14 - university of pennsylvaniawziller/math114s14/ch14-3-4.pdf · lim ( , ) x y a b f x y l o if...
TRANSCRIPT
![Page 1: 14 - University of Pennsylvaniawziller/math114s14/ch14-3-4.pdf · lim ( , ) x y a b f x y L o If the values of f(x, y) approach the number L as the point (x, y) approaches the point](https://reader033.vdocuments.us/reader033/viewer/2022042104/5e8208a077357d27eb2aae01/html5/thumbnails/1.jpg)
14.2
Continuity
![Page 2: 14 - University of Pennsylvaniawziller/math114s14/ch14-3-4.pdf · lim ( , ) x y a b f x y L o If the values of f(x, y) approach the number L as the point (x, y) approaches the point](https://reader033.vdocuments.us/reader033/viewer/2022042104/5e8208a077357d27eb2aae01/html5/thumbnails/2.jpg)
( , ) ( , )lim ( , )
x y a bf x y L
If the values of f(x, y) approach the number L as the point (x, y) approaches the point (a, b) along any path that stays within the domain of f.
Definition:
We can let (x, y) approach (a, b) from an infinite number of directions in any manner whatsoever as long as (x, y) stays within the domain of f.For all of these the limit must be the same.
![Page 3: 14 - University of Pennsylvaniawziller/math114s14/ch14-3-4.pdf · lim ( , ) x y a b f x y L o If the values of f(x, y) approach the number L as the point (x, y) approaches the point](https://reader033.vdocuments.us/reader033/viewer/2022042104/5e8208a077357d27eb2aae01/html5/thumbnails/3.jpg)
Examples:
2 2( , ) (0,0)a) lim
x y
xy
x y If 0 or 0 then x y
If y then x2
2 2 2( , ) (0,0) 0
1lim lim
2 2x y x
xy x
x y x
Limit does not exist2
2 4( , ) (0,0)b) lim
x y
xy
x y
If y then mx2 2 3
2 4 2 4 4( , ) (0,0) 0lim lim
x y x
xy m x
x y x m x
0
2If y then x
2If x then y
2 5
2 4 2 8( , ) (0,0) 0lim lim
x y x
xy x
x y x x
2 4
2 4 4( , ) (0,0) 0lim lim
2x y y
xy y
x y y
0
1
2
Limit does not exist
2 2( , ) (0,0)lim 0
x y
xy
x y
2
4 20lim
1x
m x
m x
![Page 4: 14 - University of Pennsylvaniawziller/math114s14/ch14-3-4.pdf · lim ( , ) x y a b f x y L o If the values of f(x, y) approach the number L as the point (x, y) approaches the point](https://reader033.vdocuments.us/reader033/viewer/2022042104/5e8208a077357d27eb2aae01/html5/thumbnails/4.jpg)
Example:2
2 2( , ) (0,0) lim
x y
xy
x y
If y then mx2 2 3
2 2 2 2 2( , ) (0,0) 0lim lim
x y x
xy m x
x y x m x
2
20lim 0
1x
mx
m
2If y then x
2If x then y
2 5
2 2 2 4( , ) (0,0) 0lim lim
x y x
xy x
x y x x
2 4
2 2 4 2( , ) (0,0) 0lim lim
x y y
xy y
x y y y
2
20lim
1y
y
y 0
0
Is the limit 0? Use polar coordinates: cos( ), sin( )x r y r
2 3 2
2 2 2( , ) (0,0) 0
cos( )sin ( )lim lim
x y y
xy r
x y r
2
0lim cos( )sin ( )r
r
0 2
since for all
cos( )sin ( )
"Sandwich theorem"
r r
Limit is 0!
