Limits and Continuity/Partial Derivatives
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1 Christopher Croke University of Pennsylvania Limits and Continuity/Partial Derivatives Math 115
2 Limits For (x 0, y 0 ) an interior or a boundary point of the domain of a function f (x, y). Definition: lim f (x, y) = L (x,y) (x 0,y 0 ) if for every ɛ > 0 there is a δ > 0 such that: for all (x, y) in the domain of f if 0 < (x x 0 ) 2 + (y y 0 ) 2 < δ then f (x, y) L < ɛ.
3 Limits For (x 0, y 0 ) an interior or a boundary point of the domain of a function f (x, y). Definition: lim f (x, y) = L (x,y) (x 0,y 0 ) if for every ɛ > 0 there is a δ > 0 such that: for all (x, y) in the domain of f if 0 < (x x 0 ) 2 + (y y 0 ) 2 < δ then f (x, y) L < ɛ.
4 This definition is really the same as in one dimension and so satisfies the same rules with respect to +,,,. See for example page 803 for a list.
5 This definition is really the same as in one dimension and so satisfies the same rules with respect to +,,,. See for example page 803 for a list. For example: if lim f (x, y) = L and lim (x,y) (x 0,y 0 ) g(x, y) = K (x,y) (x 0,y 0 )
6 This definition is really the same as in one dimension and so satisfies the same rules with respect to +,,,. See for example page 803 for a list. For example: if then lim f (x, y) = L and lim (x,y) (x 0,y 0 ) g(x, y) = K (x,y) (x 0,y 0 ) lim f (x, y) + g(x, y) = L + K (x,y) (x 0,y 0 ) x 2 + 2y + 3 lim =? (x,y) (1,0) x + y
7 This definition is really the same as in one dimension and so satisfies the same rules with respect to +,,,. See for example page 803 for a list. For example: if then lim f (x, y) = L and lim (x,y) (x 0,y 0 ) g(x, y) = K (x,y) (x 0,y 0 ) lim f (x, y) + g(x, y) = L + K (x,y) (x 0,y 0 ) x 2 + 2y + 3 lim =? (x,y) (1,0) x + y x 2 y 2 lim (x,y) (0,0) x y =?
8 Continuity(The definition is really the same.) f is continuous at (x 0, y 0 ) if This means three things: lim f (x, y) = f (x 0, y 0 ). (x,y) (x 0,y 0 )
9 Continuity(The definition is really the same.) f is continuous at (x 0, y 0 ) if This means three things: f is defined at (x 0, y 0 ). lim f (x, y) = f (x 0, y 0 ). (x,y) (x 0,y 0 )
10 Continuity(The definition is really the same.) f is continuous at (x 0, y 0 ) if This means three things: f is defined at (x 0, y 0 ). lim f (x, y) = f (x 0, y 0 ). (x,y) (x 0,y 0 ) lim (x,y) (x0,y 0 ) f (x, y) exists.
11 Continuity(The definition is really the same.) f is continuous at (x 0, y 0 ) if This means three things: f is defined at (x 0, y 0 ). lim f (x, y) = f (x 0, y 0 ). (x,y) (x 0,y 0 ) lim (x,y) (x0,y 0 ) f (x, y) exists. They are equal.
12 Continuity(The definition is really the same.) f is continuous at (x 0, y 0 ) if This means three things: f is defined at (x 0, y 0 ). lim f (x, y) = f (x 0, y 0 ). (x,y) (x 0,y 0 ) lim (x,y) (x0,y 0 ) f (x, y) exists. They are equal. There is an added complication here. Consider f (x, y) = { y x 2 +y 2 if (x, y) (0, 0) 0 if (x, y) = (0, 0)
13 If f (x, y) has different limits along two different paths in the domain of f as (x, y) approaches (x 0, y 0 ) then lim (x,y) (x0,y 0 ) f (x, y) does not exist.
14 If f (x, y) has different limits along two different paths in the domain of f as (x, y) approaches (x 0, y 0 ) then lim (x,y) (x0,y 0 ) f (x, y) does not exist. It is not enough to check only along straight lines! See the example in the text.
15 If f (x, y) has different limits along two different paths in the domain of f as (x, y) approaches (x 0, y 0 ) then lim (x,y) (x0,y 0 ) f (x, y) does not exist. It is not enough to check only along straight lines! See the example in the text. Continuity acts nicely under compositions: If f is continuous at (x 0, y 0 ) and g (a function of a single variable) is continuous at f (x 0, y 0 ) then g f is continuous at (x 0, y 0 ).
