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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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