3
20lim
1x
x
x
![Page 5: 14 - University of Pennsylvaniawziller/math114s14/ch14-3-4.pdf · lim ( , ) x y a b f x y L o If the values of f(x, y) approach the number L as the point (x, y) approaches the point](https://reader033.vdocuments.us/reader033/viewer/2022042104/5e8208a077357d27eb2aae01/html5/thumbnails/5.jpg)
![Page 6: 14 - University of Pennsylvaniawziller/math114s14/ch14-3-4.pdf · lim ( , ) x y a b f x y L o If the values of f(x, y) approach the number L as the point (x, y) approaches the point](https://reader033.vdocuments.us/reader033/viewer/2022042104/5e8208a077357d27eb2aae01/html5/thumbnails/6.jpg)
Also: a) Polynomials ( , ) are continuous (they contain terms like )n mP x y cx y
0 0 0 0
( , )b) Rational functions (quotients of polynomials) are continuous
Q( , )
at ( , ) if Q( , ) 0
P x y
x y
x y x y
![Page 7: 14 - University of Pennsylvaniawziller/math114s14/ch14-3-4.pdf · lim ( , ) x y a b f x y L o If the values of f(x, y) approach the number L as the point (x, y) approaches the point](https://reader033.vdocuments.us/reader033/viewer/2022042104/5e8208a077357d27eb2aae01/html5/thumbnails/7.jpg)
2
2 2( , ) (0,0) earlier we showed lim 0
x y
xy
x y
2
2 2a) Where is ( , ) continuous?
xyf x y
x y
2
2 2( , ) (1,1)hence lim
x y
xy
x y
1
2
Examples:
is continuous for all ( , ) (0,0)f x y
can be extended continuously for all ( , ) by defining (0,0) 0f x y f
3 3
b) Can ( , ) tan be extended continuously?x y
f x yx y
3 3
limy x
x y
x y
2 2( )( )limy x
x y x xy y
x y
2Yes, by defining ( , ) tan(3 )f x x x
c) as a last resort, we can try some sample path as in the examples, or use
polar coordinates to decide if a limit exists.
2 2 2lim 3y x
x xy y x
![Page 8: 14 - University of Pennsylvaniawziller/math114s14/ch14-3-4.pdf · lim ( , ) x y a b f x y L o If the values of f(x, y) approach the number L as the point (x, y) approaches the point](https://reader033.vdocuments.us/reader033/viewer/2022042104/5e8208a077357d27eb2aae01/html5/thumbnails/8.jpg)
14.3
Partial Derivatives
![Page 9: 14 - University of Pennsylvaniawziller/math114s14/ch14-3-4.pdf · lim ( , ) x y a b f x y L o If the values of f(x, y) approach the number L as the point (x, y) approaches the point](https://reader033.vdocuments.us/reader033/viewer/2022042104/5e8208a077357d27eb2aae01/html5/thumbnails/9.jpg)
Recall:
0
'( ) limh
f a h f af a
h
Functions of one variable ( ).y f x
What is the derivative at ?x a
|
( ( ))x a
dy df x
dx dx
![Page 10: 14 - University of Pennsylvaniawziller/math114s14/ch14-3-4.pdf · lim ( , ) x y a b f x y L o If the values of f(x, y) approach the number L as the point (x, y) approaches the point](https://reader033.vdocuments.us/reader033/viewer/2022042104/5e8208a077357d27eb2aae01/html5/thumbnails/10.jpg)
h
bafbhafbaf
hx
,,lim,
0
Partial derivatives:
baxf ,at respect to with of derivative Partial
Regard as a constant and differentiate , with respect to y f x y x
Example:2
2 2( , )
xf x y
x y
Compute . xf
2 2
d x
dx x y
2 2
d x
dx x b
2 2
2 2 2
1( ) (2 x)
( )
x b x
x b
2 2
2 2 2( )
x b
x b
2 2
2 2 2( )y
y xf
x y
Regard as if it were a constant : y y b
replace: b y
( , )z f x y
![Page 11: 14 - University of Pennsylvaniawziller/math114s14/ch14-3-4.pdf · lim ( , ) x y a b f x y L o If the values of f(x, y) approach the number L as the point (x, y) approaches the point](https://reader033.vdocuments.us/reader033/viewer/2022042104/5e8208a077357d27eb2aae01/html5/thumbnails/11.jpg)
Partial derivative of with respect to at ,f y a b
