3. The domain of a function of 2 or 3 variables is a set of pts in the plane or space respectively.

Transcription

1 Math 2204 Multivariable Calculus Chapter 14: Partial Derivatives Sec. 14.1: Functions of Several Variables I. Functions and Variables A. Def n : Suppose D is a set of n-tuples of real numbers (x 1, x 2, x 3,, x n ). A real-valued function f on D is a rule that assigns a unique (single) real number w=f(x 1, x 2, x 3,, x n ) to each element in D. The set D is the function s domain. The set of w-values taken on by f is the function s range. The symbol w is the dependent variable of f, and f is said to be a function of the n independent variables x 1 to x n. We also call the x j s the function s input variables and call w the function s output variable. B. One Variable y=f(x) 1. The statement y=f(x) or y is a function of x means y depends on x. 2. x is the independent variable and y is the dependent variable 3. Domain: {x f(x) is defined} Range: {f(x) x domain} 4. Point: (x,y) C. Two Variables z=f(x,y) Def n : A function f of two variables is rule that assigns to each ordered pair of real numbers (x,y) in a set D a unique real number denoted by f(x,y). The set D is the domain of f and its range is the set of values that f takes on, i.e., { f (x, y) (x, y) D} 1. The statement z=f(x,y) or z is a function of x and y means z depends on both x & y. 2. x,y are the independent variables and z is the dependent variable (values assigned independently to x and y) 3. Domain: {(x,y) f(x,y) is defined} Range: {f(x,y) (x,y) domain} 4. Point: (x,y,z) II. Domain A. Definitions concerning functions of one variable 1. Def n : A point (x 0,y 0 ) in a region (set) R in the xy-plane is an interior point of R if it is the center of a disk that lies entirely in R. A point (x 0,y 0 ) is a boundary point of R if every disk centered at (x 0,y 0 ) contains points that lie outside R as well as points that lie in R. (The boundary point itself need not belong to R.) The interior points of a region, as a set, make up the interior of the region. The region s boundary points make up its boundary. A region is open if it consists entirely of interior points. A region is closed if it contains all of its boundary points. 2. Def n : A region in the plane is bounded if it lies inside a disk of fixed radius. A region is unbounded if it is not bounded. 3. The domain of a function of 2 or 3 variables is a set of pts in the plane or space respectively.

2 B. Examples: For the following, find and sketch the domain. Determine if the domain is open, closed or neither, and if the domain id bounded or unbounded. Find the range. 1. f (x, y) = y2 x 2 x 2y 2. f (x, y) = y x 2 3. z = sin xy 4. w = tan 1 ( y x ) 5. f (x, y,z) = 1 x 2 + y 2 + z 2 +1

3 C. Definitions concerning functions of two variables 1. Def n : A closed ball consists of the region of points inside a sphere together with the sphere. An open ball is the region of points inside a sphere without the bounding sphere. 2. Def n : A point (x 0,y 0,z 0 ) in a region D in the space is an interior point of D if it is the center of a ball that lies entirely in D. A point (x 0,y 0,z 0 ) is a boundary point of D if every sphere centered at (x 0,y 0,z 0 ) encloses points that lie outside D as well as points that lie in D. The interior of D is the set of interior points of D. The boundary of D is the set of boundary points of D. A region D is open if it consists entirely of interior points. A region is closed if it contains its entire boundary. D. Examples C=W(t,s) is the wind chill in F based on air temperature t F and wind speed s mph. s t Find the value of W(5,15) and interpret what it means. 2. For what value of s is W(25,s)= -14? For what value of t is W(t,35)= -43? III. Graphs and Level Curves of Functions of Two Variables A. Definitions 1. Def n : The set of points in the plane where a function f(x,y) has a constant value f(x,y)=c is called a level curve of f. The set of all points (x,y,f(x,y)) in space for (x,y) in the domain of f, is called the graph of f. The graph of f is also called the surface z=f(x,y). i.e., A level curve (or contour) for a function z=f(x,y) is the curve of intersection of the surface z=f(x,y) and the horizontal plane z=k. 2. Def n : The set of points (x,y,z) in space where a function of three independent variables has a constant value f(x,y,z)=c is called a level surface of f. B. Note: 1. The equation of a contour (level curve) is f(x,y)=k, plotted in xy coordinate system. 2. All points on a particular contour have the same z value, i.e., f is constant on a level curve. 3. Contours cannot intersect.

4 C. Examples Graph and describe the level curves for the following: 1. f (x, y) = x 2 + y 2 2. f (x, y) = x 3. f (x, y) = y x 2

5 4. f (x, y) = 4x 2 + y f (x, y,z) = 2x + y z that passes through the pt P(2,1,1).