1.
(10 pts) Show that the curvature
T
2.
is related to the tangent and normal vectors by the equation1)
N
(20 pts) The binormal vector B
B
T
of a space curve is defined by
×N
The torsion
of the curve is a number defined by
N⋅B ′ .
The torsion measure the degree of twisting of a curve.2)
1)
Show that ∥B ′∥
2)
Show that
.
r ′⋅ r ″ × r ″ ′
∥r ′ × r ′
,
where r is a position vector of the curve.
3.
Consider the circular helix
r
cos i
where
1)
and
sin j
k
are positive constant.3)
(10 pts) Find the formula of tangent, normal, binormal vectors, and torsion of the given circular
helix.
2)
4.
(10 pts) Find the length of one turn of the given circular helix.
(15 pts) Assume that two vectors v and v in
consider two lines
given by
v for
are not parallel. For two points
and
in
,
and ∈ . Show that the distance between the two
lines is
⋅v × v
v ×v
5.
.
(10 pts) For the smooth curve given by the parametric equations
curvature is given
′
by4)
″
′
′
6.
.
′
(30 pts) Consider the curve represented by the vector valued function
r
cos i
sin j
Find the following
7.
″
k
quantities.5)
1)
Arc length parameter
2)
Unit tangent vector T
.
3)
Principal unit vector, N
4)
Binormal Vector, B
5)
Curvature
6)
Torsion
.
.
.
.
.
(15 pts) Find the limit, if exists, or show that the limit does not exist.6)
and
, prove that the
ㅋ
lim
1)
→
¥
lim
2)
90
→
lim
3)
in
ln
→
8.
가면
k ( 1-0 ) 가능 ?
( 1 一喆 )
(20 pts) Determine the differentiability of following function at the indicated point.
1)
at every point in the plane.
if
2)
≠
at the origin.
Not
if
9.
(10 pts) Show that if
, then
is a function of
is differentiable on
10. (10 pts) Let
. at
and
, where
and
di H
i
E
→
.
3
.
Et 3
,
are continuous in an open region
H1%% ) ER.dk
, where
cos
.
and
sin . Prove the following equality.
계산
11. (10 pts) The temperature at the point
on a metal plate is modeled by
.7)
1)
Find the direction of greatest increase in heat from the point
2)
Find the direction of no change in heat from the point
.
.
dcrectadderivo.ae
a
12. (10 pts) Find the tangent plane to the quadratic surface given by
at the point
.8)
13. (10 pts) Find all extreme points of
given by
and classify the extreme points.9)
☆
14. (15 pts) Find the extreme values of
subject to the constraint
$0
≦ .10)
15. (10 pts) Consider upper half sphere given by
.
We will divide the upper half sphere by two planes given by
be resulting surfaces obtained by dividing.
and
,
. Let
%
Find
and
such that
,
where
denote the surface area of
.11)
16. (20 pts) Evaluate the following integrals
&
sin
1)
2)
∞
3)
sin
豪
∞
4)
17. (20 pts)
1)
Find the area of the plane region bounded above by the spiral curve
polar axis, between
2)
Maybe
and
and below by the
.
Find the volume of the solid region bounded by the surface
and the planes
and
,
18. (10 pts) Find the volume of the solid region bounded by two elliptic paraboloids given by
and
.
,
"
0m에서
19. (15 pts) Let
be the rectangle with vertices
,
,
and
in the plane. Evaluate
the integral
cos
.
o
20. (15 pts) Let
be the ellipsoidal solid given by
≦ ,
≠ .
Evaluate the triple integral
.
21. (15 pts) A plane is capable of flying at a speed of
in still air. The pilot takes off from an
airfield and heads due north according to the plane’s compass. After
notices that, due to the wind, the plane has actually traveled
minutes of flight time, the pilot
at an angle
east of north. In
what direction should the pilot have headed to reach the intended destination? Find a direction vector.
22. Let
and
be the lines given by
1)
(10 pts) Show that the two lines can lie on two parallel planes.
2)
(5 pts) Find the distance between the two lines.
23. (20 pts)
1)
Let a b and c be vectors
2)
Let v ,
k
. Show that if a⋅b
be noncoplanar vectors in
v ×v
v ⋅ v ×v
a⋅c and a× b
, and let k ,
a× c, then b c.
be given by
,
Show that k ⋅ k × k
v ⋅ v ×v
.
24. (20 pts) Lissajous curves are the family of curve described by the parametric equations
sin
sin ,
where
and
are constants. For what value of
, is the curve closed?
⟨
25. (15 pts) A particle moves with position function r
components of acceleration at
⟩. Find the tangential and normal
.
26. (20 pts) The cornu spiral is given by
, and
cos
.
sin
1)
Find the arc length of this curve from
2)
Find the curvature of this curve at
to
.
.
