Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Functions of Several Variables Lecture 21 November 6, 2006 Lecture 21 Functions of Several Variables Functions of two variables Lecture 21 Functions of Several Variables Functions of two variables A function of two variables is a 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 ). Lecture 21 Functions of Several Variables Functions of two variables A function of two variables is a 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. Lecture 21 Functions of Several Variables Functions of two variables A function of two variables is a 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. We also write z = f (x , y ) Lecture 21 Functions of Several Variables Functions of two variables A function of two variables is a 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. We also write z = f (x , y ) The variables x and y are independent variables and z is the dependent variable. Lecture 21 Functions of Several Variables Examples Examples Lecture 21 Functions of Several Variables Examples Examples Find the domain of the function f (x , y ) = Lecture 21 x 2x + 3y 2 + y2 − 9 Functions of Several Variables Examples Examples Find the domain of the function f (x , y ) = x 2x + 3y 2 + y2 − 9 Find the domain and range of f (x , y ) = Lecture 21 p 4 − x2 − y2 Functions of Several Variables Graphs Denition If f is a function of two variables with domain , then the graph of f is the set of all points (x , y , z ) ∈ R3 such that z = f (x , y ) and (x , y ) is in D . Lecture 21 Functions of Several Variables Examples Example Lecture 21 Functions of Several Variables Examples Example A linear function is a function f (x ) = ax + by + c Lecture 21 Functions of Several Variables Examples Example A linear function is a function f (x ) = ax + by + c The graph of such a function is a plane. Lecture 21 Functions of Several Variables Examples Example f (x , y ) = sin(x ) + sin(y ) Lecture 21 Functions of Several Variables Examples Example f (x , y ) = sin(x ) + sin(y ) 2 1.5 1 0.5 0 -0.5 -1 -1.5 -2 6 4 -6 2 -4 0 -2 0 -2 2 Lecture 21 4 -4 6 -6 Functions of Several Variables Examples Example f (x , y ) = (x 2 + y 2 )e −x 2 −y 2 Lecture 21 Functions of Several Variables Examples Example f (x , y ) = (x 2 + y 2 )e −x 2 −y 2 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 -2 -1.5 -1 -0.5 0 0.5 Lecture 21 1 1.5 2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 Functions of Several Variables 2 The Cobb-Douglas production function Example P (L, K ) = bLα K 1−α Lecture 21 Functions of Several Variables The Cobb-Douglas production function Example P (L, K ) = bLα K 1−α P (L, K ) = 1.01L0.75 K 0.25 350 300 250 200 150 100 50 0 300 250 0 200 50 150 100 150 100 200 Lecture 21 250 50 3000 Functions of Several Variables Level Curves Denition The level curves of a function f of two variables are the curves with equations f (x , y ) = k , where k is constant. Lecture 21 Functions of Several Variables Examples Example f (x , y ) = sin(x ) + sin(y ) Lecture 21 Functions of Several Variables Examples Example f (x , y ) = sin(x ) + sin(y ) sin(y)+sin(x) 2 1.5 1 0.5 0 -0.5 -1 -1.5 -2 2 1.5 1 0.5 0 -0.5 -1 -1.5 -2 6 4 -6 2 -4 0 -2 0 -2 2 Lecture 21 4 -4 6 -6 Functions of Several Variables Examples Example f (x , y ) = (x 2 + y 2 )e −x 2 −y 2 Function 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 -2 -1.5 -1 -0.5 0 0.5 Lecture 21 1 1.5 2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 Functions of Several Variables 2 The Cobb-Douglas production function Example P (L, K ) = 1.01L0.75 K 0.25 1.01*x0.75*y0.25 350 300 250 200 150 100 50 0 350 300 250 200 150 100 50 0 300 250 0 200 50 150 100 150 100 200 Lecture 21 250 50 3000 Functions of Several Variables Limits and Continuity Denition Lecture 21 Functions of Several Variables Limits and Continuity Denition We say that a function f (x , y ) has limit L as (x , y ) approaches a point (a, b) and we write lim (x ,y )→(a,b ) f (x , y ) = L if we can make the values of f (x , y ) as close to L as we like by taking the point (x , y ) suciently close to the point (a, b), but not equal to (a, b). Lecture 21 Functions of Several Variables Limits and Continuity Denition We say that a function f (x , y ) has limit L as (x , y ) approaches a point (a, b) and we write lim (x ,y )→(a,b ) f (x , y ) = L if we can make the values of f (x , y ) as close to L as we like by taking the point (x , y ) suciently close to the point (a, b), but not equal to (a, b). We write also f (x , y ) → L as (x , y ) → (a, b) and lim x →a,y →b Lecture 21 f (x , y ) = L Functions of Several Variables Important Lecture 21 Functions of Several Variables Important If f (x , y ) → L1 as (x , y ) → (a, b) along a path C1 and f (x , y ) → L2 as (x , y ) → (a, b) along a path C2 , where L1 6= L2 , then lim(x ,y )→(a,b) f (x , y ) does not exists. Lecture 21 Functions of Several Variables Important If f (x , y ) → L1 as (x , y ) → (a, b) along a path C1 and f (x , y ) → L2 as (x , y ) → (a, b) along a path C2 , where L1 6= L2 , then lim(x ,y )→(a,b) f (x , y ) does not exists. Example: Show that x2 − y2 (x ,y )→(0,0) x 2 + y 2 lim does not exist. Lecture 21 Functions of Several Variables Continuity Denition Lecture 21 Functions of Several Variables Continuity Denition A function f of two variables is called lim (x ,y )→(a,b ) Lecture 21 continuous at f (x , y ) = f (a, b) Functions of Several Variables (a, b) if Continuity Denition A function f of two variables is called lim (x ,y )→(a,b ) continuous at (a, b) if f (x , y ) = f (a, b) polynomials, rational, trigonometric, exponential, logarithmic functions are continuous on theirs domain. Examples: Lecture 21 Functions of Several Variables Examples Examples Lecture 21 Functions of Several Variables Examples Examples Find the limit lim (x ,y )→(0,0) p Lecture 21 2x 2 − 4y 2x 2 − 4y + 1 − 1 Functions of Several Variables Examples Examples Find the limit lim (x ,y )→(0,0) p 2x 2 − 4y 2x 2 − 4y + 1 − 1 Find the largest set on which the function 2xy 9 − x2 − y2 is continuous. Lecture 21 Functions of Several Variables