Functions of Two variables.

Transcription

1 Functions of Two variables. Ferdinánd Filip siva.banki.hu/jegyzetek 27 February 217 Ferdinánd Filip 27 February 217 Functions of Two variables. 1 / 36

2 Table of contents Functions of Several Variables Ferdinánd Filip 27 February 217 Functions of Two variables. 2 / 36

3 Definition Suppose D is a set of n-tuples of real numbers (x 1, x 2,..., 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 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. Ferdinánd Filip 27 February 217 Functions of Two variables. 3 / 36

4 Interior and Boundary Points, Open, Closed Definition A point (x, y )in a region (set) R in the xy-plane is an interior point of R if it is the center of a disk of positive radius that lies entirely in R. A point (x, y ) is a boundary point of R if every disk centered at (x, y ) contains points that lie outside of 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 its boundary points. Ferdinánd Filip 27 February 217 Functions of Two variables. 4 / 36

5 Interior and Boundary Points, Open, Closed (x, y ) Interior point of R (x, y ) R (x 1, y 1 ) (x 1, y 1 ) Boundary pointof R Ferdinánd Filip 27 February 217 Functions of Two variables. 5 / 36

6 Bounded and Unbounded Regions in the Plane Definition 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. Example: bounded sets: line segments, triangles, interiors of triangles, rectangles, circles, and disks bounded sets: lines, coordinate axes, the graphs of functions defined on infinite intervals, quadrants, half-planes, and the plane itself. Ferdinánd Filip 27 February 217 Functions of Two variables. 6 / 36

7 Graphs, Level Curves, and Contours of Functions of Two Variables Definition:Level Curve, Graph, Surface The set of points in the plane where a function f (x, y) has a constant valuef (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) Ferdinánd Filip 27 February 217 Functions of Two variables. 7 / 36

8 Graphs, Level Curves, and Contours of Functions of Two Variables f (x, y) = x + y f (x, y) = x 2 + y x y x y 2 Ferdinánd Filip 27 February 217 Functions of Two variables. 8 / 36

9 Graphs, Level Curves, and Contours of Functions of Two Variables f (x, y) = x 2 + y 2 f (x, y) = xe x2 y x y x y 2 Ferdinánd Filip 27 February 217 Functions of Two variables. 9 / 36

10 Graphs, Level Curves, and Contours of Functions of Two Variables f (x, y) = y 2 y 4 x 2 f (x, y) = cos x cos ye x 2 +y x y 1 5 x 5 5 y 5 Ferdinánd Filip 27 February 217 Functions of Two variables. 1 / 36

11 Definition: Limit of a Function of Two Variables We say that a function f (x, y) approaches the limit L as (x, y) approaches (x, y ), and write lim f (x, y) = L (x,y) (x,y ) if, for every number ε >, there exists a corresponding number δ > such that for all (x, y in the domain of f f (x, y) L < ε wheneever < (x x ) 2 + (y y ) 2 < δ Ferdinánd Filip 27 February 217 Functions of Two variables. 11 / 36

12 Properties of Limits of Functions of Two Variablesi The following rules hold if L, M, and k are real numbers and lim f (x, y) = L, lim (x,y) (x,y ) g(x, y) = M (x,y) (x,y ) lim (f (x, y) + g(x, y)) = L + M (x,y) (x,y ) lim (f (x, y) g(x, y)) = L M (x,y) (x,y ) lim kf (x, y) = kl (x,y) (x,y ) f (x,y) lim (x,y) (x,y ) g(x,y) = L M, ha M. Ferdinánd Filip 27 February 217 Functions of Two variables. 12 / 36

13 Definition: Continuous Function of Two Variables A function f (x, y) is continuous at the point (x, y ) if 1 f is defined at (x, y ), 2 lim (x,y) (x,y ) f (x, y) exists 3 lim (x,y) (x,y ) f (x, y) = f (x, y ) A function is continuous if it is continuous at every point of its domain. Ferdinánd Filip 27 February 217 Functions of Two variables. 13 / 36

14 Definition: Partial Derivative with Respect to x The partial derivative of f (x, y) with respect to x at the point (x, y ) is f f (x + h, y ) f (x, y ) x = lim (x,y ) h h provided the limit exists. The equivalent expressions for the partial derivative are: f x (x, y ) f x(x, y ) Ferdinánd Filip 27 February 217 Functions of Two variables. 14 / 36

