Chapter 11-Functions of Several Variables

Chapter 11-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) = x2+y2, g(x, y) = a/(x2 + y2) and h(x, y, z) = z/(x2 + y2). Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

Combining Functions EXAMPLE: Let f(x, y) = 2x+3y2, g(x, y) = 5+ x3y, and h (x, y, z) = xyz2. Compute (f +g)(1, 2), (f g)(1, 2), (f/g)(1,2), and (1/9 h2)(1,2,3). Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

Graphing Functions of Several Variables DEFINITION: Let (x, y)  f(x, y) be a function of two variables. If c is a constant, then we call the set Lc = {(x, y) : f(x, y) = c} a level set of f. EXAMPLE: Let f(x, y) = x2 + y Calculate and graph the level sets that correspond to horizontal slices at heights 20, 13, 5, 4, and 2. Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

Graphing Functions of Several Variables EXAMPLE: Sketch the graph of f(x, y) = x2 + y. EXAMPLE: Sketch the graph of f(x, y) = y2 - x2. Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

Quick Quiz 1. Describe the domain of 2. When the graph of f (x, y) = x − y2 is sliced with planes that are parallel to the yz-plane, what curves result? 3. Describe the level sets of f (x, y) = x2 − y2. 4. Describe the level sets of F (x, y, z) = x + 2y − 3z. Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

11.2 Cylinders and Quadric Surfaces Chapter 11-Functions of Several Variables Cylinders EXAMPLE: Sketch the set of points in three dimensional space satisfying the equation x2 + 4y2 = 16. Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

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

11.2 Cylinders and Quadric Surfaces Chapter 11-Functions of Several Variables Quadric Surfaces-Ellipsoids EXAMPLE: Sketch the set of points satisfying the equation 4x2 + y2 + 2z2 = 4. Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

11.2 Cylinders and Quadric Surfaces Chapter 11-Functions of Several Variables Quadric Surfaces-Elliptic Cones EXAMPLE: Sketch the set of points satisfying the equation x2 + 2z2 = 2y2. Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

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

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

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

11.2 Cylinders and Quadric Surfaces Chapter 11-Functions of Several Variables Quadric Surfaces-Hyperboloids of Two Sheets EXAMPLE: Sketch the set of points satisfying the equation x2 − 2z2 − 4y2 = 4. Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

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

11.2 Cylinders and Quadric Surfaces Chapter 11-Functions of Several Variables Quadric Surfaces- Hyperbolic Paraboloid EXAMPLE: Sketch the set of points satisfying the equation z = 2y2 − 4x2. Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

11.2 Cylinders and Quadric Surfaces Chapter 11-Functions of Several Variables Quick Quiz 1. The graphs of which of the following equations are cylinders in space? a) x = y2 + z2; b) y = y2 + z2; c) ex+2z = y; d) ex+2z = z2 2. The graphs of which of the following equations are cones in space? a) x2 = y2 − 2z2; b) x2 = y2 + 2z2; c) x = y2 + z2; d) x = y2 − z2 3. The graphs of which of the following equations are hyperboloids of one sheet in space? a) x2 − y2 + 2z2 = −1; b) x2 − y2 + 2z2 = 0; c) x2 − y2 + 2z2 = 1; d) x2 − y2 − 2z2 = 1 4. The graphs of which of the following equations are hyperbolic paraboloids in space? a) x = y2 − 2z2; b) x = y2 + 2z2; c) z2 + y2 = x2; d) z + y2 = x2 Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

11.3 Limits and Continuity Chapter 11-Functions of Several Variables Limits EXAMPLE: Define f(x, y) = x2 + y2. 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). Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

11.3 Limits and Continuity Chapter 11-Functions of Several Variables Rules for Limits EXAMPLE: Define f(x, y) = (x + y + 1) /(x2 − y2). What is the limiting behavior of f as (x, y) tends to (1, 2)? EXAMPLE: Evaluate the limit Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

11.3 Limits and Continuity Chapter 11-Functions of Several Variables Continuity DEFINITION : Suppose that f is a function of two variables that is defined at a point P0 = (x0, y0). If f(x, y) has a limit as (x, y) approaches (x0, y0), and if then we say that f is continuous at P0. If f is not continuous at a point in its domain, then we say that f is discontinuous there. Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

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

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

11.3 Limits and Continuity Chapter 11-Functions of Several Variables Functions of Three Variables EXAMPLE: Show that V (x, y, z) = z3 cos (xy2) is a continuous function. Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

11.4 Partial Derivatives Chapter 11-Functions of Several Variables THEOREM: Let P0 = (x0, y0) be a point in the xy-plane. Suppose that f is a function that is defined on a disk D(P0, r). We say that f is differentiable with respect to x at P0 if exists. We call this limit the partial derivative of f with respect to x at the point P0, and denote it by Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

