Bruce Mayer, PE Licensed Electrical &amp; Mechanical Engineer BMayer@ChabotCollege.edu

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Chabot Mathematics. §7.2 Partial Derivatives. Bruce Mayer, PE Licensed Electrical &amp; Mechanical Engineer BMayer@ChabotCollege.edu. 7.1. Review §. Any QUESTIONS About §7.1 → MultiVariable Functions Any QUESTIONS About HomeWork §7.1 → HW-03. §7.2 Learning Goals.

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Chabot Mathematics

§7.2 PartialDerivatives

Bruce Mayer, PE

7.1

Review §
• §7.1 → MultiVariable Functions
• §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 indepVarx
• The “Partial” Derivative of a 2Var Fcn with respect to indepVary
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 fat (a, b) arethe Tangent-Lineslopes of the Linesof 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  Another Tangent Line
• Find Slope for Constant x at (1,1,1)
• Then the Slope at (1,1,1)
• Then the Line Eqn

y&zChange; x does NOT

% 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([xminxmaxyminymaxzpminzpmax]), 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([xminxmaxyminymaxzpminzpmax])

%

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
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 theΔ% 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 over 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 asPredicted 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
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 is approximated by
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 IntersectingTangent Lines Definethe 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 ptP(xo,yo,zo) 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)
• The Linearization produces the Linear Approximation of f about (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
in 3D dzvsΔz

Linear Approximation

WhiteBoard Work
• Problems From §7.2
• P62 → Hybrid AutoMobile Demand
WhiteBoard Work
• Problems From §7.2
• P62 → Hybrid AutoMobile Demand
All Done for Today

PartialDerivatives

Chabot Mathematics

Appendix

Do On