0
, ,, limy
h
f a b h f a bf a b
h
Regard as a constant and differentiate
, with respect to
x
f x y y
0 0, :xf x y
0 0, :yf x y
![Page 12: 14 - University of Pennsylvaniawziller/math114s14/ch14-3-4.pdf · lim ( , ) x y a b f x y L o If the values of f(x, y) approach the number L as the point (x, y) approaches the point](https://reader033.vdocuments.us/reader033/viewer/2022042104/5e8208a077357d27eb2aae01/html5/thumbnails/12.jpg)
Notation:
fDx
zyxf
xx
ffyxf xxx
,,
2
,xy xy
f ff x y f
y x y x
( , )z f x y
Example: 1( , ) arctan tany y
z f x yx x
2
recall:
1arctan( )
1
dx
dx x
2
1arctan
1
y y
x x x xy
x
2 2
1
1
y
xy
x
2 2
y
x y
The derivative with respect to first,
then the derivative with respect to of that
x
y
2
1arctan
1
y y
y x y xy
x
2
1 1
1xy
x
2
1
yx
x
We have four second order derivatives: , , , , , , ,xx xy yx yyf x y f x y f x y f x y
2 2
x
x y
Find ,f f
x y
![Page 13: 14 - University of Pennsylvaniawziller/math114s14/ch14-3-4.pdf · lim ( , ) x y a b f x y L o If the values of f(x, y) approach the number L as the point (x, y) approaches the point](https://reader033.vdocuments.us/reader033/viewer/2022042104/5e8208a077357d27eb2aae01/html5/thumbnails/13.jpg)
2 3, lnx
g x y x yy
3 1 12x x
y
g xyy
3 12xy
x
2 2
2
13y x
y
xg x y
y
2 2 13x y
y
3
2
12xxg y
x 2
2
16yyg x y
y
26xyg xy
Mixed partials are equal, or the order of differentiation does not matter.
Example:
26yxg xy
![Page 14: 14 - University of Pennsylvaniawziller/math114s14/ch14-3-4.pdf · lim ( , ) x y a b f x y L o If the values of f(x, y) approach the number L as the point (x, y) approaches the point](https://reader033.vdocuments.us/reader033/viewer/2022042104/5e8208a077357d27eb2aae01/html5/thumbnails/14.jpg)
14.4
Chain Rule
![Page 15: 14 - University of Pennsylvaniawziller/math114s14/ch14-3-4.pdf · lim ( , ) x y a b f x y L o If the values of f(x, y) approach the number L as the point (x, y) approaches the point](https://reader033.vdocuments.us/reader033/viewer/2022042104/5e8208a077357d27eb2aae01/html5/thumbnails/15.jpg)
Recall: Functions of one variable z (y).f
and depends on : y (x).y x g
so depends on : z ( ) ( (x))z x f y f g
Chain rule:
'( ) '( ) '( ( ) '( )dz dz dy
f y g x f g x g xdx dy dx
![Page 16: 14 - University of Pennsylvaniawziller/math114s14/ch14-3-4.pdf · lim ( , ) x y a b f x y L o If the values of f(x, y) approach the number L as the point (x, y) approaches the point](https://reader033.vdocuments.us/reader033/viewer/2022042104/5e8208a077357d27eb2aae01/html5/thumbnails/16.jpg)
yxfz , , x g t y h t
dz z dx z dy
dt x dt y dt
Use the chain rule to find at 1.dz
tdt
2 2cos sin 2xy xyye x e x x
2cosxyxe x
1
2
dx
dt t
1dy
dt t
2 2 21 1cos sin 2 cos
2
xy xy xydzye x e x x xe x
dt tt
If 1,
then 1, 0
t
x y
1
10 cos 1 1 2sin 1 1 cos 1 1
2t
dz
dt
cos1 sin1
Chain rule for a function of two variables:
Example:
z
x
z
y
2Let cos with and ln . xyz e x x t y t
![Page 17: 14 - University of Pennsylvaniawziller/math114s14/ch14-3-4.pdf · lim ( , ) x y a b f x y L o If the values of f(x, y) approach the number L as the point (x, y) approaches the point](https://reader033.vdocuments.us/reader033/viewer/2022042104/5e8208a077357d27eb2aae01/html5/thumbnails/17.jpg)
yxfz , , , ,x g s t y h s t
s
y
y
z
s
x
x
z
s
z
t
y
y
z
t
x
x
z
t
z
2, 2 , 7 . z x y x x s t y s t
1
2
zy
x x
2
z x
y y
1x
t
7y
t
4 and 1s t
x
y
9
9
1
1 72 2
z xy
t x y
1 9
3 76 6
18 1 63
6
44
6
22
3
Another chain rule:
Example:
Find if 4 and 1.z
s tt