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Name
Math 260
Summer 2021
Homework Set 3
Show all work!
1. Show that
(x;
lim
)!(0
y
;
+ y3
does not exist.
3
x+y
2
x
0)
(5 pts.)
2
x + 2y
2. Determine the set of points at which the function F (x; y ) =
1+e
(5 pts.)
x
1
y2
is continuous.
3. Find the rst partial derivatives of f (x; y ) = 3y sin(x2 + y 2 ).
4. Find an equation of the tangent plane to the surface
(1; 2; 5). (5 pts.)
2
z
=
3
x
(5 pts.)
y
2
+ 12x + 2y at the point
@z
5. Let z = x2 y 4 y 3 , where x = 3s 4t and y = 2s + 5t. Use the Chain Rule to nd . Be sure
@s
that your nal answer is fully in terms of s and t. Beyond that, you do not have to simplify.
(5 pts.)
6. Di erentiate implicitly to nd
@z
@x
and
@z
@y
if x2 yz + ln(xy ) + sin(yz 3 ) = 10. (5 pts.)
3
7. Find the directional derivative of f (x; y ) = 4×3 + 3y 2 at the point (2;
v = h3; 4i. (5 pts.)
8. Find an equation for the tangent plane to the surface given by
( 2; 1; 2). (5 pts.)
4
3
x yz
1) in the direction of
2
=
32 at the point
Consider the surface given by x2 + y2 + z2 = a?. Show
that every normal line to this surface passes through the
origin.
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