tangent Line Slope

BMayer@ChabotCollege.edu MTH15_Lec-08_sec_2-3_Higher_Order_Derivatives_.pptx 18 Bruce Mayer, PE Chabot College Mathematics Example  CellPhone Revenue $0.1695k $1.2597M

BMayer@ChabotCollege.edu MTH15_Lec-08_sec_2-3_Higher_Order_Derivatives_.pptx 19 Bruce Mayer, PE Chabot College Mathematics MATLAB Code MATLAB Code % Bruce Mayer, PE % MTH-15 05Jul13 % XYfcnGraph6x6BlueGreenBkGndTemplate1306.m % % The Limits xmin = 0; xmax = 0.3; ymin =0; ymax = 1.4; % The FUNCTION x = linspace(xmin,xmax,500); y1 = x.*(12-10*x-100*x.^2); y2 = -4*(x-.2) +1.2 % % 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,y1, 'LineWidth', 4),axis([xmin xmax ymin ymax]),... grid, xlabel('\fontsize{14}p ($k/Ph)'), ylabel('\fontsize{14}R ($M)'),... title(['\fontsize{16}MTH15 CellPh Revenue Sensitivity',]),... annotation('textbox',[.15.05.0.1], 'FitBoxToText', 'on', 'EdgeColor', 'none', 'String', 'XYfcnGraph6x6BlueGreenBkGndTemplate1306.m','FontSize',7) hold on plot(x,y2, '-- m', 0.2,1.2, 'd r', 'MarkerSize', 10,'MarkerFaceColor', 'r', 'LineWidth', 2) set(gca,'XTick',[xmin:.05:xmax]); set(gca,'YTick',[ymin:.2:ymax]) hold off % disp('showning first plot - HIT ANY KEY to continue') pause axes; set(gca,'FontSize',12); whitebg([0.8 1 1]); % Chg Plot BackGround to Blue-Green plot(x,y1, 'LineWidth', 4),axis([xmin xmax ymin ymax]),... grid, xlabel('\fontsize{14}p ($k/Ph)'), ylabel('\fontsize{14}R ($M)'),... title(['\fontsize{16}MTH15 CellPh Max Revenue',]),... annotation('textbox',[.15.05.0.1], 'FitBoxToText', 'on', 'EdgeColor', 'none', 'String', 'XYfcnGraph6x6BlueGreenBkGndTemplate1306.m','FontSize',7) hold on plot([0.1695,0.1695], [0,1.2597], '-- m', [0,0.1695], [1.2597,1.2597], '-- m', 'LineWidth', 2) set(gca,'XTick',[xmin:.05:xmax]); set(gca,'YTick',[ymin:.2:ymax]) % [C,I] = max(y1) x(I)

BMayer@ChabotCollege.edu MTH15_Lec-08_sec_2-3_Higher_Order_Derivatives_.pptx 20 Bruce Mayer, PE Chabot College Mathematics Rule Roster – Quotient Rule  If f(x) and g(x) are differentiable functions with g(x) ≠ 0, then  In particular, the derivative of the quotient of f(x) and g(x) is NOT df/dx divided by dg/dx.

BMayer@ChabotCollege.edu MTH15_Lec-08_sec_2-3_Higher_Order_Derivatives_.pptx 21 Bruce Mayer, PE Chabot College Mathematics QuotientRule Proof Do On White Board

BMayer@ChabotCollege.edu MTH15_Lec-08_sec_2-3_Higher_Order_Derivatives_.pptx 22 Bruce Mayer, PE Chabot College Mathematics Example  RoC in a Population  One population model for deer on an island suggests that t years after initial observation, the population Where P is the fraction of the carrying capacity on the island. –e.g.; P(0) = 2/5 = 0.4, meaning 40% of the Island’s total carrying capacity  Find, and Interpret the Meaning of:

BMayer@ChabotCollege.edu MTH15_Lec-08_sec_2-3_Higher_Order_Derivatives_.pptx 23 Bruce Mayer, PE Chabot College Mathematics Example  RoC in a Population  SOLUTION  The function’s formula is a ratio of expressions containing variables (and there’s no nice way to simplify the fraction), so use the quotient rule:

BMayer@ChabotCollege.edu MTH15_Lec-08_sec_2-3_Higher_Order_Derivatives_.pptx 24 Bruce Mayer, PE Chabot College Mathematics Example  RoC in a Population  Simplifying:  Now need to compute P’(1) and interpret the result

BMayer@ChabotCollege.edu MTH15_Lec-08_sec_2-3_Higher_Order_Derivatives_.pptx 25 Bruce Mayer, PE Chabot College Mathematics Example  RoC in a Population  Units Analysis for dP/dt  Thus the Interpretation of  After 1 year the Deer population is growing at a rate of about 14.06% of the carrying capacity per year.

BMayer@ChabotCollege.edu MTH15_Lec-08_sec_2-3_Higher_Order_Derivatives_.pptx 26 Bruce Mayer, PE Chabot College Mathematics Higher Order Derivatives  Q) What is the Derivative of a Derivative?  A) Just another Function  Quick Example recalling that the 1 st Derivative is just the Slope, m  The Derivative of the Slope is Called the “Curvature” or “Concavity”

BMayer@ChabotCollege.edu MTH15_Lec-08_sec_2-3_Higher_Order_Derivatives_.pptx 27 Bruce Mayer, PE Chabot College Mathematics Higher Order Derivatives  From the Previous Example  Following the Derivation Sequence  If we “fudge” and treat the differentials “d” and “dx” as algebraic quantities…

BMayer@ChabotCollege.edu MTH15_Lec-08_sec_2-3_Higher_Order_Derivatives_.pptx 28 Bruce Mayer, PE Chabot College Mathematics Higher Order Derivatives  Then  Thus  Conventionally (dx) 2 is written as dx 2  Thus if y = f(x) the 2 nd Derivative of y W.R.T. x:

BMayer@ChabotCollege.edu MTH15_Lec-08_sec_2-3_Higher_Order_Derivatives_.pptx 29 Bruce Mayer, PE Chabot College Mathematics Higher Order Derivatives  In general the conventional notation for the n th derivative of y W.R.T. x  Some Examples

BMayer@ChabotCollege.edu MTH15_Lec-08_sec_2-3_Higher_Order_Derivatives_.pptx 30 Bruce Mayer, PE Chabot College Mathematics Higher Order Derivatives  Back to the Previous Example  Then the 2 nd derivative  Then the 3 rd derivative

BMayer@ChabotCollege.edu MTH15_Lec-08_sec_2-3_Higher_Order_Derivatives_.pptx 31 Bruce Mayer, PE Chabot College Mathematics WhiteBoard Work  Problem From §2.3 P54 → Profit Sensitivity With Respect to the Product Production Rate

BMayer@ChabotCollege.edu MTH15_Lec-08_sec_2-3_Higher_Order_Derivatives_.pptx 32 Bruce Mayer, PE Chabot College Mathematics All Done for Today UNconventional Liebniz Notation

BMayer@ChabotCollege.edu MTH15_Lec-08_sec_2-3_Higher_Order_Derivatives_.pptx 33 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics Appendix –

BMayer@ChabotCollege.edu MTH15_Lec-08_sec_2-3_Higher_Order_Derivatives_.pptx 34 Bruce Mayer, PE Chabot College Mathematics

BMayer@ChabotCollege.edu MTH15_Lec-08_sec_2-3_Higher_Order_Derivatives_.pptx 49 Bruce Mayer, PE Chabot College Mathematics Alternative Quotient Rule  Restate Quotient as rational Exponent, then apply Product rule; to whit:  Then  Putting 2 nd term over common denom

MTH15_Lec-08_sec_2-3_Higher_Order_Derivatives_.pptx 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical.

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MTH15_Lec-08_sec_2-3_Higher_Order_Derivatives_.pptx 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical.

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