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Chapter 11-Functions of Several Variables Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved.

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1 Chapter 11-Functions of Several Variables Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

2 Chapter 11-Functions of Several Variables 11.1 Functions of Several Variables Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved EXAMPLE: Express the surface area A and volume V of a rectangular box as functions of the side lengths. EXAMPLE: Let a be any constant. Discuss the domains of the functions f(x, y) = x 2 +y 2, g(x, y) = a/(x 2 + y 2 ) and h(x, y, z) = z/(x 2 + y 2 ).

3 Chapter 11-Functions of Several Variables 11.1 Functions of Several Variables Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved EXAMPLE: Let f(x, y) = 2x+3y 2, g(x, y) = 5+ x 3 y, and h (x, y, z) = xyz 2. Compute (f +g)(1, 2), (f g)(1, 2), (f/g)(1,2), and (1/9 h 2 )(1,2,3). Combining Functions

4 Chapter 11-Functions of Several Variables 11.1 Functions of Several Variables Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved DEFINITION: Let (x, y)  f(x, y) be a function of two variables. If c is a constant, then we call the set L c = {(x, y) : f(x, y) = c} a level set of f. Graphing Functions of Several Variables EXAMPLE: Let f(x, y) = x 2 + y Calculate and graph the level sets that correspond to horizontal slices at heights 20, 13, 5, 4, and 2.

5 Chapter 11-Functions of Several Variables 11.1 Functions of Several Variables Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved EXAMPLE: Sketch the graph of f(x, y) = x 2 + y. Graphing Functions of Several Variables EXAMPLE: Sketch the graph of f(x, y) = y 2 - x 2.

6 Chapter 11-Functions of Several Variables 11.1 Functions of Several Variables Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved EXAMPLE: Discuss the level sets of the function F(x, y, z) = x 2 + y 2 + z 2. More on Level Sets

7 Chapter 11-Functions of Several Variables 11.1 Functions of Several Variables Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved 1. Describe the domain of 2. When the graph of f (x, y) = x − y 2 is sliced with planes that are parallel to the yz-plane, what curves result? 3. Describe the level sets of f (x, y) = x 2 − y Describe the level sets of F (x, y, z) = x + 2y − 3z. Quick Quiz

8 Chapter 11-Functions of Several Variables 11.2 Cylinders and Quadric Surfaces Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved EXAMPLE: Sketch the set of points in three dimensional space satisfying the equation x 2 + 4y 2 = 16. Cylinders

9 Chapter 11-Functions of Several Variables 11.2 Cylinders and Quadric Surfaces Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Quadric Surfaces Ellipsoids

10 Chapter 11-Functions of Several Variables 11.2 Cylinders and Quadric Surfaces Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Quadric Surfaces-Ellipsoids EXAMPLE: Sketch the set of points satisfying the equation 4x 2 + y 2 + 2z 2 = 4.

11 Chapter 11-Functions of Several Variables 11.2 Cylinders and Quadric Surfaces Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Quadric Surfaces-Elliptic Cones EXAMPLE: Sketch the set of points satisfying the equation x 2 + 2z 2 = 2y 2.

12 Chapter 11-Functions of Several Variables 11.2 Cylinders and Quadric Surfaces Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Quadric Surfaces-Hyperboloids of One Sheet

13 Chapter 11-Functions of Several Variables 11.2 Cylinders and Quadric Surfaces Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Quadric Surfaces-Hyperboloids of One Sheet EXAMPLE: Sketch the set of points satisfying the equation

14 Chapter 11-Functions of Several Variables 11.2 Cylinders and Quadric Surfaces Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Quadric Surfaces-Hyperboloids of Two Sheets

15 Chapter 11-Functions of Several Variables 11.2 Cylinders and Quadric Surfaces Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Quadric Surfaces-Hyperboloids of Two Sheets EXAMPLE: Sketch the set of points satisfying the equation x 2 − 2z 2 − 4y 2 = 4.

