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

2. 7.1 Review § Any QUESTIONS About Any QUESTIONS About HomeWork
§7.1 → MultiVariable Functions
Any QUESTIONS About HomeWork
§7.1 → HW-03

3. §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

4. 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

5. 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)

6. 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

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

8. 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 →

9. Example  2Var Exponential
For

10. 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

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

12. 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 *R/2; zpmax = zmid *R/2;
% the Domain Plot
axes; set(gca,'FontSize',12);
whitebg([ ]); % 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',[ ], '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

13. 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”)

14. 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.

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

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

17. 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 “Satisfaction Units”
But…is a LOT, or a little???

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

19. 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.

20. 2nd Order Partial Derivatives
If z=f (x, y), use the following notation:

21. 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

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

23. 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

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

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

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

27. 𝜕 𝜕𝑥
𝜕 𝜕𝑦 𝟐

29. Another Chain Rule Example

30. 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

31. 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

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

33. ∆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 =
>> dQdL = 30*0.7*K^(0.3)*L^(-0.3)
dQdL =

34. 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

35. 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)

36. 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)

37. 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

38. 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

39. in 3D dz vs Δz
Linear Approximation

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

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

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

43. Cobb-Douglas Model

48. 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 → PD ChainRule, Incremental Approx., Chain Rule WhtBd Example
File: MTH16_Lec-06_Sp16_sec_7-3_2Var_Optimization.pptx