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Bruce Mayer, PE Licensed Electrical & Mechanical Engineer [email protected]

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§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

% 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 CodeReCall 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

Example Chain Rule (Case-I)

- Let
- Then Find dz/dt

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

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