(a) The function f ( x , y ) = { x y x 2 + y 2 if ( x , y ) ≠ (0, 0) 0 if ( x , y ) = (0, 0) was graphed in Figure 4. Show that f x (0, 0) and f y (0, 0) both exist but f is not differentiable at (0, 0). [Hint: Use the result of Exercise 45.] (b) Explain why f x and f y are not continuous at (0, 0).
(a) The function f ( x , y ) = { x y x 2 + y 2 if ( x , y ) ≠ (0, 0) 0 if ( x , y ) = (0, 0) was graphed in Figure 4. Show that f x (0, 0) and f y (0, 0) both exist but f is not differentiable at (0, 0). [Hint: Use the result of Exercise 45.] (b) Explain why f x and f y are not continuous at (0, 0).
Solution Summary: The author explains that f(x,y)=
f
(
x
,
y
)
=
{
x
y
x
2
+
y
2
if
(
x
,
y
)
≠
(0, 0)
0
if
(
x
,
y
)
=
(0, 0)
was graphed in Figure 4. Show that fx(0, 0) and fy(0, 0) both exist but f is not differentiable at (0, 0). [Hint: Use the result of Exercise 45.]
(b) Explain why fx and fy are not continuous at (0, 0).
With integration, one of the major concepts of calculus. Differentiation is the derivative or rate of change of a function with respect to the independent variable.
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