2210-90 Final Exam
Summer 2013
Name
K
Show all work arid include appropriate explanations when necessary. Correct answers
Instructions.
unaccompanied by work may not receive full credit. Please try to do all all work in the space provided and
circle your final answer.
1. (l5pts) Consider the three vectors u
(a) (3pts) Find u -t- v
—
=
2i
+j
—
k, v
=
i—j + 2k, and w
=
3i
+j
—
k
w
(2*)
+(-+4)
(3I÷1-)
(b) (3pts) Find vH
--1
(c) (3pts) Find w u
(d) (2pts) Which two vectors are orthogonal? Circle one:
B. u. w
A. u. v
(e) (4pts) Find u x w
J
1
-1
-I
-J—-.
2. (9pts) Find an equation for the plane which contains the points (1,0, 1), (2, —1, 1), and (3, 1, 2).
A
=
C-
t)1,
1
J
r
I
I.
-?-j3fr
3. (2Opts) Consider the function
f(x, y)
=
—
(a) (4pts) Find Vf(i, y), the gradient of
f)
x
=
.
3
zy + y
f.
j)X
(b) (4pts) Find (Df)(2, 1), the directional derivative of
f
at (2, 1) in the direction u
i
—
j.
=-
f)(,t)I,-I>>
(c) (4pts) Find the equation of the tangent plane to the graph of
(I
£
...J...
-
—
f
3
I(x-)-i (a-i)
(d) (4pts) Find the two critical points of
at the point (2, 1).
x_I_
f.
-
—
‘I
(e) (4pts) Is the critical point with the larger i-coordinate a local minimum, a local maximum, or a
saddle point?
I
I)
1
L
)_(
-
fyy2ty
t)
1
tt
t
VJJ4.
t.
4. (lOpts) Use the method of Lagrange multipliers to find the point on the hyperbola x
2
= 1 in
the first quadrant which is closest to the point (0, 1). Do this by finding the point (x, y) in the first
quadrant which minimizes the function f(x, y) = x + (y 1)2, the distance squared from the point
(0, 1), subject to the constraint g(x, y) = 2
1 = 0.
—
—
—
—
,J
÷
xç)
1
/XLL
U
J
J
1c)
•
J
I
0
1cv2
vø&M/&
5. (24pts) Evaluate the following double or triple integrals:
(a) (6pts)
+ 1)
fIR
dA, where R is the rectangle 0 <x
f+x
c
(x
< 2, 0 < y < 3.
flDX
ox 3x
fit
÷
k
3
(b) (6pts) [[(x2 + 3y) dA, where T is the triangle in the xy-plane with vertices (0,0), (2,0) and
J JT
(2,2).
-
ff
-
0
0
(oo)
(3
L
( o)
j
(xy t
-
“
(c) (6pts)fff(2x_4y+z2)dV,whereBisthebox0x1,_1y1,and0z3.
B
ii3
j
2
YXf
c
CL-i
I [[ I
(d) (6pts)
(z(x +
2
-
2
)) dV, where H is the hemisphere 2
2
x +y
2 +z
+z
1, y
0.
J J JH
iA
cJ
(
0
c
(
—
I
‘2..
2.1
P
-
]
A)°
\)
l
3
-t--/
2-1
3
“—I
6. (8pts) Match tile following vector fields with the graphs below by writing the letter in the blank
provided.
&
A
//// •\\\i
A ////
FQzy)=yi+xj
\\\N
F(x,y)=xi—yj
•7/ /
D
.‘
‘%
FQc,y)=—yi+xj
B
:\
\\\N
7//f
\
\SN
—
7
ill,
‘1 //
//7
-S\\
///
N\\\
c- c
•/
“
/ —
.1 /
D
/ / 7.z
-—--7
-
V.., /
/
7/
7. (lopts) Consider the vector field F(x, y)
(a) (4pts) Compute
101
=
i
2
zy
1fr
/2
/ / /
Ji’-
\
I’
.‘
-
•%S \
.
t
1
t.
41’
‘ii!
\
\
.7//I
\\\\
.7//I
.1 t t.
\
\
—
—
.——-
“
\“._-
/ /
I
— 2xyj.
F dr, where C
1 is the cnrve connecting (0, 0) to (1, 1) parameterized by
.
y(t) =t
x(t) =t
for 0 < t ; 1.
I
k3IitZy). (fr)
a
J--a
(b) (4pts) Compute
I
F dr, where C
2 is the cnrve connecting (0,0) to (1, 1) parameterized by
.
J C,
x(t)
for 0
L
<
t
=
2
y(t)
=
1.
) (wI
1
(C1-u
I
(
fjja
)at
3
(zt-it
Jo
-
3
(c) (2pts) Is F a conservative vector field? Circle one:
2-
YES
8. (6pts) Consider the vector field and the closed oriented curve C drawn below. In the blank provided,
write ‘P’ if the quantity is positive, ‘N’ if the quantity is negative, and ‘Z’ if the quantity is zero.
2:
P
\ \
/ //
F
7
7-
/ //
4
\ \\
9. (l6pts) Let F(x, y)
=
3
(z
—
y)i ± (y
3 + x)j and let C denote the unit circle traversed counterclockwise.
J’, F
(a) (Spts) Compute the line integral
.
dr using Green’s Theorem.
F’
u-rIF
—
C.
-y
3
x
Føtr
LA(s)
S
(b) (8pts) Compute the line integral
a form of Green’s Theorem).
J, F
n ds using the plane Divergence Theorem (which is also
3
tLF
x-y)
3 (i+y)ctA
C
S
Ucc..
‘e
f
zwr
f3r3o
L
0
10. (8pts) Consider the vector field F(i. p. z)
compute
=
j + xyk. Use the Divergence Theorem and to
3
i + zy
3
zx
ff
where S is the surface of the cylinder
vector.
fL
vF
4/
x2
F n
+p
2 < 1, 0 < z < 1 and n is the outward pointing unit normal
1i.
3
3
cc
+
(xy
(x+y)
R
Ov1 O&2ir
3 r o1&ci
r
5
z
(
3,tr)
11. (8pts) Let F(x, y,
z)
=
y
2
(x
2 + xz)j
+ 1)i + (3y
—
k.
2
xz
(a) (4pts) Compute div F
(xt ) +
(,2)
(3+,) .f
-
JiE
(b) (4pts) Compute curiF
A
A
/ix
•iP
x
2
y
3
I
-?<
:‘z3
+
(
Aj
12. (l6pts) Let S dente the top half of the unit sphere, a surface determined by the graph of
z
—
—
(a) (Spts) Find the surface area of S by evaluating an integral.
)CXL1
y)
fv
A
I
\i
\)4e-
—.Y
—2-
4 (x
?0
er
r(i-rj
2-Tr(j
o
fr,c2y2M
HJ
D
ç
‘/t
i%)
2I
IL)
(b) (8pts) Use Stokes’s Theorem to evaluate
ff
(curl F) n
where F denotes the vector field
F
.
=
2 + z)i + yzj + xyk.
(x
r
v-Lki
L,
COf
çF
S
6
+
cj-
7.
-=
r’L±)