Geometry of systems of equations
1. Write a 3by3 system of equations
a) with no solutions and where all the planes are parallel;
b) where two planes are parallel and the other intersects them;
c) where the planes are all dierent and all intersect in a l
18.02 Practice Exam 3 A
1. Let (, y ) be the center of mass of the triangle with vertices at (2, 0), (0, 1), (2, 0) and
x
uniform density = 1.
a) (10) Write an integral formula for y . Do not evaluate the integral(s), but write explicitly the
integrand a
Solutions for PSet 10
1. (E7:1,2)
1 Let l be the arclength of C, and parameterize C by its arclength: (t) =
(x(t), y(t). Then (t) = 1 thus n(t) = (y (t), x (t).
We have
l
C
f n ds =
0
(P (t), Q(t) (y (t), x (t) dt =
C
Q dx + P dy
2 Using Greens Theorem fo
Partial derivatives
Partial derivatives
Let w = f (x, y) be a function of two variables. Its graph is a surface in xyzspace, as
w
pictured.
w=f(x,y)
Fix a value y = y0 and just let x vary. You get a function of one variable,
(1)
w = f (x, y0 ),
w=f(x,y 0
18.02 Practice Exam 2 B
Problem 1. Let f (x, y) = x2 y 2 x.
a) (5) Find f at (2, 1)
b) (5) Write the equation for the tangent plane to the graph of f at (2, 1, 2).
c) (5) Use a linear approximation to nd the approximate value of f (1.9, 1.1).
d) (5) Find
18.02 Practice Exam 2 B Solutions
Problem 1. a) f = 2xy 2 1, 2x2 y = 3, 8 = 3 8
i
j.
b) z 2 = 3(x 2) + 8(y 1)
or z = 3x + 8y 12.
c) x = 1.92 = 1/10 and y = 1.11 = 1/10. So z 2+3x+8y = 23/10+8/10 = 2.5
d)
df
ds
u
= f u = 3, 8
1, 1
3 + 8
5
=
=
2
2
2
Probl
Uses of dot product
1. Find the angle between i + j + 2k and 2i
j + k.
Answer: We call the angle and use both ways of computing the dot product.
Algebraically we have
(i + j + 2k) (2i
j + k) = 2
1 + 2 = 3.
Geometrically
(i + j + 2k) (2i
j + k) = i + j +
18.02 Practice Exam 3 A Solutions
1
2
1. a) The area of the triangle is 2, so y =
1
22y
y dxdy.
0
2y2
b) By symmetry x = 0.
2. = x = r cos . I0 =
2
0
1
0
r2 rdrd =
D
r2 r cos rdrd = 4
/2
1
0
/2
r4 cos drd = 4
0
0
1
4
cos d =
5
5
3. a) Nx = 6x2 + by
Solutions for PSet 7
1. (9.8:7) Hint: It might help to dene a scalar eld F (x, y, z) = f (u(x, y, z), v(x, y, z)
where u, v are as needed. We rst assume that x = 0. Given g(x, y) = z , we
know
g
F/x
g
F/y
=
;
=
.
x
F/z
y
F/z
Now, we need only use the chai
Solutions for PSet 9
1. (11.9:8) Using Fubinis Theorem (we assumed that the double integral exists):
tx
[0,t][1,t]
t
1
ey
dx dy =
y3
t
t
1
0
tx
ey
dx
y3
dy =
t2
t y
e 1
y tx t
dy =
dy =
y 3 e y
t
ty 2
x=0
1
t
1 t2
1
1
1
1
1 2
3e y +
= 2 3 et + 3 et
ty y=
18.02 Practice Exam 3 B
1
Problem 1. a) Draw a picture of the region of integration of
2x
dydx.
0
x
b) Exchange the order of integration to express the integral in part (a) in terms of integration
in the order dxdy. Warning: your answer will have two piec
The

inverse of a matrix
We now consider the pro5lern of the existence of multiplicatiave
inverses for matrices. A t this point, we must take the noncommutativity
of matrix.multiplicationinto account.Fc;ritis perfectly possible, given
a matrix A, that t
18.02 Practice Exam 4A  Solutions
lb) We begin with
fz = e5yz
fy = exz + 2yz
f, = e5y y2 1
+ +
Integrating f, we get f = e"yz
with the above equations we get
+ g(y, z).