16 Chapter 11-Functions of Several Variables 11.2 Cylinders and Quadric Surfaces Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Quadric Surfaces- Hyperbolic Paraboloid

17 Chapter 11-Functions of Several Variables 11.2 Cylinders and Quadric Surfaces Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Quadric Surfaces- Hyperbolic Paraboloid EXAMPLE: Sketch the set of points satisfying the equation z = 2y 2 − 4x 2.

18 Chapter 11-Functions of Several Variables 11.2 Cylinders and Quadric Surfaces Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Quick Quiz 1. The graphs of which of the following equations are cylinders in space? a) x = y 2 + z 2 ; b) y = y 2 + z 2 ; c) e x+2z = y; d) e x+2z = z 2 2. The graphs of which of the following equations are cones in space? a) x 2 = y 2 − 2z 2 ; b) x 2 = y 2 + 2z 2 ; c) x = y 2 + z 2 ; d) x = y 2 − z 2 3. The graphs of which of the following equations are hyperboloids of one sheet in space? a) x 2 − y 2 + 2z 2 = −1; b) x 2 − y 2 + 2z 2 = 0; c) x 2 − y 2 + 2z 2 = 1; d) x 2 − y 2 − 2z 2 = 1 4. The graphs of which of the following equations are hyperbolic paraboloids in space? a) x = y 2 − 2z 2 ; b) x = y 2 + 2z 2 ; c) z 2 + y 2 = x 2 ; d) z + y 2 = x 2

19 Chapter 11-Functions of Several Variables 11.3 Limits and Continuity Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Limits

20 Chapter 11-Functions of Several Variables 11.3 Limits and Continuity Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Limits EXAMPLE: Define f(x, y) = x 2 + y 2. Verify that lim (x,y)  (0,0) f(x, y) = 0. EXAMPLE: Define Discuss the limiting behavior of f(x, y) as (x, y)  (0, 0).

21 Chapter 11-Functions of Several Variables 11.3 Limits and Continuity Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Rules for Limits EXAMPLE: Define f(x, y) = (x + y + 1) /(x 2 − y 2 ). What is the limiting behavior of f as (x, y) tends to (1, 2)? EXAMPLE: Evaluate the limit

22 Chapter 11-Functions of Several Variables 11.3 Limits and Continuity Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Continuity DEFINITION : Suppose that f is a function of two variables that is defined at a point P 0 = (x 0, y 0 ). If f(x, y) has a limit as (x, y) approaches (x 0, y 0 ), and if then we say that f is continuous at P 0. If f is not continuous at a point in its domain, then we say that f is discontinuous there.

23 Chapter 11-Functions of Several Variables 11.3 Limits and Continuity Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Continuity EXAMPLE: Suppose that is f continuous at (0, 0)?

24 Chapter 11-Functions of Several Variables 11.3 Limits and Continuity Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Rules for Continuity EXAMPLE: Discuss the continuity of

25 Chapter 11-Functions of Several Variables 11.3 Limits and Continuity Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Functions of Three Variables EXAMPLE: Show that V (x, y, z) = z 3 cos (xy 2 ) is a continuous function.

26 Chapter 11-Functions of Several Variables 11.3 Limits and Continuity Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Quick Quiz

27 Chapter 11-Functions of Several Variables 11.4 Partial Derivatives Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved THEOREM: Let P 0 = (x 0, y 0 ) be a point in the xy-plane. Suppose that f is a function that is defined on a disk D(P 0, r). We say that f is differentiable with respect to x at P 0 if exists. We call this limit the partial derivative of f with respect to x at the point P 0, and denote it by

28 Chapter 11-Functions of Several Variables 11.4 Partial Derivatives Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

29 Chapter 11-Functions of Several Variables 11.4 Partial Derivatives Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved EXAMPLE: Calculate f x and f y for the function f defined by f(x, y) = ln (x) e x cos(y). EXAMPLE: A string in the xy-plane vibrates up and down in the y-direction and has endpoints that are fixed at (0, 0) and (1, 0). Suppose that the displacement of the string at point x and time t is given by y (x, t) = sin (  x) sin (2t). What is the instantaneous rate of change of y with respect to time at the point x = 1/4?

