r(print) Name S~co~
EID~
M427L Fall 2015 Exam 1A
Si-rerlc'Gk:
NO NOTES. NO CALCULATORS.
1. If f(x, y, z)
=
(x2
Of(\jl
+ y2 + 4z2,
xy
2)' and Xo
=
)~
2 ') .:
~1Jc'.1
X
7-L
C1\
(3,1,2), find Df(xo).
~~
l
l~
[:
1
z
2'1.
-:z1-
J.
\)f (xo)
'2
Ib
..,
-
I
- '$
l-t
l
19
2. If a, b, and c are vectors in R3, show that a· (b + c)
D...::
c
0\
I
I
c..
:1
1
01.
b'::
3 /
= a b + a· c.
cb
I
)
b
;l.. I
h -:/
.5
C =: «:
C,
I
c. ~I~
C"'---;;.
-'
CD
1'--
I a , C~r&} -:I
0\
,1 +
IT",
L.
3. For P(2,1,-1),
and 5.
Q(l,l,l),
R(-2,5,3),
and 5(0,4,5),
find the distance from P to the plane determined by Q, R,
tOL + (OJ -
to
(Ox
1-
IU[!) +
IOj - S-z
<to
0::0
rOC,) - 5(1)
.¥D~O
5k
p
0::-15'
IA yo
+ Q,~oof
C
'2-"
+
---------------I
--
t ~ ~ f?-J;-c
2-
---
4. Let 5 be the sphere of radius 2 with center at (0,0,2). Find an equation is spherical coordinates for 5.
------------K -
pm-:s
J :::-/)
0c::'1 j
"i.9); ~ fY
z: P [.Dsct
VI
I
5. Find parametric
equations of the curve which is the intersection
\u
x -;:;'2
I.j ~
\CO
of the surfaces z = x2
+ y2
and z = 4.
s-l
:1 ~;,V\t
z.-::\..1
6.
Let
1:]R2 -+ ]R2
and 9 : ]R2 -+
g(O,O)
Dg(O,O)
o
Find D(f
0
=
(!1 ~)
]R2
be differentiable functions such that:
= (1,2), g(l, 2) = (3,5),1(0,0) = (3,5), f(4, 1) = (1,2),
,Dg(1,2)
= (~
~) ,Df(3,5)
g)(l, 2). (You may not need all of the date above.)
O(~oJ)(1121
: lJ
-J
oJ (I! z) -: ( '3/
(j~iJ)l ,D~{IJ£- )
i)
------------------
fW~(~~)
-~~-~
= (~
~),
and Df(4,1)
=
(~1 ~).