Fall 2010: MATH 2210-003
Calculus Ill
Wed. 6th October
Midterm Examination
Name:
Student ID:
Instructions.
1. Attempt all questions.
2. No calculators or cheat. sheets allowed.
3. Scratch paper is on the last page. Ask the instructor if you need some more.
Question
Points Your Score
QI
25
Q2
25
Q3
20
Q4
30
TOTAL
100
Dr. R. Lodh
60 minutes
Qi).
..
[25 points) Consider the vectors
u
v
=
=
—i+3j+2k
5i+j—k.
(i) Calculate the product u x v.
(ii) Find the angle between u and v (you may write your answer using an inverse trigonometric function).
(iii) Find the equation of the plane which is parallel to both
F9=(l.2.3).
11
Ui
-7
41
~j)<’
—S
N4fr. +
~
~ —j
~ k
r
xk
115J
c7N
I
~ *
—
and v and passes through the point
J
>c~j
L ~
—~
~
~I~T
-
co
‘a
s~G
-~
(i~i~
D ~
vaH’
)
J
9 (j-2D-14
1
7=
OVER
D~
Q2].
-
.
[25 points] A curve in 3-space is parameterized by the function
r(t)
=
ti +
~t2j
+ ~.~L3I2k
for 2 ≥ 0.
(i) Find the velocity v(t)
=
r’(t) of r(t).
(ii) Calculate the arc length of r(t) for 0 ≤ t ~ 2 (hint: complete the square).
(iii) Find the parametric equations of the line which is tangent to this curve at the point P0
(iv) Find the equation of the plane which is perpendicular to this curve at the point P0
;(E)=r’ee)=
Cii) L~ JI
Lft
=
(22, ~).
(2.2. ~).
=
f÷a
H-
at
~
(3
aLL
C-)
=
1
)
-~
f~j
~—
cc
~
(~)
V~QL~
F
C~NL
~
0_vt
N~’e~J
~~
I
I
<2,2
J v~
C’°~
1-
3
2H:
~ (2)
cx-2t 2(~i~
0
~;
Q3].
..
[20 points] A curve in 3-space is parameterized by a function rQ).
~O (i) Define the curvature
)Q (ii)
If r(t)
=
tt(t)
4sin(t)i + 3tj
—
of this curve.
4cos(t)k. find a formula for
ic(t).
J—.
6)
0(1
w~rt
L
L
_ I
a
=
T
=
t
~J~J
\\~~(A~\\ 44
Vt
Cw
p
CL4
Lh N
oLc
—
s~ft) ~
U’
—~
I
=
~Il
ft
~3- J
4/‘25
L
(~≤~ (~)~4c~(+)
i6s-ft)÷i6c~(u’
‘Q4].
..
[30 points] Let the equation z
=
(y
—
x)2 define a surface in 3-space.
(1) Write this equation in
(a) cylindrical coordinates
(b) sphericai coordinates.
(ii) Find the partial derivatives
Dz Dz 82z
Dx Dy DyDx
—.
—,
Sketch the level curves to this surface for z
=
z~, and
=
82z
DxDy
=
z~.
=
4 at the point (1. —1,4).
0, 1,2,3,4.
~ (iv) Find the equation of the line tangent to the level curve at z
Show that the vector u
=
~(L —1)i + ~(1, —1)j is perpendicular to the line in (iv).
Dx
Dy
2
=
= ~
~
(
—
~q)
~9)2
n
0
~
L~
3
‘~—~(=~
~
~{
4
~
‘C
~(
2
Ci)
—
(b)
C-—
3
Yz&t=
Ri;
S~~A”
r~
#
e
r
19
CU)
9
)
2
~3=x—2
[v~ ~~rO4 9j~. L~
—H
=
jU’
C
<~
~‘>