30 Chapter 11-Functions of Several Variables 11.4 Partial Derivatives Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved EXAMPLE: Calculate the partial derivatives of F(x, y, z) = xz sin(y 2 z) with respect to x, with respect to y, and with respect to z. Functions of Three Variables

31 Chapter 11-Functions of Several Variables 11.4 Partial Derivatives Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved EXAMPLE: Calculate all the second partial derivatives of f(x, y) = xy − y 3 + x 2 y 4. Higher Partial Derivatives

32 Chapter 11-Functions of Several Variables 11.4 Partial Derivatives Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Higher Partial Derivatives

33 Chapter 11-Functions of Several Variables 11.4 Partial Derivatives Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Quick Quiz

34 Chapter 11-Functions of Several Variables 11.5 Differentiability and the Chain Rule Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved DEFINITION: Suppose that P 0 = (x 0, y 0 ) is the center of an open disk D(P 0, r) on which a function f of two variables is defined. Suppose that f x (P 0 ) and f y (P 0 ) both exist. We say that f is differentiable at the point P 0 if we can express f (x, y) by the formula where and We say that f is differentiable on a set if it is differentiable at each point of the set.

35 Chapter 11-Functions of Several Variables 11.5 Differentiability and the Chain Rule Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved THEOREM: If f is differentiable at P 0, then f is continuous at P 0. EXAMPLE: Show that the function f defined by is not differentiable at the origin even though both partial derivatives f x (0, 0) and f y (0, 0) exist.

36 Chapter 11-Functions of Several Variables 11.5 Differentiability and the Chain Rule Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved THEOREM: Suppose that P 0 = (x 0, y 0 ) is the center of an open disk D(P 0, r) on which a function f of two variables is defined. If both f x (x, y) and f y (x, y) exist and are continuous on D(P 0, r), then f is differentiable at P 0. In other words, if f is continuously differentiable at P 0, then f is differentiable at P 0. EXAMPLE: Show that f (x, y) = y/(1 + x 2 ) is differentiable on the entire xy-plane.

37 Chapter 11-Functions of Several Variables 11.5 Differentiability and the Chain Rule Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved THEOREM: Let z = f(x, y) be a differentiable function of x and y. Suppose that are differentiable functions of s. Then z = f (  (s),  (s)) is a differentiable function of s and When written entirely in terms of variables, the above equation takes the form The Chain Rule for a Function of Two Variables Each Depending on Another Variable

38 Chapter 11-Functions of Several Variables 11.5 Differentiability and the Chain Rule Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved EXAMPLE: Define z = f(x, y) = x 2 + y 3, x = sin (s), and y = cos (s). Calculate dz/ds. The Chain Rule for a Function of Two Variables Each Depending on Another Variable

39 Chapter 11-Functions of Several Variables 11.5 Differentiability and the Chain Rule Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved THEOREM: Let z = f(x, y) be a differentiable function of x and y. Furthermore, assume that are differentiable functions of s and t. Then the composition z = f(  (s, t),  (s, t)) is a differentiable function of s and t. The Chain Rule for a Function of Two Variables Each Depending on Two Other Variables

40 Chapter 11-Functions of Several Variables 11.5 Differentiability and the Chain Rule Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Furthermore, The Chain Rule for a Function of Two Variables Each Depending on Two Other Variables When written entirely in terms of variables, these equations take the form

41 Chapter 11-Functions of Several Variables 11.5 Differentiability and the Chain Rule Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved THEOREM: Suppose that w = f(x, y, z) is a function of x, y, and z, and that these variables are functions of the variable s. That is, suppose that there are functions , , and  such that x =  (s), y =  (s), and z =  (s). If the functions f, , , and  are differentiable, then w = f (  (s),  (s),  (s)) is a differentiable function of s and The Chain Rule for a Function of Two Variables Each Depending on Two Other Variables

42 Chapter 11-Functions of Several Variables 11.5 Differentiability and the Chain Rule Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved EXAMPLE: Suppose that w = xy 4 +y 2 z, x = s 2, y = s 1/2, and z = s −1. Calculate dw/ds. The Chain Rule for a Function of Two Variables Each Depending on Two Other Variables