Differentiating and comparing
+
+
Integrating g, we get g = y2z h(z). Then g , = y2 h
Intersection of a line and a plane
1. Consider the plane P = 2x + y 4z = 4.
a) Find all points of intersection of P with the line
x = t,
y = 2 + 3t,
z = t.
b) Find all points of intersection of P with the line
x = 1 + t,
y = 4 + 2t,
z = t.
c) Find all poi
18.02 Practice Exam 4 8
Problem 1. (10 poir~ts)
+
Let C 1)e the portion of the cylinder z2 y2 2 1 lying in the f i ~ octant (:L 2 0, y 2 0, z 2 0)
t
which gives the
and 1)elow t,he plane z = 1. Set 1111a triple integral in cglCri,i.rlr(ctr.lcoo~r.rlCri,i.
Solutions to linear systems
1. Consider the system
x + y + 2z = 0
2x + y + cz = 0
3x + y + 6z = 0.
a) Take c = 1 and nd all the solutions.
b) Take c = 4 and nd all the solutions.
Answer: a) In matrix form we have
0
10 1 0 1
1 1 2
x
0
@ 2 1 1 A@ y A = @ 0
Areas and Determinants
1. Compute
Answer:
6 5
.
1 2
6 5
1 2
=62
5 1 = 7.
2. Compute the area of the parallelogram shown.
y
(2, 3)
( 1, 2)
x
(1, 0)
Answer: The area is given by the determinant of the vectors determining the parallelogram.
y
A = h1, 3i
B =
Solutions to linear systems
1. Consider the system of equations
x + 2y + 3z = 1
4x + 5y + 6z = 2
7x + 8y + cz = 3.
a) Write the system in matrix form.
b) For which values of c is there exactly one solution?
c) For which values of c are there either 0 or i
18.02 Practice Exam 4A
Problem 1. (15)
a) Show that the vector eld
is conservative.
F = ex yz, ex z + 2yz, ex y + y 2 + 1
b) By a systematic method, nd a potential for F.
c) Show that the vector eld G = y, x, y is not conservative.
Problem 2. (20) Let S b
Solutions for PSet 8
1. (10.5:11) Parameterize the sides of the square C by maps si : [0, 1] R2 by
s1 (t)
s2 (t)
s3 (t)
s4 (t)
=
=
=
=
(1 t, t);
(t, 1 t);
(t 1, t);
(t, t 1).
1 + 1
dt+
(1 t) + t
1
1 1
dt+
t + (1 t)
With this parametrization:
C
dx + dy
=

18.02 Practice Exam 3 B Solutions
q (1,2)
y = 2x
x=1
q
1. a)
(1,1)
y = x
q
2. a) dA =
1
b)
0
2
dxdy +
y/2
1
dxdy.
1
y/2
(the rst integral corresponds to the bottom half 0 y 1, the second
integral to the top half 1 y 2.)
r sin
rdrd = sin drd.
r2
M=
d
18.02 Practice Exam 4B Solutions
Problem 1.
/2
1
0
1
0
2
0
0
Problem 2.
a) sphere: = 2a cos .
c)
r2 r dz dr d.
/4
0
2a cos
b) plane: = a sec .
2 sin d d d.
a sec
Problem 3.
a) y (2xy + z 3 ) = 2x =
2
z (x
2
3
2
2
x (x + 2yz);
z (2xy + z ) = 3z = x (y
2
Speed and arc length
1. A rocket follows a trajectory
r(t) = x(t) i + y(t) j = 10t i + ( 5t2 + 10t)j.
Find its speed and the arc length from t = 0 to t = 1.
Answer:
p
dr
ds p 2
= 10i+( 10t+10)j )
= 10 + ( 10t + 10)2 = 10 1 + (1
dt
dt
Z 1
Z 1p
ds
Arc lengt