Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics §7.2 Partial Derivatives Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu

7.1 Review § Any QUESTIONS About Any QUESTIONS About HomeWork §7.1 → MultiVariable Functions Any QUESTIONS About HomeWork §7.1 → HW-03

§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

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

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 (C1) and Constant-x (C2)

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

Surface Tangent Line SOLUTION Use the Point Slope Equation In this case Use Algebra to Simplify:

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 →

Example  2Var Exponential For

Example  Another Tangent Line Find Slope for Constant x at (1,1,1) Then the Slope at (1,1,1) Then the Tan Line Eqn Slope is DOWNward in a plane parallel to the y-z plane y&z Change; x does NOT

Example  Another Tangent Line (1,1,1) 1

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 - 1.025*R/2; zpmax = zmid + 1.025*R/2; % the Domain Plot axes; set(gca,'FontSize',12); whitebg([0.8 1 1]); % 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',[.73 .05 .0 .1 ], '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 MATLAB Code

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”)

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.

Example  Chg in Satisfaction SOLUTION: The Approximate Change, ΔU Using Differentials ΔU = [Change due to Δx] + [Change due to Δy]

Example  Chg in Satisfaction Simplifying ΔU Now SubStitute in x = $0.30h & Δx = −$0.01h y = $0.50h & Δy = −$0.02h

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 +0.00012 “Satisfaction Units” But…is +0.00012 a LOT, or a little???

Example  Chg in Satisfaction Calculate the PreChange, or Original Value of U, Uo(xo,yo) ReCall the Δ% Calculation Thus the Δ% for U

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.

2nd Order Partial Derivatives If z=f (x, y), use the following notation:

Clairaut’s Theorem Consider z = f(x,y) which is defined on Domain, D, that contains the point (a, b). If the functions ∂2f/∂x∂y and ∂2f/∂y∂x are both continuous on D, then That is, the “Mixed 2nd Partials” are EQUAL regardless of Sequencing

Example  2nd Partials The last two “mixed” partials are equal as Predicted by Clairaut’s Theorem

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

Example  Chain Rule (Case-I) Let Then Find dz/dt → 𝜕𝑧 𝜕𝑥 𝑑𝑥 𝑑𝑡 𝜕𝑧 𝜕𝑦 𝑑𝑥 𝑑𝑡 Recall Quotient Rule

Example  Chain Rule (Case-I) Let Sub Out 𝑥 and 𝑦 to Find dz/dt →

Previous Example In-Detail Do On Doc Cam Find 𝑑𝑧 𝑑𝑡 For:

𝜕 𝜕𝑥 𝜕 𝜕𝑦 𝟐

Another Chain Rule Example

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

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, Δz, is approximated by

∆𝒛 Example based on P7.48 For Δ𝐾=3 & Δ𝐿=2 Cobb-Douglas Model

∆𝒛 Example based on P7.48 Run the Numbers on MATLAB >> K = 630; L = 830; >> dQdK = 30*0.3*K^(-0.7)*L^(0.7) dQdK = 10.9159 >> dQdL = 30*0.7*K^(0.3)*L^(-0.3) dQdL = 19.3329

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

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(x0,y0,z0) is given by z−z0=Σm(u-u0)

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) (constants) The Linearization produces the Linear Approximation of f about point (a,b)

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

ReCall in 2D dx&dy vs Δx&Δy 𝑑𝑦 is a Projection along a Slope While ∆𝑦 is the Difference Between two Known Points If ∆𝑥 is “small” Then 𝑑𝑦≈∆𝑦 dx & Δx are the SAME. dy & Δy are NOT the same

in 3D dz vs Δz Linear Approximation

WhiteBoard Work Problems From §7.2 P62 → Hybrid AutoMobile Demand Chevy Volt WindTunnel Test

All Done for Today Partial Derivatives Google: “third derivative name”

Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics Appendix Wht/Blk Borad Do On Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu –

Cobb-Douglas Model

Game Plan  20Feb17 HandOut Exam-1 Study Guide Note that the Sp14 Exam (the Study Guide) is TOO HARD – The Sp16 edition will be slightly less “Pressure Packed” Do On Board Solution to MTH16_MQ_7-1a.docx File: MTH16_Lec-05_Sp16_sec_7-2_Partial_Derivatives.pptx Slides 23-41 → PD ChainRule, Incremental Approx., Chain Rule WhtBd Example File: MTH16_Lec-06_Sp16_sec_7-3_2Var_Optimization.pptx