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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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slide1

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 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

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
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
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

slide36

Chabot Mathematics

Appendix

Do On

Wht/BlkBorad

Bruce Mayer, PE

Licensed Electrical & Mechanical [email protected]

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