FaIl 2010: MATH 2210-003
Calculus Ill
Wed. 17th November
Name:
Midterm Examination
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Student ID:
Instructions.
1. Attempt all questions.
2. No calculators or cheat sheets allowed.
& Scratch paperis on the last page. AsIc the instructor if you need some more.
Question
Points
Qi
20
Q2
30
Q3
25
Q4
25
TOTAL
100
Your Score
Dr. R. Lodh
60 minutes
Qi].
.
.
[20 points] Suppose the equation
3x2z3 + 2xyz5 + 5
=
0
defines a surface S in 3-space.
(i) Find the tangent space to S at the point (1,1, —1).
(ii) The equation implicitly defines z as a function of the variables x, y (but one cannot solve for z!). By
implicitly differentiating the equation, determine
‘2
-~
~I~~)(
C)
~
~
Q~
~O~ff~UPE
dz
dx
in terms of x. y, z.
7 5
pttn’_~
7~(i
1d
VF(? 1t)_
0~
So
~‘)+
~1t
-a?
n~tI
~r
n
5
I’
~ 41OX\t~
EP
Q2]
[30 points] Consider the function
-
f(x,y)= 2z2+y2+5.
(i) Find all local extrema of this function, if any, and determine ~Thether they are local maxima or local
minima..
(ii) Use the method of Lagrange multipliers to find the maximum and minimum of
with equation
1+y2=1.
CU
x~ 4-2
‘ycQ
SQ
(fY%L-\
~
~
(o,D)
=4
7~tA
~0 i D~~
n
~ r~t
coi~
4~Lt
vç~V~}
6-ct ~
~
(0
2
(~)
f(z, y)
on the curve
Rct
co~J*crvi~:
b~
~
—
~ C~)
§0
(3N
~=
~
~ oj
% =z~—
____
\~J~A ~0sç~L(
~
fr(t~LE,oJ=
&
I
~
a)
2~rE ~
Cv~o~
t
(a,
V)
Q3].
[25 points] Find the volume of the tetrahedron bound by the planes
2z+3x+y = 6
x=O
y=Q
z=O.
C3D’
7,
~0
I
(flr
6
‘hi ~tJ\ h±€
~t4
o
S
o
0
ff1
3
0
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0
ad
0
2
I
Q4].
.
.
[25 points] Find the surface area of
z
=
~2z2 + 2(y
—
1)2
over the triangle with vertices (0,0), (1, 2), (0,2).
2
I Q Ct
=
~
\/
—
C
a
2
2
2-
—.7
n
2.
~j
‘~2~
-~
co
~
2
)
~2
~;f~4-k
4Lt(~-1
2L
S1~ ~
2y
2~1~
3