Math 250, Spring 2007
Exam
1
Page
1
of
4
1.
[11
points total] Answer each of the following as true (T) or false (F). Do not justify your
answers.
[£]
The speed of the curve r
=<
cos
t,
sin
t, 3t
>
is <  sin
t,
cos
t,
3
>.
OJ
The curvature of a circle of radius 2007 is 1/2007.
OJ
The gradient of
f(x, y)
is normal to the level curves of
f.
m
If
(1,1). is a critical point for .the fu?ction
!(:J;,'j/)
then (1, 1) is also a
<;:.it~c~
P?i?t
+lK
forthefunctlOng(x,Y)=f(x~y2).
(~t(
I}I'
_
(t~(\ll)
i.fll~J
l),  l
;
VI
~))
tV
3
t'
, \ \
:
( 1)(
(I
I
I )
2.
X
I
(1
l'
,,)
21 )
~
l
VI
..:>,
[1]
The function
u(x, t)
=
~2
+
t
satisfies the equation
Ut
=
U
xx
(heat equation).
[1]
If
u is a unit vector tangent at
(x, y)
to a, level curve of
f(x, y)
then
t~
directional
derivative satisfies
Duf(x,y)
=
o.
0
V\

.(<:'tJ
t.
I
l>A{.te
I ),
\J
e
A.!\ l
t
'v7
d
~
e)
,,.;J \
(.
h
.:".01\
S
e { \,. ('
...

JI'
[
i
J
(~1H
.....
,
~
b 1
(,II ('
f"1
)
[£]
::.
fx(x, y)
=
Jy(x, y)
for
~l
x, y,
~hen
i(x: y)
is_a constant.
I
.
V
(
[,
f
,~
.;):
'\t
f
~ ~
\.
'f'
"1> ._ .
.,.
t
1
m
If
in spherical coordinates a point is given by
(p,
(),
¢)
(2,
7r
/2,
7r
/2), then its rect
angular coordinates are
(x,y,z)
=
(0,2,0).
[Jj