43 Chapter 11-Functions of Several Variables 11.5 Differentiability and the Chain Rule Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved THEOREM: Suppose that w = f(x, y, z) is a function of x, y, and z, and that these variables are functions of the variables s and t. That is, suppose that there are functions , , and  such that x =  (s, t), y =  (s, t), and z =  (s, t). If the functions f, , , and  are differentiable, then w = f (  (s, t),  (s, t),  (s, t)) is a differentiable function of s and t. The Chain Rule for a Function of Three Variables Each Depending on Two Other Variables

44 Chapter 11-Functions of Several Variables 11.5 Differentiability and the Chain Rule Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Furthermore, The Chain Rule for a Function of Three Variables Each Depending on Two Other Variables Schematically, we may write this as

45 Chapter 11-Functions of Several Variables 11.5 Differentiability and the Chain Rule Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved THEOREM: Suppose that P = (x, y) is a point in a rectangle I that is centered at P 0 = (x 0, y 0 ). Set h = x−x 0 and k = y − y 0. If f is twice continuously differentiable on I, then where for some point Q 1 on the line segment between P 0 and P. Taylor’s Formula in Several Formulas

46 Chapter 11-Functions of Several Variables 11.5 Differentiability and the Chain Rule Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved EXAMPLE: Find a quadratic polynomial T 2 (x, y) that approximates the function f(x, y) = cos (x) cos (y) near the origin. Taylor’s Formula in Several Formulas

47 Chapter 11-Functions of Several Variables 11.5 Differentiability and the Chain Rule Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved 1. True or false: If both f x (P 0 ) and exist f y (P 0 ), then f is continuous at P True or false: If f is differentiable at P 0, then f is continuous at P If f x (−1, 2) = 3, f y (−1, 2) = −5, and z = f (7t − 8, 2t), then what is dz/dt when t = 1? 4. Give a quadratic polynomial that approximates the function x/ (1 + y) near the origin. Quick Quiz

48 Chapter 11-Functions of Several Variables 11.6 Gradients and Directional Derivatives Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved The Directional Derivative DEFINITION: The directional derivative of the function f in the direction u = u 1 i + u 2 j at the point P 0 is defined to be

49 Chapter 11-Functions of Several Variables 11.6 Gradients and Directional Derivatives Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved The Directional Derivative THEOREM: Let P 0 be a point in the plane, u = u 1 i + u 2 j a unit vector, and f a differentiable function on a disk centered at P 0. Then the directional derivative of f at P 0 in the direction u is given by the formula EXAMPLE: Let f(x, y) = 1 + 2x + y 3. What is the directional derivative of f at P = (2, 1) in the direction from P to Q = (14, 6)?

50 Chapter 11-Functions of Several Variables 11.6 Gradients and Directional Derivatives Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved The Gradient EXAMPLE: Let f(x, y) = x sin(y). Calculate  f(x, y). If u = (−3/5)i + (4/5)j then what is D u f(2,  /6)? DEFINITION: Let f be a differentiable function of two variables. The gradient function of f is the vector-valued function  f defined by

51 Chapter 11-Functions of Several Variables 11.6 Gradients and Directional Derivatives Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved The Directions of Greatest Increase and Decrease THEOREM: Suppose that f is a differentiable function for which  f(P 0 ) ≠ 0. Then D u f(P 0 ) is maximal when the unit vector u is the direction of the gradient  f(P 0 ). For this choice of u, the directional derivative is D u f(P 0 ) = ||  f(P 0 )||. Also, D u f(P 0 ) is minimal when u is opposite in direction to  f(P 0 ). For this choice of u, the directional derivative is D u f(P 0 ) = −||  f(P 0 )||.

52 Chapter 11-Functions of Several Variables 11.6 Gradients and Directional Derivatives Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved The Directions of Greatest Increase and Decrease EXAMPLE: At the point P 0 = (−2, 1), what is the direction that results in the greatest increase for f(x, y) = x 2 + y 2 and what is the direction of greatest decrease? What are the greatest and least values of the directional derivative at P 0 ?

