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

1. Chabot Mathematics §7.2 PartialDerivatives Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

2. 7.1 Review § • Any QUESTIONS About • §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 indepVarx • The “Partial” Derivative of a 2Var Fcn with respect to indepVary

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 fat (a, b) arethe Tangent-Lineslopes of the Linesof 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 Line Eqn y&zChange; x does NOT

11. Example  Another Tangent Line

12. % 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

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

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 +0.00012 “Satisfaction Units” • But…is +0.00012 a LOT, or a little???

18. Example  Chg in Satisfaction • Calculate the PreChange, or Original Value of U, Uo(xo,yo) • ReCall theΔ% 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 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

22. Example  2nd Partials • The last two “mixed” partials are equal asPredicted 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. 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

25. Example  Chain Rule (Case-I) • Let • Then Find dz/dt

26. 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

27. 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

28. 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)

29. 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)

30. 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

31. ReCall in 2D dx&dyvsΔx&Δy

32. in 3D dzvsΔz Linear Approximation

33. WhiteBoard Work • Problems From §7.2 • P62 → Hybrid AutoMobile Demand

34. WhiteBoard Work • Problems From §7.2 • P62 → Hybrid AutoMobile Demand

35. All Done for Today PartialDerivatives

36. Chabot Mathematics Appendix Do On Wht/BlkBorad Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu –