y
y
e
d
R
`
R
y
f
R
t
y
v
v R
t
| y
R
y d vx
` v v cfw_ n n n y ` v v cfw_ e v x v x ` P v x v cfw_ y ` v x v cfw_ P
| y
y v x v cfw_ y yn q ` v x v cfw_ v ` P v cfw_ v cfw_ y ~ n r u r ~ I r ~ q v cfw_ n P
| y
y vx v cfw_ v ` vx v cfw_ e v cfw_
Practice Exam 2 B Solutions
Problem 1.
a) f (x, y) = x(y x2 ) is zero on the yaxis (x = 0) and on the
parabola y = x2 .
+
+
b) saddle point.
c) f = (y 3x2 ) + x; at P , f = 2, 1
j
.
d) w 2 x + y.
Problem 2.
a) By measuring, h = 100 for s 500, so
dh
ds
.2
Practice Exam 3A
1. Let ( y) be the center of mass of the triangle, with vertices at (2, 0),
x,
(0, 1), (2, 0) and uniform density = 1.
a) Write an integral formula for y . Do not evaluate the integral(s), but write
explicitly the integrand and limits of
Practice Exam 3A
1. Let ( y) be the center of mass of the triangle, with vertices at (2, 0),
x,
(0, 1), (2, 0) and uniform density = 1.
a) Write an integral formula for y . Do not evaluate the integral(s), but write
explicitly the integrand and limits of
Practice Exam 1B
Problem 1.
Let P , Q and R be the points at 1 on the x-axis, 2 on the y-axis and 3 on the z-axis, respectively.
i,
a) (6) Express QP and QR in terms of j and k.
b) (9) Find the cosine of the angle P QR.
Problem 2. Let P = (1, 1, 1), Q =
Practice Exam 1A
Problem 1. (15 points)
A unit cube lies in the rst octant, with a vertex at the origin (see gure).
a) Express the vectors OQ (a diagonal of the cube) and OR (joining O to the center of a face)
in terms of , k.
, j
b) Find the cosine of t
y
G
w
6
4
w
9
w
m
y
g
y
w
y
1
9
s
y
l
2200
G
y
G
h
9
g
w
2
f
w
9
U
8
m
9
U
B
U
U
S
S
1
l
y
1
y
9
m
g
j
j
l
9
w
9
y
s
g
1
m
4
9
D
u
h
g
g
G
y
1
6
2100
I
1000
1900
2000
P
y
m
w
w
l
4
l
y
8
g
1
m
l
y
w
f
y
w
4
l
m
y
2
1
6
l
w
w
4
1
w
f
y
f
l
B
9
w
l
4
g
8
9
Practice Exam 4A
Problem 1.
Let R be the solid region dened by the inequalities
2
x2 + y + z 2 a2 ,
x 0,
y0
(a) (15) Set up a triple integral in cylindrical coordinates which gives the volume of R. (Put in integrand and
limits, but DO NOT EVALUATE.)
(b)
Problem 5. a) f (x, y, z) = x; the constraint is g(x, y, z) = x4 + y 4 + z 4 + xy + yz + zx = 6. The
Lagrange multiplier equation is:
1 = (4x3 + y + z)
0 = (4y 3 + x + z)
f = g
0 = (4z 3 + x + y)
b) The level surfaces of f and g are tangent at (x0 , y0
Solutions of Practice Final A
k
Problem 1. P Q = 2, 0, 3; P R = 1, 2, 2; P Q P R = 2 0 3 = 6 4k
1 2 2
Equation of the plane: 6x y 4z = d. Plane passing through P : 6 0 1 4 0 = d.
Equation of the plane: 6x y 4z = 1.
Problem 2. Parametric equati
Practice Final A
Problem 1. Let P = (0, 1, 0), Q = (2, 1, 3), R = (1, 1, 2). Compute P Q P R and nd
the equation of the plane through P , Q, and R, in the form ax + by + cz = d.
Problem 2. Find the point of intersection of the line through P1 = (1, 2, 1
Practice Exam 4A
Problem 1.
Let R be the solid region dened by the inequalities
2
x2 + y + z 2 a2 ,
x 0,
y0
(a) (15) Set up a triple integral in cylindrical coordinates which gives the volume of R. (Put in integrand and
limits, but DO NOT EVALUATE.)
(b)