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Unformatted text preview: x, 0
D 1 f 0, 0 lim
x0 4 2 x x f 0, 0
0 0 and, similarly,
D 2 f 0, 0 0.
Thus, if
r x, y f x, y f 0, 0 x, y Df 0, 0 0, 0 for each point x, y then we have
r x, y
x, y x2 x2y2
y2 3/2 and, since the latter expression approaches 0 as x, y
0, 0 . 0, 0 , the function f is differentiable at b. We define
x2y2
2
x 2 y 2 f x, y
0 if x, y 0, 0
. if x, y 0, 0 1 0.2 x 0.1
0
1 y
0 0.5 1
1 This function is not continuous at 0, 0 .
c. We define
x3y3
2
x 2 y 2 f x, y
0 414 if x, y 0, 0
. if x, y 0, 0 5
0
5
2
y0
2
4 4 x 2 4 This function is continuous at 0, 0 . Now we have
f x, 0
D 1 f 0, 0 lim
x0 x f 0, 0
0 0 and, similarly,
D 2 f 0, 0 0.
Thus, if
r x, y f x, y f 0, 0 x, y Df 0, 0 0, 0 for each point x, y then we have
r x, y
x, y x3y3
x2 y2 5/2 and, since the latter expression approaches 0 as x, y
0, 0 . 0, 0 , the function f is differentiable at d. We define
1
x 2 y 2 exp f x, y if
if 0 x, y 0, 0 x, y 0, 0 . 0.6
0.5
0.4
0.3
0.2
0.1
0
1 0.5 0
x
11 This function is continuous at 0, 0 . Now we have
f x, 0
D 1 f 0, 0 lim
x0 0.5 x 0
y f 0, 0
0 0 and, similarly,
D 2 f 0, 0 0.
Thus, if
r x, y f x, y f 0, 0 Df 0, 0 x, y 0, 0 for each point x, y then we have
r x, y
x, y exp x2 y2 and, since the latter expression approaches 0 as x, y 415 1
x 2 y 2 0, 0 , the function f is differentiable at 0, 0 .
e. We define
1
x 2 y 2 x 2 y 2 sin f x, y if
if 0 x, y 0, 0 x, y 0, 0 . 12
10
8
6
4
2
0
5
0
y
5 2 4 This function is continuous at 0, 0 . Now we have
f x, 0
D 1 f 0, 0 lim
x0 4 x x f 0, 0
0 0 and, similarly,
D 2 f 0, 0 0.
Thus, if
r x, y f x, y f 0, 0 x, y Df 0, 0 0, 0 for each point x, y then we have
r x, y
x, y x 2 y 2 sin 1
x 2 y 2 x2 y2 and, since the latter expression approaches 0 as x, y
0, 0 .
2. Prove that if A is an m
f x A.
We see easily that n matrix and if f x Ax for every x 0, 0 , the function f is differentiable at
R n then for every such point x we have D j f i x a ij
at each point x and for all i and j.
3. Suppose that U is a neighborhood of a point a in R n and that f : U R m . Suppose that A is a k m matrix
and that g x Af x for every x U. Prove that if f is differentiable at the point a then so is g and we have
g a Af a . (Your proof should be very short.)
This assertion follows at once from the chain rule.
4. State true or false: If the partial derivatives of a function f are all zero at every point in an open connected set
U then f must be constant in U.
This statement is true. If the partial derivatives are all zero then they are all continuous. Thus the
function f is differentiable with a zero derivative at every point of U.
5. Given an example of a nonconstant function f on an open set U
We take n 1 and define
fx R n such that f x O for every x U. 1 if x 0
0 if x 0. 6. State true of false: if U is a neighborhood of a point a in R n and if the directional derivative of a function f 416 exists at a in every direction then f is differentiable at a.
This statement is false. The function does not even have to be continuous at a.
7. Prove that if f and g are both differentiable at a point a in R n then so is the function f g and we have
fg a f a g a .
This assertion is very easy. 417...
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This note was uploaded on 11/26/2012 for the course MATH 2313 taught by Professor Staff during the Fall '08 term at Texas El Paso.
 Fall '08
 STAFF
 Math, Calculus

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