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Math22lO Midterm 2 Summer, 2012 Dylan Zwick
Name Date
Instructions: Please show all of your work as partial credit will be given where appropriate, and there may be no credit given for problems where there is no work shown. All answers should be completely simplified, unless otherwise stated.
1. (20 pts) Find the directional derivative of [(x,y)=x at p=(—l,2) in the direction of a=2i—j yyI
IltH
=
(-i,z)
J7-
-1
-
I I
1
Answer:

3. (8 pts) Draw the set S=(x,y)xE(—2,2)y[—3,3fl
7/
3
(5 points) Is the set open, closed, or neither?
—
(3 points) Is the point (O,-3) an interior point?
-
/C)
(3 points) Is the point (2,-3) an interior point?
—
(3 points) Is the point (O,-3) a boundary point?
(3 points) Is the point (2,-3) a boundary point?

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5. For
(a) (10 pts)
— at (1, 1)
, find
Z + Ze
(b) (10 pts) f
1
Answer:
5
Answer:
Zy

6
6. (20 points) Find the minimum of x—y—6=0 x subject to the constraint
7
+Yy y-y_ y
V
Au
Z7 y y-y y y-A
6+6
CI
_,?(—iy
Answer:
L
I
/

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7. (25 points) Find the dimensions of the rectangular box of volume V which the sum of the edge lengths is least.
+(iy)
X*y
Yy7
/7
I
7 p
V
V’J yy
-
/
YY
)i)
X
—
1/
-xY
Answer:
A1/
5,)ej
/fr?
V y
-

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8. (20 pts) Find be only in terms of t.) if w(x,y)=x
, x=t
, yzzt
.
(Your answer must ay.
2( z\/)
(e) (c
()()+ 15(et(t(z÷) i1l
8
Answer:

9. Find the limit, if it exists. (Show all your reasoning.)
(a) urn
(x,y)—(1,1) y—2x
(10 pts)
FC+IOñ/ i/
(1 i)
/ o
i
-!
/
-
TA M / m
/
*
Answer
(b) urn
(x,y)-(O,O)
/
(o4vet
(7L)
(15 pts)
Z
‘ e c
L)
/
Z i) (
)
Answer:
9