16 If f (x, y) has different limits along two different paths in the domain of f as (x, y) approaches (x 0, y 0 ) then lim (x,y) (x0,y 0 ) f (x, y) does not exist. It is not enough to check only along straight lines! See the example in the text. Continuity acts nicely under compositions: If f is continuous at (x 0, y 0 ) and g (a function of a single variable) is continuous at f (x 0, y 0 ) then g f is continuous at (x 0, y 0 ). Example: y 2 sin( x 2 1 )
17 Partial Derivatives of f (x, y) f x partial derivative of f with respect to x
18 Partial Derivatives of f (x, y) f x partial derivative of f with respect to x Easy to calculate: just take the derivative of f w.r.t. x thinking of y as a constant. f y partial derivative of f with respect to y
19 Examples: Compute f x and f y for f (x, y) = 2x 2 + 4xy
20 Examples: Compute f x and f y for f (x, y) = 2x 2 + 4xy (xy 3 + xy) 3
21 Examples: Compute f x and f y for f (x, y) = 2x 2 + 4xy (xy 3 + xy) 3 sin(xy + y)
22 Examples: Compute f x and f y for f (x, y) = 2x 2 + 4xy (xy 3 + xy) 3 sin(xy + y) x 3 y y + x 2
23 Example: For f (x, y) = 5 + 2xy 3x 2 y 2 find f (1, 2) x
24 Example: For f (x, y) = 5 + 2xy 3x 2 y 2 find f (1, 2) x f (2, 0) y
25 Example: For f (x, y) = 5 + 2xy 3x 2 y 2 find f (1, 2) x f (2, 0) y This notion shows up in many fields and many notations have been developed for the same thing. f x f x z x z x x f
26 What does f x (a, b) mean geometrically?
27 What does f x (a, b) mean geometrically?
28 Definition: f x (a, b) = d dx x=af (x, b) =
29 Definition: f x (a, b) = d dx x=af (x, b) = = lim h 0 f (a + h, b) f (a, b) h
30 Definition: f x (a, b) = d dx x=af (x, b) = = lim h 0 f (a + h, b) f (a, b) h
31 Similarly f y (a, b) = d dy f (a, b + h) f (a, b) y=bf (a, y) = lim h 0 h
32 Similarly f y (a, b) = d dy f (a, b + h) f (a, b) y=bf (a, y) = lim h 0 h
33
34 You can do implicit differentiation in the same way as for functions of one variable.
35 You can do implicit differentiation in the same way as for functions of one variable. Example: Find z x when z is is defined implicitly as a function of x and y by the formula; x 2 + y 2 + z 2 = 5.
36 For f (x, y) since f x and f y are functions of x and y we can take partial derivatives of these to yield second partial derivatives.
37 For f (x, y) since f x and f y are functions of x and y we can take partial derivatives of these to yield second partial derivatives. Notation ( f x x ) = 2 f x x.
38 For f (x, y) since f x and f y are functions of x and y we can take partial derivatives of these to yield second partial derivatives. Notation ( f x x ) = ( f y x ) = 2 f x x. 2 f y x.
39 For f (x, y) since f x and f y are functions of x and y we can take partial derivatives of these to yield second partial derivatives. Notation ( f x x ) = ( f y x ) = ( f x y ) = 2 f x x. 2 f y x. 2 f x y.
40 For f (x, y) since f x and f y are functions of x and y we can take partial derivatives of these to yield second partial derivatives. Notation ( f x x ) = ( f y x ) = ( f x y ) = ( f y y ) = 2 f x x. 2 f y x. 2 f x y. 2 f y y.
41 Example: Compute all second partials of f (x, y) = x 3 + 2xy 2 y
42 Example: Compute all second partials of f (x, y) = x 3 + 2xy 2 y Theorem Mixed derivative theorem If f (x, y),f x,f y,f xy,and f yx are defined in an open region about (a, b) and all are continuous at (a, b) then f xy (a, b) = f yx (a, b)
43 Example: Compute all second partials of f (x, y) = x 3 + 2xy 2 y Theorem Mixed derivative theorem If f (x, y),f x,f y,f xy,and f yx are defined in an open region about (a, b) and all are continuous at (a, b) then f xy (a, b) = f yx (a, b) Moral: Can differentiate in any order
44 Example: Compute all second partials of f (x, y) = x 3 + 2xy 2 y Theorem Mixed derivative theorem If f (x, y),f x,f y,f xy,and f yx are defined in an open region about (a, b) and all are continuous at (a, b) then f xy (a, b) = f yx (a, b) Moral: Can differentiate in any order This also holds for more derivatives: f xyxx 3 f x y 2
45 This all works for functions with more variables.
46 This all works for functions with more variables. Example: Compute f xyz for f = xyz + e xz xsin(z) + x 3 ln(z).
47 This all works for functions with more variables. Example: Compute f xyz for f = xyz + e xz xsin(z) + x 3 ln(z). It turns out that just because f x (a, b) and f y (a, b) exist does not mean that f is continuous at (a, b):
48 This all works for functions with more variables. Example: Compute f xyz for f = xyz + e xz xsin(z) + x 3 ln(z). It turns out that just because f x (a, b) and f y (a, b) exist does not mean that f is continuous at (a, b): { 1 if xy = 0 f (x, y) = 0 if xy 0
49 This all works for functions with more variables. Example: Compute f xyz for f = xyz + e xz xsin(z) + x 3 ln(z). It turns out that just because f x (a, b) and f y (a, b) exist does not mean that f is continuous at (a, b): { 1 if xy = 0 f (x, y) = 0 if xy 0 It turns out that one has to b more careful (see text) about what it means to be differentiable at (a, b) but it is enough if f x and f y exist and are continuous in a disk about (a, b).
50 This all works for functions with more variables. Example: Compute f xyz for f = xyz + e xz xsin(z) + x 3 ln(z). It turns out that just because f x (a, b) and f y (a, b) exist does not mean that f is continuous at (a, b): { 1 if xy = 0 f (x, y) = 0 if xy 0 It turns out that one has to b more careful (see text) about what it means to be differentiable at (a, b) but it is enough if f x and f y exist and are continuous in a disk about (a, b). If f is differentiable at (a, b) then it will be continuous at (a, b).
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