Chabot Mathematics
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Chabot Mathematics. §7.2 Partial Derivatives. Bruce Mayer, PE Licensed Electrical & Mechanical Engineer [email protected] 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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Bruce mayer pe licensed electrical mechanical engineer bmayer chabotcollege edu

Chabot Mathematics

§7.2 PartialDerivatives

Bruce Mayer, PE

Licensed Electrical & Mechanical [email protected]


Review

7.1

Review §

  • Any QUESTIONS About

    • §7.1 → MultiVariable Functions

  • Any QUESTIONS About HomeWork

    • §7.1 → HW-03


7 2 learning goals

§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

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

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

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 line1

Surface Tangent Line

  • SOLUTION

  • Use the Point Slope Equation

  • In this case

  • Use Algebra to Simplify:


Partial derivative practically

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

Example  2Var Exponential

  • For


Example another tangent line

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


Example another tangent line1

Example  Another Tangent Line


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([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

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

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 satisfaction1

Example  Chg in Satisfaction

  • SOLUTION:

  • The Approximate Change, ΔU

  • Using Differentials


Example chg in satisfaction2

Example  Chg in Satisfaction

  • Simplifying ΔU

  • Now SubStitute in

    • x = $0.30h & Δx = −$0.01h

    • y = $0.50h & Δy = −$0.02h


Example chg in satisfaction3

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 satisfaction4

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 satisfaction5

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.


2 nd order partial derivatives

2nd Order Partial Derivatives

  • If z=f (x, y), use the following notation:


Clairaut s theorem

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 2 nd partials

Example  2nd Partials

  • The last two “mixed” partials are equal asPredicted by Clairaut’s Theorem


The chain rule case i

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

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


Example chain rule case i

Example  Chain Rule (Case-I)

  • Let

  • Then Find dz/dt


Incremental approximation

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

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 variables1

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 variables2

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 variables3

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 x y

ReCall in 2D dx&dyvsΔx&Δy


In 3d dz vs z

in 3D dzvsΔz

Linear Approximation


Whiteboard work

WhiteBoard Work

  • Problems From §7.2

    • P62 → Hybrid AutoMobile Demand


Whiteboard work1

WhiteBoard Work

  • Problems From §7.2

    • P62 → Hybrid AutoMobile Demand


All done for today

All Done for Today

PartialDerivatives


Bruce mayer pe licensed electrical mechanical engineer bmayer chabotcollege edu

Chabot Mathematics

Appendix

Do On

Wht/BlkBorad

Bruce Mayer, PE

Licensed Electrical & Mechanical [email protected]


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