# Document 11890544 ```2210-90 Exam 1
Fall 2013
/
Name
Show all work and include appropriate explanations when space is provided.
Instructions.
1. (l0pts) Consider the vectors u
=
(1.5,—i) and v
Correct
(4,1,1). Find
(a) (2pts) v + 3u
&gt;
1
q,i,
(b) (2pts) Iu
tU
(c) (3pts) u v
3
(iq)
-i-(c)(’+ (—0(i)
(d) (3pts) u x v
fri
(i
&ccedil;
4
-
1
2. (7pts) Find a unit vector that is orthogonal to the plane containing the points P
(2,0,1), and R= (0,0,5).
=
(1, 1, —1),
Q
=
—
)
-If_l1)
)— (tt-i&gt;
&lt;0
c
o
= 1
I
/
—1
j
1-1
-I
I
.
‘1z))
—
3. (6pts) The following equation determines a sphere. Find the center and radius of the sphere. Hint:
Complete the squares.
9
x 2 &plusmn;y++2x—6y=—6
Center=
(1,
(
—l
1
,
3
,
0
(+
)*
ii
—
L-y)
(
1
3
(xi)(
1
2-3
4. (6pts) Find a vector equation for a line that passes through the point (1, 2, 3) at t
(—1,4,—2) at t = 1. Note, your answer should be a curve
r(t)
=
=
0 and the point
(x(t),y(t),z(t))
ic
-l-t)-i,3&gt;+
1 3-5t&gt;
12t)2+2t
z,-c&gt;
1
t2-
5. (l6pts) Suppose a particle’s position at time t is given by the curve
5ti + sin (3t)j
r(t)
—
cos (3t)k.
2 x + cos
2x
For this problem, it may be helpful if you remember the trig identity sin
=
1.
(a) (3pts) Find the velocity v(t) of the particle at time t.
3
3€&ccedil;(3t)3
i-
3si
(b) (3pts) Find the acceleration a(t) of the particle at time t.
3
&amp;t —stu(3t)j-Ftos(3t)fr-
Ift
(c) (3pts) Find the arc length of the curve between times t
tt5 4c ;:;;
[si tt II.
3
L--
r
3
iiv lk)l)
—
—
r-LIi
(e) (4pts) Find the curvature
—I
J—
/
[0
t(t)
4
=
0 and t
=
3.
3
u
0
(d) (3pts) Find the unit tangent vector T(t)
=
v(t)
IIv(t)IV
ct3cos()fr?st[3),)
HT’(t)II of the particle’s path at time t.
11vtI
-I(3t) (3t)
2
6. (2lpts, 1.5pts each) Match the equation with the type of surface it determines by writing the appro
priate capital letter (A-I) in the provided blank. The last six equations are written in cylindrical or
spherical coordinates. Some letters may be used more than once.
9
9
x- + y
A
I3
1,
9
X
9
+4
9
—
B Ellipsoid
—Y=l
2 +y
x
2 +
2
A Elliptic Paraboloid
1
z
=
C Hyperboloid of one sheet
1
2
2
D Hyperboloid of two sheets
x —2y —z =1
9
z
&amp;
=
9
x-
9
E Hyperbolic Paraboloid
—
9
9
x- + y- + zx
‘4
=
—
9
9
2
.2
x + yr
=
F Plane
1
G Sphere
z
H Cylinder
1
=
I Cone
=
r=1
3
I
&amp;
2r 2 +z 2 =1
7r
Y
5
2
p=
F
pcos
=
1
7. (9pts) Convert between Cartesian, cylindrical, and spherical coordinates as indicated. Please simplify
as much as possible.
(a) Find the spherical coordinates of the point with Cartesian coordinates (—1, 1,
4
37r/
2—
1L
k2&plusmn; t*ii’)
(b) Find the cylindrical coordinates of the point with Cartesian coordinates (1,
9=
—2)
-7__
--
4—
173
(c) Find the Cartesian coordinates of the point with spherical coordinates (4,
()
I
=
1
()s)
I.. 2-/i 2-)
3
,)
=
Ll-L
1
T(
-
8. (l2pts) Evaluate the following limits. If they do
(a)
urn
(x,y)(O,O)
)
2
cos(x
i
I
toS(0)
—
—
‘DNE’ and explain why.
riot exist, write
2
y
I
—
4
(b)
lirn
(x.y)—*(O.O)
x + 2y
X
p
(o t.Ij -(S (o)i t-&lt;.i
—
+
lII.4.k
)-(VgO
1
(
1
Li
!..&plusmn;_J
1o
to
)-(u
(x
u
1
(c)
lirn
(xy)—(OO)
—y
2 + y
X
2
0 )
X-
x-j
‘j
‘
—z
---
Hint: Use polar coordinates.
i&amp;
(—J)
r
‘
o
cD
—
iit9
&deg;&deg;
h
9. (l3pts) Consider the function
=
2
sin (xy)
—
x.
(a) (7pts) Find the equation of the tangent plane to the graph of
sy)
cy)
=
f(x, y)
at the point (1,0, —1).
l.
ftiio)
fx
z
2x
SIH
Ixy)
cs(xy)
3
X
co
+
Ocy)
I
(1, o)
— .fcit&ocirc;)
-x#y
(b) (6pts) Find the following second derivatives:
Z
.
f(x,y)=
ZsaLxy)
+vq ‘/xytc(fy) —xyii-(xy)
ii. f(x,y)=
iii. f(xy)=
toS(xy) _x
1
X
5
3
(xy)
4
-5.
```