# Worksheet #8 - Iowa State University

```Leader:
Course:
Instructor:
Date:
Worksheet #8
12.1-12.4
Supplemental Instruction
Iowa State University
Jon
Math 265
Poon &amp; Weber
09/22/11
Section 12.1
1) Sketch the level curve z  k for z 
x2
and k  4, 1, 0,1, 4 [#19]
y
Section 12.2
2) Find all first partial derivatives of the following functions.
A) g ( x, y)  e xy [#9]
B) f ( x, y ) 
x2  y 2
[#3]
xy
C) f ( x, y)  y cos( x 2  y 2 ) [#13]
3) The volume V of a cylinder is given by V   hr 2 . If h is held constant where h  10'' ,
find the rate of change of V with respect to r when r  6'' [#29].
4) Show that the function f ( x, y)  x3 y  xy 3 is harmonic [
2 f 2 f

 0 ] [#33].
x 2 y 2
Supplemental Instruction
1060 Hixson-Lied Student Success Center  294-6624  www.si.iastate.edu
Section 12.3
5) Find the indicated limit or state that is does not exist.
 sin( x 2  y 2 ) 
A) lim ( x , y )(0,0) 
 [#7]
2
2
 x y

B)
 x2  y 2 
lim( x, y )(0,0)  x4  y 4  [#9]
1
x2 y
6) If f ( x, y )  4
, show that f  as ( x, y )  (0, 0) along the parabola y  x 2 [#37].
2
2
x y
Section 12.4
7) Find the gradient of f ( x, y)  x 2 y  xy 2 and the tangent plane @ the point p  (2,3) [#11].
8) Find the equation of the tangent hyper-plane to the surface f  x, y, z   3x2  2 y 2  xz 2 at
the point p  (1, 2, 1) [#15].
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