15 Definition: Partial Derivative with Respect to y The partial derivative of f (x, y) with respect to y at the point (x, y ) is f f (x, y + h) f (x, y ) y = lim (x,y ) h h provided the limit exists. The equivalent expressions for the partial derivative are:: f y (x, y ) f y (x, y ) Ferdinánd Filip 27 February 217 Functions of Two variables. 15 / 36

16 Calculations f x f y differentiating f with respect to x in the usual way while treating y as a constant y differentiating f with respect to x in the usual way while treating y as a constant Examples: Find f x and f y f (x, y) = x 2 + y 3 f (x, y) = x 2 y 3 f (x, y) = x 3 sin(x + y) Ferdinánd Filip 27 February 217 Functions of Two variables. 16 / 36

17 Differentiable Function A function z = f (x, y) is differentiable at (x, y ) if f x(x, y ) and f y (x ), y ) exist and z = f (x + x, y + y) f (x, y ) satisfies an equation of the form z = f x(x, y ) x + f y (x, y ) y + ε 1 x + ε 2 y in which each of ε 1, ε 2 as both x, y. We call f differentiable if it is differentiable at every point in its domain. Ferdinánd Filip 27 February 217 Functions of Two variables. 17 / 36

18 Theorem: If the partial derivatives f x and f y of a function f (x, y) are continuous throughout an open region R, then f is differentiable at every point of R. Tétel: If a function f (x, y) is differentiable at (x, y ), then f is continuous at (x, y ). Ferdinánd Filip 27 February 217 Functions of Two variables. 18 / 36

19 Linearization, Linear Approximation The linearization of a function f at a point (x, y ) where f is differentiable L(x, y) = f (x, y ) + f x(x, y )(x x ) + f y (x, y )(y y ) The approximation f (x, y) L(x, y) is the standard linear approximation of f at ((x, y ). Find the linearization of f (x, y) = x 2 + y 3 at point ( 2, 1) f (x, y) = x 3 sin(x + y) at point ( π 2, π 2 ) Ferdinánd Filip 27 February 217 Functions of Two variables. 19 / 36

20 Plane Tangent to a Surface The plane tangent to the surface z = f (x, y) of a differentiable function f at the point P (x, y, z ) = (x, y, f (x, y )) is z = z + f x(x, y )(x x ) + f y (x, y )(y y ) Ferdinánd Filip 27 February 217 Functions of Two variables. 2 / 36

21 Total Differential If we move from (x, y ) to a point (x + dx, y + dy) nearby, the resulting change df = f x(x, y )dx + f y (x, y )dy in the linearization of f f is called the total differential of f. Ferdinánd Filip 27 February 217 Functions of Two variables. 21 / 36

22 Estimating Absolut and Percentage Error Let function f (x, y) is differentiable at (x, y ). The absolute error: f df = f x(x, y )dx + f y (x, y )dy f x(x, y ) dx + f y (x, y ) dy Percentage Error: δf = f f f x(x, y ) f (x, y ) dx + f y (x, y ) f (x, y ) dy Ferdinánd Filip 27 February 217 Functions of Two variables. 22 / 36

23 Second-Order Partial Derivatives When we differentiate a function f (x, y) twice, we produce its second-order derivatives. These derivatives are usually denoted by 2 f x 2 vagy f xx; 2 f y x vagy f xy 2 f x y vagy f yx; 2 f y 2 vagy f yy Ferdinánd Filip 27 February 217 Functions of Two variables. 23 / 36

24 The defining equations are 2 f x 2 = ( ) f x x 2 f x y = ( ) f x y 2 f y x = ( ) f y x 2 f y 2 = ( ) f y y Ferdinánd Filip 27 February 217 Functions of Two variables. 24 / 36

25 Examples: Find all the second-order partial derivatives of the functions f (x, y) = x 2 y 3 f (x, y) = x 3 sin(x + y) f (x, y) = x 2 y 3 f (x, y) = (x 2 1)(y + 2) f (x, y) = x 2 + y 2 f (x, y) = (x 2 y 1)(y + 2) f (x, y) = x 3 sin(x + y) f (x, y) = x y f (x, y) = x2 y 1 y 3 f (x, y) = xe x2 y Ferdinánd Filip 27 February 217 Functions of Two variables. 25 / 36