27. (15 pts) Find the limit, if exist, or show that the limit does not exist.
lim
1)
t
→
¥
lim
2)
→
3)
0
cos
lim
→
t 0
28. (20 pts) The plane
intersects the paraboloid
and lowest points on this ellipse with respect to the
in an ellipse. Find the highest
-axis, and thereby determine the lengths of its
axes.
29. (20 pts) Determine the differentiability of following functions.
1)
, use the definition of differentiability.
if
2)
≠
.
if
30. (20 pts) Let
.
1)
Find the critical point(s) and test for local extrema.
2)
Does the function have an absolute maximum or minimum? Justify your answer.
31. (15 pts) Suppose that the second partial derivatives of
, and suppose that
are continuous on a disk with center
. Given a unit vector u ⟨
⟩, let
.
1)
Express
2)
Show that if
′
and
″
in terms of the partial derivatives of
maximum.
N
32. (15 pts) Evaluate following integrals.
g_fyma.be
1)
2)
3)
∞
and
.
, then
is a local
111
33. (20 pts) Let
be the surface given by
,
≦
∞.
Th
Evaluate the volume and surface area of
Maybe
34. (20 pts) Let
9
.
~
?
be the region in the first quadrant bounded by the curves
,
,
, and
.
Evaluate
.
☆
35. (20 pts) Let
be the solid generated by the curve whose equation is
about its axis, and
be a right cylinder whose equation is
in
-plane, revolving
. Find the volume common to
v4 a.
both.
⟨
36. (20 pts) Let r
⟩ be the position vector of a moving particle.
1)
Calculate the tangential component aT , the normal component aN and the curvature
2)
Find the linear equation of the osculating plane and normal plane of the curve at
when
.
.
37. (20 pts) Consider the tetrahedron as the following figure which is obtained by cutting a rectangular
parallelepiped. For a triangle ∆
with vertices
and
, we denote the area of the ∆
.
Show that the following holds.
.
38. (15 pts) The DNA molecule has the shape of a double helix. The radius of helix is
. Each helix rises about
during each complete turn. For
×
, where
complete turns,
estimates the length of each helix.
39. (15 pts) The position function of a spaceship is
r
⟨
⟩
ln
and the coordinates of a space station are
. The captain wants the spaceship to coast into
space station. When should the engines be turned off?
40. (15 pts) Does
exist? Explain why?
lim
→
1)
lim
→
2)
lim
where
→
sin
3)
lim
→
≠
by
41. (15 pts) Show that
is differentiable by definition.
≠
42. (20 pts) Let
Show that there exist
and
43. (20 pts) Let
.
, but not differentiable at
,
,
and
.
. Find
by using the chain rule.
44. (15 pts) Find the symmetric equation of the tangent line at a common point
and
.
of surfaces
ftp.dwntn
45. (15 pts) Find the shortest distance between the origin
and the point of the plane
by lagrange’s multiplier.
46. (15 pts) Let
be a differentiable function and
Prove that
cos i sin j .
sin .
cos
47. (5 pts for each one) Evaluate the following
1)
2)
sin
3)
4)
where
∈
∣ ≧
≧
≦ ≦
48. (10 pts) Show that
∞
1)
∞
2)
.
49. (15 pts) Find the volume of the solid that lies within the sphere
and below the cone
above the
非母 倒崎ist
.
⇒
de de de
倒崎
'
=
50. (15 pts) Let
be the region enclosed by the lines
Compute the double integral
,
,
-plane,
mode e
砒斗 E. 判爬神
, and
.
sin
51. (10 pts) For
nn
꼭폐
and
, show that
ln
cos
ln
뺘
.
.
彬心判
do.it
∞
52. (10 pts) Evaluate the integral
.
53. (20 pts) Consider the solid bounded below by the hemisphere
,
≧ , and above by the equation
cos .
1)
Sketch the solid in the space
2)
Find the volume of the solid.
.
54. (10 pts) Find the curvature of the curve with parametric equations
,
sin
.
cos
55. (15 pts) Show that
∞
.
cos
56. (15 pts) Prove the following equalities
1)
2)
a× b ⋅ c× d
a× b× c
a⋅c b⋅d
b× c× a
a⋅d b⋅c
.
c× a× b
57. (5 pts for each one) Find the following limit if it exists. If not, disprove it.
1)
lim
→
2)
lim
→
3)
4)
sin
∞
∞
58. (10 pts) By using the change of variables evaluate the following
where
∣
≦
59. Consider the function
1)
2)
.
For a unit vector u ⟨
⟩, find the directional derivative
Find the gradient ∇
60. (15 pts) Let
.
,
Show that
61. (15 pts) Solve the following
.
cos
and
sin .
.
u
.
1)
Show that
2) Find
∞
∞
∞
∞
∞
.
∞
.
∞
62. (15 pts) Find the absolute minimum value and absolute maximum value of
points on and within the triangle with vertices
,
63. (15 pts) Consider the solid bounded below by the hemisphere
cos .
1)
Sketch the solid in the space
2)
Find the volume of the solid.
.
and
at
.
,
≧ , and above by the equation