11.4 Partial Derivatives Chapter 11-Functions of Several Variables EXAMPLE: Calculate fx and fy for the function f defined by f(x, y) = ln (x) ex 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 (px) sin (2t). What is the instantaneous rate of change of y with respect to time at the point x = 1/4? Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

11.4 Partial Derivatives Chapter 11-Functions of Several Variables Functions of Three Variables EXAMPLE: Calculate the partial derivatives of F(x, y, z) = xz sin(y2z) with respect to x, with respect to y, and with respect to z. Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

11.4 Partial Derivatives Chapter 11-Functions of Several Variables Higher Partial Derivatives EXAMPLE: Calculate all the second partial derivatives of f(x, y) = xy − y3 + x2y4. Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

11.5 Differentiability and the Chain Rule Chapter 11-Functions of Several Variables DEFINITION: Suppose that P0 = (x0, y0) is the center of an open disk D(P0, r) on which a function f of two variables is defined. Suppose that fx (P0) and fy (P0) both exist. We say that f is differentiable at the point P0 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. Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

11.5 Differentiability and the Chain Rule Chapter 11-Functions of Several Variables THEOREM: If f is differentiable at P0, then f is continuous at P0. EXAMPLE: Show that the function f defined by is not differentiable at the origin even though both partial derivatives fx(0, 0) and fy(0, 0) exist. Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

11.5 Differentiability and the Chain Rule Chapter 11-Functions of Several Variables THEOREM: Suppose that P0 = (x0, y0) is the center of an open disk D(P0, r) on which a function f of two variables is defined. If both fx (x, y) and fy (x, y) exist and are continuous on D(P0, r), then f is differentiable at P0. In other words, if f is continuously differentiable at P0, then f is differentiable at P0. EXAMPLE: Show that f (x, y) = y/(1 + x2) is differentiable on the entire xy-plane. Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

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

11.5 Differentiability and the Chain Rule Chapter 11-Functions of Several Variables The Chain Rule for a Function of Two Variables Each Depending on Another Variable EXAMPLE: Define z = f(x, y) = x2 + y3, x = sin (s), and y = cos (s). Calculate dz/ds. Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

11.5 Differentiability and the Chain Rule Chapter 11-Functions of Several Variables The Chain Rule for a Function of Two Variables Each Depending on Two Other Variables 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(r(s, t), s(s, t)) is a differentiable function of s and t. Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

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

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

11.5 Differentiability and the Chain Rule Chapter 11-Functions of Several Variables The Chain Rule for a Function of Two Variables Each Depending on Two Other Variables EXAMPLE: Suppose that w = xy4 +y2z, x = s2, y = s1/2, and z = s−1. Calculate dw/ds. Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

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

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

11.5 Differentiability and the Chain Rule Chapter 11-Functions of Several Variables Taylor’s Formula in Several Formulas THEOREM: Suppose that P = (x, y) is a point in a rectangle I that is centered at P0 = (x0, y0). Set h = x−x0 and k = y − y0. If f is twice continuously differentiable on I, then where for some point Q1 on the line segment between P0 and P. Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

11.5 Differentiability and the Chain Rule Chapter 11-Functions of Several Variables Taylor’s Formula in Several Formulas EXAMPLE: Find a quadratic polynomial T2 (x, y) that approximates the function f(x, y) = cos (x) cos (y) near the origin. Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

11.5 Differentiability and the Chain Rule Chapter 11-Functions of Several Variables Quick Quiz 1. True or false: If both fx (P0) and exist fy (P0), then f is continuous at P0. 2. True or false: If f is differentiable at P0, then f is continuous at P0. 3. If fx (−1, 2) = 3, fy (−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. Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

11.6 Gradients and Directional Derivatives Chapter 11-Functions of Several Variables The Directional Derivative DEFINITION: The directional derivative of the function f in the direction u = u1i + u2j at the point P0 is defined to be Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

11.6 Gradients and Directional Derivatives Chapter 11-Functions of Several Variables The Directional Derivative THEOREM: Let P0 be a point in the plane, u = u1i + u2j a unit vector, and f a differentiable function on a disk centered at P0. Then the directional derivative of f at P0 in the direction u is given by the formula EXAMPLE: Let f(x, y) = 1 + 2x + y3. What is the directional derivative of f at P = (2, 1) in the direction from P to Q = (14, 6)? Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

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

11.6 Gradients and Directional Derivatives Chapter 11-Functions of Several Variables The Directions of Greatest Increase and Decrease THEOREM: Suppose that f is a differentiable function for which  f(P0) ≠ 0. Then Du f(P0) is maximal when the unit vector u is the direction of the gradient  f(P0). For this choice of u, the directional derivative is Duf(P0) = || f(P0)||. Also, Du f(P0) is minimal when u is opposite in direction to  f(P0). For this choice of u, the directional derivative is Du f(P0) = −||  f(P0)||. Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