53 Chapter 11-Functions of Several Variables 11.6 Gradients and Directional Derivatives Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved The Gradient and Level Curves THEOREM: Suppose that f is differentiable at P 0. Let T be a unit tangent vector to the level curve of f at P 0. Then: EXAMPLE: Consider the curve C in the xy-plane that is the graph of the equation x 2 + 6y 4 = 10. Find the line that is normal to the curve at the point (2, 1).

54 Chapter 11-Functions of Several Variables 11.6 Gradients and Directional Derivatives Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Functions of Three or More Variables EXAMPLE: Find the directions of greatest rate of increase and greatest rate of decrease for the function F(x, y, z) = xyz at the point (−1, 2, 1).

55 Chapter 11-Functions of Several Variables 11.6 Gradients and Directional Derivatives Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Quick Quiz

56 Chapter 11-Functions of Several Variables 11.7 Tangent Planes Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved DEFINITION: If f is a differentiable function of two variables and (x 0, y 0 ) is in its domain, then the tangent plane to the graph of f at (x 0, y 0, f(x 0, y 0 )) is the plane that passes through the point (x 0, y 0, f(x 0, y 0 )) and that is normal to the vector f x (x 0, y 0 )i+f y (x 0, y 0 )j−k. We say that the vector f x (x 0, y 0 )i+f y (x 0, y 0 )j−k is normal to the graph of f at the point (x 0, y 0, f(x 0, y 0 )).

57 Chapter 11-Functions of Several Variables 11.7 Tangent Planes Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved EXAMPLE: Find a Cartesian equation of the tangent plane to the graph of f(x, y) = 2x − 3xy 3 at the point (2,−1, 10).

58 Chapter 11-Functions of Several Variables 11.7 Tangent Planes Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved THEOREM: If F is a differentiable function of three variables, then  F (x 0, y 0, z 0 ) is perpendicular to the level surface of F at (x 0, y 0, z 0 ). Level Surfaces EXAMPLE: Find the tangent plane to the surface x 2 + 4y 2 + 8z 2 = 13 at the point (1,−1, 1).

59 Chapter 11-Functions of Several Variables 11.7 Tangent Planes Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Normal Lines EXAMPLE: Find symmetric equations for the normal line to the graph of f(x, y) = −y 2 − x 3 + xy 2 at the point (1, 4,−1).

60 Chapter 11-Functions of Several Variables 11.7 Tangent Planes Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Numerical Approximations Using the Tangent Plane

61 Chapter 11-Functions of Several Variables 11.7 Tangent Planes Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Numerical Approximations Using the Tangent Plane DEFINITION: The expression L(x, y) defined by equation below is called the linear approximation (or the tangent plane approximation) to f (x, y) at the point P 0.

62 Chapter 11-Functions of Several Variables 11.7 Tangent Planes Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Quick Quiz

63 Chapter 11-Functions of Several Variables 11.8 Maximum-Minimum Problems Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved The Analogue of Fermat’s Theorem

64 Chapter 11-Functions of Several Variables 11.8 Maximum-Minimum Problems Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved The Analogue of Fermat’s Theorem EXAMPLE: Let f(x, y) = 10+(x − 1) 2 +(x − y) 2. Locate all points that might be local extrema for f. Identify what type of critical points these are.

65 Chapter 11-Functions of Several Variables 11.8 Maximum-Minimum Problems Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Saddle Points EXAMPLE: Locate and analyze the critical points of f(x, y) = x 2 − y 2.

66 Chapter 11-Functions of Several Variables 11.8 Maximum-Minimum Problems Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved The Second Derivative Test DEFINITION: Let (x, y)  f (x, y) be a twice continuously differentiable function. The scalar-valued function defined by is called the discriminant of f.

67 Chapter 11-Functions of Several Variables 11.8 Maximum-Minimum Problems Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved The Second Derivative Test

68 Chapter 11-Functions of Several Variables 11.8 Maximum-Minimum Problems Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved The Second Derivative Test EXAMPLE: Locate all local maxima, local minima, and saddle points for the function f(x, y) = 2x 2 + 3xy + 4y 2 − 5x + 2y + 3. EXAMPLE: Locate and identify the critical points of the function f(x, y) = 2x 3 − 2y 3 − 4xy + 5.

