MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer Chabot Mathematics §7.2 Partial Derivatives

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 2 Bruce Mayer, PE Chabot College Mathematics Review §  Any QUESTIONS About §7.1 → MultiVariable Functions  Any QUESTIONS About HomeWork §7.1 → HW

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 3 Bruce Mayer, PE Chabot College Mathematics §7.2 Learning Goals  Compute and interpret Partial Derivatives  Apply Partial Derivatives to study marginal analysis problems in economics  Compute Second-Order partial derivatives  Use the Chain Rule for partial derivatives to find rates of change and make incremental approximations

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 4 Bruce Mayer, PE Chabot College Mathematics OrdinaryDeriv→PartialDeriv  Recall the Definition of an “Ordinary” Derivative operating on a 1Var Fcn  The “Partial” Derivative of a 2Var Fcn with respect to indep Var x  The “Partial” Derivative of a 2Var Fcn with respect to indep Var y

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 5 Bruce Mayer, PE Chabot College Mathematics Partial Derivative GeoMetry  The “Partials” compute the SLOPE of the Line on the SURFACE where either x or y are held constant (at, say, 19) The partial derivatives of f at (a, b) are the Tangent-Line slopes of the Lines of Constant-y (C 1 ) and Constant-x (C 2 )

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 6 Bruce Mayer, PE Chabot College Mathematics Surface Tangent Line  Consider z = f(x,y) as shown at Right  At the Black Point x = 1.2 inches y = −0.2 inches z = 8 °C ∂z/∂x = −0.31 °C/in  Find the Equation of the Tangent Line

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 7 Bruce Mayer, PE Chabot College Mathematics Surface Tangent Line  SOLUTION  Use the Point Slope Equation  In this case  Use Algebra to Simplify:

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 8 Bruce Mayer, PE Chabot College Mathematics Partial Derivative Practically  SIMPLE RULES FOR FINDING PARTIAL DERIVATIVES OF z=f(x, y)  To find ∂f/∂x, regard y as a constant and differentiate f(x, y) with respect to x y does NOT change →  2. To find ∂f/∂y, regard x as a constant and differentiate f(x, y) with respect to y x does NOT change →

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 9 Bruce Mayer, PE Chabot College Mathematics Example  2Var Exponential  For

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 10 Bruce Mayer, PE Chabot College Mathematics Example  Another Tangent Line  Find Slope for Constant x at (1,1,1)  Then the Slope at (1,1,1)  Then the Line Eqn y&z Change; x does NOT

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 11 Bruce Mayer, PE Chabot College Mathematics Example  Another Tangent Line

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 12 Bruce Mayer, PE Chabot College Mathematics MATLAB Code % Bruce Mayer, PE % MTH-16 19Jan14 % Sec7_2_multi3D_1419.m % clear; clc; clf; % clf clears figure window % % The Domain Limits xmin = -2; xmax = 2; % Weight ymin = -sqrt(2); ymax = sqrt(2); % Height NumPts = 20 % The GRIDs) ************************************** xx = linspace(xmin,xmax,NumPts); yy = linspace(ymin,ymax,NumPts); [x,y]= meshgrid(xx,yy); xp = ones(NumPts); % for PLANE xL = ones(1,NumPts); % for LINE xt = 1; yt =1; zt = 1; % for Tangent POINT % The FUNCTION SkinArea*********************************** z = 4 -(x.^2) - (2*y.^2); % zp = 4-xp.^2-2*y.^2 zL = 5-4*y % % the Plotting Range = 1.05*FcnRange zmin = min(min(z)); zmax = max(max(z)); % the Range Limits R = zmax - zmin; zmid = (zmax + zmin)/2; zpmin = zmid *R/2; zpmax = zmid *R/2; % % the Domain Plot axes; set(gca,'FontSize',12); whitebg([ ]); % Chg Plot BackGround to Blue-Green mesh(x,y,z,'LineWidth', 2),grid, axis([xmin xmax ymin ymax zpmin zpmax]), grid, box,... xlabel('\fontsize{14}x'), ylabel('\fontsize{14}y'), zlabel('\fontsize{14}z = 4 - x^2 - 2y^2'),... title(['\fontsize{16}MTH16 Bruce Mayer, PE',]),... annotation('textbox',[ ], 'FitBoxToText', 'on', 'EdgeColor', 'none', 'String', 'MTH16 Sec7 2 multi3D 1419.m','FontSize',7) % hold on mesh(xp,y,zp,'LineWidth', 7) plot3(xt,yt,zt,'pb', 'MarkerSize', 19, 'MarkerFaceColor', 'b') plot3(xL,y,zL, '-k', 'LineWidth', 11), axis([xmin xmax ymin ymax zpmin zpmax]) % hold off

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 13 Bruce Mayer, PE Chabot College Mathematics ReCall Marginal Analysis  Marginal analysis is used to assist people in allocating their scarce resources to maximize the benefit of the output produced That is, to Simply obtain the most value for the resources used.  What is “Marginal” Marginal means additional, or extra, or incremental (usually ONE added “Unit”)

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 14 Bruce Mayer, PE Chabot College Mathematics Example  Chg in Satisfaction  A Math Model for a utility function, measuring consumer satisfaction with a pair of products: Where x and y are the unit prices of product A and B, respectively, in hecto-Dollars, $h (hundreds of dollars), per item  Use marginal analysis to approximate the change in U if the price of product A decreases by $1, product B decreases by $2, and given that A is currently priced at $30 and B at $50.