26 Theorem: (Schwarz-Young) If f (x, y) and its partial derivatives f x, f y, f xy and f yx are defined throughout an open region containing a point (a, b) pontot, and are all continuous at (a, b), then f xy(a, b) = f yx(a, b). Ferdinánd Filip 27 February 217 Functions of Two variables. 26 / 36

27 The Chain Rule If w = f (x, y)has continuous partial derivatives f x and f y and if x = x(t), y = y(t) are differentiable functions of t then the composite w = f (x(t), y(t)) is a differ- entiable function of t and df dt = f x[x(t), y(t)]x (t) + f y [x(t), y(t)]y (t) or dw dt = f dx x dt + f dy y dt Ferdinánd Filip 27 February 217 Functions of Two variables. 27 / 36

28 A Formula for Implicit Differentiation Suppose that F (x, y) is differentiable and that the equation F (x, y) = defines y as a differentiable function of x. Then at any point where F y, dy dx = F x F y. Ferdinánd Filip 27 February 217 Functions of Two variables. 28 / 36

29 A Formula for Implicit Differentiation Examples Find the value of dy dx at the given point. x 3 2y 2 + xy =, (1, 1) x 2 + xy + y 2 7 =, (1, 2) xe y + sin xy + y ln 2 =, (, ln 2) Ferdinánd Filip 27 February 217 Functions of Two variables. 29 / 36

30 Definition: Local Maximum, Local Minimum Let f (x, y) be defined on a region R containing the point (a, b). Then 1 f (a, b) is a local maximum value of f if f (a, b) f (x, y) for all domain points (x, y) in an open disk centered at (a, b). 2 f (a, b)) is a local minimum value of f if f (a, b) f (x, y) for all domain points (x, y) in an open disk centered at (a, b). Ferdinánd Filip 27 February 217 Functions of Two variables. 3 / 36

31 First Derivative Test for Local Extreme Values If f (x, y) has a local maximum or minimum value at an interior point (a, b) of its domain and if the first partial derivatives exist there, then f x(a, b) = and f y (a, b) =. Definition: An interior point of the domain of a function f (x, y) where both f x and f y are zero or where one or both of f x and f y do not exist is a critical point of f Ferdinánd Filip 27 February 217 Functions of Two variables. 31 / 36

32 Definition: Saddle point A differentiable function f (x, y) has a saddle point at a critical point (a, b) if in every open disk centered at (a, b) there are domain points (x, y) where f (x, y) < f (a, b) and domain points (x, y) where f (x, y) > f (a, b). The corresponding point (a, b, f (a, b)) on the surface f (x, y) > f (a, b) is called a saddle point of the surface. Ferdinánd Filip 27 February 217 Functions of Two variables. 32 / 36

33 lok. max lok. min Ferdinánd Filip 27 February 217 Functions of Two variables. 33 / 36

34 1 nyeregpont Ferdinánd Filip 27 February 217 Functions of Two variables. 34 / 36

35 Second Derivative Test for Local Extreme Values Suppose that f (x, y) and its first and second partial derivatives are continuous throughout a disk centered at (a, b) and that f x(a, b) = f y (a, b) = O. Then i. f has a local maximum at (a, b) if f xxf yy ( f xy) 2 > and f xx <. ii. f has a local minimum at (a, b) if f xxf yy ( f xy) 2 > and f xx >. iii. f has a saddle point at (a, b) if f xxf yy ( f xy) 2 <. iv. The test is inconclusive at (a, b) if f xxf yy ( f xy) 2 = at (a, b). In this case, we must find some other way to determine the behavior of f at (a, b). Ferdinánd Filip 27 February 217 Functions of Two variables. 35 / 36

36 The expression f xxf yy ( f xy) 2 is called the discriminant or Hessian of f. It is sometimes easier to remember it in determinant form, f xxf yy ( f xy ) 2 f = xx f xy f yx f yy Ferdinánd Filip 27 February 217 Functions of Two variables. 36 / 36