11.6 Gradients and Directional Derivatives Chapter 11-Functions of Several Variables The Directions of Greatest Increase and Decrease EXAMPLE: At the point P0 = (−2, 1), what is the direction that results in the greatest increase for f(x, y) = x2 + y2 and what is the direction of greatest decrease? What are the greatest and least values of the directional derivative at P0? Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

11.6 Gradients and Directional Derivatives Chapter 11-Functions of Several Variables The Gradient and Level Curves THEOREM: Suppose that f is differentiable at P0. Let T be a unit tangent vector to the level curve of f at P0. Then: EXAMPLE: Consider the curve C in the xy-plane that is the graph of the equation x2 + 6y4 = 10. Find the line that is normal to the curve at the point (2, 1). Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

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

11.7 Tangent Planes Chapter 11-Functions of Several Variables DEFINITION: If f is a differentiable function of two variables and (x0, y0) is in its domain, then the tangent plane to the graph of f at (x0, y0, f(x0, y0)) is the plane that passes through the point (x0, y0, f(x0, y0)) and that is normal to the vector fx(x0, y0)i+fy(x0, y0)j−k. We say that the vector fx(x0, y0)i+fy(x0, y0)j−k is normal to the graph of f at the point (x0, y0, f(x0, y0)). Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

11.7 Tangent Planes Chapter 11-Functions of Several Variables EXAMPLE: Find a Cartesian equation of the tangent plane to the graph of f(x, y) = 2x − 3xy3 at the point (2,−1, 10). Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

11.7 Tangent Planes Chapter 11-Functions of Several Variables Level Surfaces THEOREM: If F is a differentiable function of three variables, then F (x0, y0, z0) is perpendicular to the level surface of F at (x0, y0, z0). EXAMPLE: Find the tangent plane to the surface x2 + 4y2 + 8z2 = 13 at the point (1,−1, 1). Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

11.7 Tangent Planes Chapter 11-Functions of Several Variables Normal Lines EXAMPLE: Find symmetric equations for the normal line to the graph of f(x, y) = −y2 − x3 + xy2 at the point (1, 4,−1). Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

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

11.7 Tangent Planes Chapter 11-Functions of Several Variables 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 P0. Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

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

11.8 Maximum-Minimum Problems Chapter 11-Functions of Several Variables Saddle Points EXAMPLE: Locate and analyze the critical points of f(x, y) = x2 − y2. Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

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

11.8 Maximum-Minimum Problems Chapter 11-Functions of Several Variables The Second Derivative Test EXAMPLE: Locate all local maxima, local minima, and saddle points for the function f(x, y) = 2x2 + 3xy + 4y2 − 5x + 2y + 3. EXAMPLE: Locate and identify the critical points of the function f(x, y) = 2x3 − 2y3 − 4xy + 5. Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

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

11.8 Maximum-Minimum Problems Chapter 11-Functions of Several Variables Least Squares Lines THEOREM: Suppose that N is an integer greater than or equal to 2. Given N observations (x1, y1), (x2, y2), , (xN, yN), the least squares line is y = mx + b where and Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

11.8 Maximum-Minimum Problems Chapter 11-Functions of Several Variables Quick Quiz Suppose that f (x, y) is twice continuously differentiable on an open disk centered at P0. What conclusion about the behavior of f (x, y) at P0 can be drawn from the given information. 1. fxx (P0) = 4, fyy (P0) = 9, fxy (P0) = 1 2. fx (P0) = 0, fy (P0) = 0, fxx (P0) = 4, fyy (P0) = 9, fxy (P0) = 6 3. fx (P0) = 0, fy (P0) = 0, fxx (P0) = 4, fyy (P0) = 9, fxy (P0) = 5 4. fx (P0) = 0, fy (P0) = 0, fxx (P0) = 4, fyy (P0) = 9, fxy (P0) = 7 Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

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

11.9 Lagrange Multipliers Chapter 11-Functions of Several Variables Lagrange Multipliers-A Geometric Approach EXAMPLE: Find the point on the hyperbola x2 − y2 = 4 that is nearest to the point (0, 2). Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

11.9 Lagrange Multipliers Chapter 11-Functions of Several Variables 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 l such that  f (P’) = l  g (P’) . EXAMPLE: Find the maximum and minimum values of the function f(x, y) = 2x2 −y2 on the ellipse x2 +2(y−1)2 = 2. Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

11.9 Lagrange Multipliers Chapter 11-Functions of Several Variables A Strategy for Solving Lagrange Multiplier Equations EXAMPLE: Maximize x + 2y subject to the constraint x2 + y2 = 5. A Cautionary Example EXAMPLE: What are the extreme values of f (x, y) = x2 + 2y + 16, subject to the constraint (x + y)2 = 1? Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

11.9 Lagrange Multipliers Chapter 11-Functions of Several Variables 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 P0 for which f (P0) = l  g (P0). 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, x2 + y2 subject to the constraint x + 2y = 5. 4. Minimize, if possible, x2 + y2 subject to the constraint Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

Chapter 11-Functions of Several Variables

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