Chapter 11-Functions of Several Variables Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John...

Chapter 11-Functions of Several Variables

Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

Chapter 11-Functions of Several Variables11.1 Functions of Several

Variables


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).


11.1 Functions of Several Variables


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).

Combining Functions




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.

Graphing Functions of Several Variables

EXAMPLE: Let f(x, y) = x2 + y2 + 4. Calculate and graph the level sets that correspond to horizontal slices at heights 20, 13, 5, 4, and 2.




EXAMPLE: Sketch the graph of f(x, y) = x2 + y.

Graphing Functions of Several Variables

EXAMPLE: Sketch the graph of f(x, y) = y2 - x2.




EXAMPLE: Discuss the level sets of the function F(x, y, z) = x2 + y2 + z2.

More on Level Sets




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.

Quick Quiz

Chapter 11-Functions of Several Variables11.2 Cylinders and Quadric

Surfaces


EXAMPLE: Sketch the set of points in three dimensional space satisfying the equation x2 + 4y2 = 16.

Cylinders


11.2 Cylinders and Quadric Surfaces


Quadric Surfaces

Ellipsoids




Quadric Surfaces-Ellipsoids

EXAMPLE: Sketch the set of points satisfying the equation 4x2 + y2 + 2z2 = 4.




Quadric Surfaces-Elliptic Cones

EXAMPLE: Sketch the set of points satisfying the equation x2 + 2z2 = 2y2.




Quadric Surfaces-Hyperboloids of One Sheet




Quadric Surfaces-Hyperboloids of One Sheet

EXAMPLE: Sketch the set of points satisfying the equation




Quadric Surfaces-Hyperboloids of Two Sheets




Quadric Surfaces-Hyperboloids of Two Sheets

EXAMPLE: Sketch the set of points satisfying the equation x2 − 2z2 − 4y2 = 4.




Quadric Surfaces- Hyperbolic Paraboloid




Quadric Surfaces- Hyperbolic Paraboloid

EXAMPLE: Sketch the set of points satisfying the equation z = 2y2 − 4x2.




Quick Quiz1. 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 = 14. 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

Chapter 11-Functions of Several Variables11.3 Limits and Continuity


Limits


11.3 Limits and Continuity


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).




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




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 isdiscontinuous there.




Continuity

EXAMPLE: Suppose that

is f continuous at (0, 0)?




Rules for Continuity

EXAMPLE: Discuss the continuity of




Functions of Three Variables

EXAMPLE: Show that V (x, y, z) = z3 cos (xy2) is a continuous function.




Quick Quiz

Chapter 11-Functions of Several Variables11.4 Partial Derivatives


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


11.4 Partial Derivatives





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 (x) sin (2t). What is the instantaneous rate of change of y with respect to time at the point x = 1/4?




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.

Functions of Three Variables




EXAMPLE: Calculate all the second partial derivatives of f(x, y) = xy − y3 + x2y4.

Higher Partial Derivatives




Higher Partial Derivatives




Quick Quiz

Chapter 11-Functions of Several Variables11.5 Differentiability and the

Chain Rule


DEFINITION: Suppose that P0 = (x0, y0) is the center of an open disk D(P0, r) on which a function f of two variablesis 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.


11.5 Differentiability and the Chain Rule


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.




THEOREM: Suppose that P0 = (x0, y0) is the center of an open disk D(P0, r) on which a function f of two variablesis 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.




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




EXAMPLE: Define z = f(x, y) = x2 + y3, x = sin (s), and y = cos (s). Calculate dz/ds.

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. 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




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




THEOREM: Suppose that w = f(x, y, z) is a function of x, y, and z, and that these variables are functions of thevariable s. That is, suppose that there are functions , , and such that x = (s), y = (s), and z = (s). If thefunctions f, , , and are differentiable, then w = f ( (s) , (s), (s)) is a differentiable function of s and





EXAMPLE: Suppose that w = xy4 +y2z, x = s2, y = s1/2, and z = s−1. Calculate dw/ds.





THEOREM: Suppose that w = f(x, y, z) is a function of x, y, and z, and that these variables are functions of thevariables 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 OtherVariables




Furthermore,

The Chain Rule for a Function of Three Variables Each Depending on Two OtherVariables

Schematically, we may write this as




THEOREM: Suppose that P = (x, y) is a point in a rectangle I that is centered at P0 = (x0, y0). Set h = x−x0 andk = y − y0. If f is twice continuously differentiable on I, then

where

for some point Q1 on the line segment between P0 and P.

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.

Taylor’s Formula in Several Formulas




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.

Quick Quiz

Chapter 11-Functions of Several Variables11.6 Gradients and Directional

Derivatives


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


11.6 Gradients and Directional Derivatives


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)?




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 Duf(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




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)||.




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?




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).




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).




Quick Quiz

Chapter 11-Functions of Several Variables11.7 Tangent Planes


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)).


11.7 Tangent Planes


EXAMPLE: Find a Cartesian equation of the tangent plane to the graph of f(x, y) = 2x − 3xy3 at the point (2,−1, 10).


11.7 Tangent Planes


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).

Level Surfaces

EXAMPLE: Find the tangent plane to the surface x2 + 4y2 + 8z2 = 13 at the point (1,−1, 1).


11.7 Tangent Planes


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).


11.7 Tangent Planes


Numerical Approximations Using the Tangent Plane


11.7 Tangent Planes


Numerical Approximations Using the Tangent Plane

DEFINITION: The expression L(x, y) defined by equation below is called the linear approximation (or the tangentplane approximation) to f (x, y) at the point P0.


11.7 Tangent Planes


Quick Quiz

Chapter 11-Functions of Several Variables11.8 Maximum-Minimum

Problems


The Analogue of Fermat’s Theorem


11.8 Maximum-Minimum Problems


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.




Saddle Points

EXAMPLE: Locate and analyze the critical points of f(x, y) = x2 − y2.




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.









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.




Applied Maximum-Minimum Problems

EXAMPLE: A rectangular box, with a top, is to hold 20 cubic inches. The material used to make the top andbottom 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?




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




Quick QuizSuppose that f (x, y) is twice continuously differentiable on an open disk centered at P0. What conclusion aboutthe 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



Chapter 11-Functions of Several Variables11.9 Lagrange Multipliers


Lagrange Multipliers-A Geometric Approach


11.9 Lagrange Multipliers


Lagrange Multipliers-A Geometric Approach

EXAMPLE: Find the point on the hyperbola x2 − y2 = 4 that is nearest to the point (0, 2).




Why the Method of Lagrange Multipliers WorksTHEOREM: (Method of Lagrange Multipliers) Suppose that

(x, y) f (x, y) and (x, y) g (x, y) are differentiablefunctions. Let c be a constant. If f has an extreme value at a point P’ on the constraint curve g (x, y) = cthen 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) = 2x2 −y2 on the ellipse x2 +2(y−1)2 = 2.




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?




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?




Quick Quiz1. 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) = 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 x + 2y = 5.

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