69 Chapter 11-Functions of Several Variables 11.8 Maximum-Minimum Problems Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Applied Maximum-Minimum Problems EXAMPLE: A rectangular box, with a top, is to hold 20 cubic inches. The material used to make the top and bottom costs 2 cents per square inch, while the material used to make the front and back and the sides costs 3 cents per square inch. What dimensions will yield the most economical box?

70 Chapter 11-Functions of Several Variables 11.8 Maximum-Minimum Problems Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Least Squares Lines THEOREM: Suppose that N is an integer greater than or equal to 2. Given N observations (x 1, y 1 ), (x 2, y 2 ),..., (x N, y N ), the least squares line is y = mx + b where and

71 Chapter 11-Functions of Several Variables 11.8 Maximum-Minimum Problems Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Quick Quiz Suppose that f (x, y) is twice continuously differentiable on an open disk centered at P 0. What conclusion about the behavior of f (x, y) at P 0 can be drawn from the given information. 1. f xx (P 0 ) = 4, f yy (P 0 ) = 9, f xy (P 0 ) = 1 2. f x (P 0 ) = 0, f y (P 0 ) = 0, f xx (P 0 ) = 4, f yy (P 0 ) = 9, f xy (P 0 ) = 6 3. f x (P 0 ) = 0, f y (P 0 ) = 0, f xx (P 0 ) = 4, f yy (P 0 ) = 9, f xy (P 0 ) = 5 4. f x (P 0 ) = 0, f y (P 0 ) = 0, f xx (P 0 ) = 4, f yy (P 0 ) = 9, f xy (P 0 ) = 7

72 Chapter 11-Functions of Several Variables 11.9 Lagrange Multipliers Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Lagrange Multipliers-A Geometric Approach

73 Chapter 11-Functions of Several Variables 11.9 Lagrange Multipliers Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Lagrange Multipliers-A Geometric Approach EXAMPLE: Find the point on the hyperbola x 2 − y 2 = 4 that is nearest to the point (0, 2).

74 Chapter 11-Functions of Several Variables 11.9 Lagrange Multipliers Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Why the Method of Lagrange Multipliers Works THEOREM: (Method of Lagrange Multipliers) Suppose that (x, y)  f (x, y) and (x, y)  g (x, y) are differentiable functions. Let c be a constant. If f has an extreme value at a point P’ on the constraint curve g (x, y) = c then either  g (P’) = 0 or there is a constant such that  f (P’) =  g (P’). EXAMPLE: Find the maximum and minimum values of the function f(x, y) = 2x 2 −y 2 on the ellipse x 2 +2(y−1) 2 = 2.

75 Chapter 11-Functions of Several Variables 11.9 Lagrange Multipliers Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved A Strategy for Solving Lagrange Multiplier Equations EXAMPLE: Maximize x + 2y subject to the constraint x 2 + y 2 = 5. A Cautionary Example EXAMPLE: What are the extreme values of f (x, y) = x 2 + 2y + 16, subject to the constraint (x + y) 2 = 1?

76 Chapter 11-Functions of Several Variables 11.9 Lagrange Multipliers Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved A Strategy for Solving Lagrange Multiplier Equations EXAMPLE: Maximize x + 2y subject to the constraint x 2 + y 2 = 5. A Cautionary Example EXAMPLE: What are the extreme values of f (x, y) = x 2 + 2y + 16, subject to the constraint (x + y) 2 = 1?

77 Chapter 11-Functions of Several Variables 11.9 Lagrange Multipliers Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Quick Quiz 1. True or false: If f and g are differentiable functions of x and y, then an extreme value of f (x, y) subject to the constraint g (x, y) = c must occur at a point P 0 for which  f (P0) =  g (P 0 ). 2. True or false: If two critical points arise in the solution of a constrained extremum problem, then a maximum occurs at one of the points and a minimum occurs at the other. 3. Maximize, if possible, x 2 + y 2 subject to the constraint x + 2y = Minimize, if possible, x 2 + y 2 subject to the constraint x + 2y = 5.


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