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 15 Bruce Mayer, PE Chabot College Mathematics Example  Chg in Satisfaction  SOLUTION:  The Approximate Change, ΔU  Using Differentials ΔUΔU= [Change due to Δx] + [Change due to Δy]

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 16 Bruce Mayer, PE Chabot College Mathematics Example  Chg in Satisfaction  Simplifying ΔU  Now SubStitute in x = $0.30h & Δx = −$0.01h y = $0.50h & Δy = −$0.02h

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 17 Bruce Mayer, PE Chabot College Mathematics Example  Chg in Satisfaction  Thus DROPPING PRICES Product-A: $30→$29  A −1/30 = −3.33% change (a Decrease) Product-B: $50→$48  A −2/50 = −1/25 = −4.00% change (a Decrease)  IMPROVES Customer Satisfaction by “Satisfaction Units”  But…is a LOT, or a little???

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 18 Bruce Mayer, PE Chabot College Mathematics Example  Chg in Satisfaction  Calculate the PreChange, or Original Value of U, U o (x o,y o )  ReCall the Δ% Calculation  Thus the Δ% for U

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 19 Bruce Mayer, PE Chabot College Mathematics Example  Chg in Satisfaction  The Avg Product-Cost = (30+50)/2 = 40  The Avg Price Drop = (1+2)/2 = 1.5  The Price %Decrease = 1.5/40 = 3.75%  Thus 3.75% Price-Drop Improves Customer Satisfaction by only 0.653%; a ratio of 0.653/3.75 = 1/5.74 Why Bother with a Price Cut? It would be better to find ANOTHER way to Improve Satisfaction.

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 20 Bruce Mayer, PE Chabot College Mathematics 2 nd Order Partial Derivatives  If z=f (x, y), use the following notation:

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 21 Bruce Mayer, PE Chabot College Mathematics Clairaut’s Theorem  Consider z = f(x,y) which is defined on over Domain, D, that contains the point (a, b). If the functions ∂ 2 f/∂x∂y and ∂ 2 f/∂y∂x are both continuous on D, then  That is, the “Mixed 2 nd Partials” are EQUAL regardless of Sequencing

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 22 Bruce Mayer, PE Chabot College Mathematics Example  2nd Partials  The last two “mixed” partials are equal as Predicted by Clairaut’s Theorem

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 23 Bruce Mayer, PE Chabot College Mathematics The Chain Rule (Case-I)  Let z=f(x, y) be a differentiable function of x and y, where x=g(t) and y=h(t) and are both differentiable functions of t. Then z is a differentiable function of t such that: Case-I is the More common of the 2 cases

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 24 Bruce Mayer, PE Chabot College Mathematics The Chain Rule (Case-II)  Let z=f(x, y) be a differentiable function of x and y, where x=g(s, t) and y=h(s, t) are differentiable functions of s and t. Then Case-II is the Less common of the 2 cases

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 25 Bruce Mayer, PE Chabot College Mathematics Example  Chain Rule (Case-I)  Let  Then Find dz/dt

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 26 Bruce Mayer, PE Chabot College Mathematics Incremental Approximation  Let z = f(x,y)  Also Let Δx denote a small change in x Δy denote a small change in y,  then the Corresponding change in z is approximated by

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 27 Bruce Mayer, PE Chabot College Mathematics Linearization in 2 Variables  The incremental Approximation Follows from the Mathematical process of Linearization  In 3D, Linearization amounts to finding the Tangent PLANE at some point of interest Note that Two Intersecting Tangent Lines Define the Tangent Plane

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 28 Bruce Mayer, PE Chabot College Mathematics Linearization in 2 Variables  Suppose f has continuous partial derivatives. An equation of the tangent plane to the surface z=f (x,y) at the pt P(x o,y o,z o ) is given by z−z 0 =Σm(u-u 0 )

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 29 Bruce Mayer, PE Chabot College Mathematics Linearization in 2 Variables  Now the Linear Function whose graph is Described by the Tangent Plane  The above Operation is called the LINEARIZATION of f at (a,b)  The Linearization produces the Linear Approximation of f about (a,b)

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 30 Bruce Mayer, PE Chabot College Mathematics Linearization in 2 Variables  In other words, NEAR Pt (a,b)  The Above is called the Linear Approximation or the Tangent Plane Approximation of f at (a,b)  Note that

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 31 Bruce Mayer, PE Chabot College Mathematics ReCall in 2D dx&dy vs Δx&Δy

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 32 Bruce Mayer, PE Chabot College Mathematics in 3D dz vs Δz Linear Approximation

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 33 Bruce Mayer, PE Chabot College Mathematics WhiteBoard Work  Problems From §7.2 P62 → Hybrid AutoMobile Demand

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 34 Bruce Mayer, PE Chabot College Mathematics WhiteBoard Work  Problems From §7.2 P62 → Hybrid AutoMobile Demand

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 35 Bruce Mayer, PE Chabot College Mathematics All Done for Today Partial Derivatives

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 36 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer Chabot Mathematics Appendix –

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 37 Bruce Mayer, PE Chabot College Mathematics

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 38 Bruce Mayer, PE Chabot College Mathematics

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 39 Bruce Mayer, PE Chabot College Mathematics