Determinants and areas
1 2
.
3 4
1. a) Compute
b) Compute
1
3
2
.
4
c) Compute
3 4
.
1 2
Answer: a)
1 2
3 4
b)
1
3
c)
3 4
1 2
2
4
=14
=14
=32
23=
2.
( 2) ( 3) =
2.
4 1 = 2.
2. Find the area of the quadrilateral shown.
y
(4, 3)
(1, 2)
x
(3, 1)
Answer:
y
B
18.02 Practice Exam 2 A
Problem 1. (10 points: 5, 5)
Let f (x, y) = xy
x4 .
a) Find the gradient of f at P : (1, 1).
b) Give an approximate formula telling how small changes
w in the value of w = f (x, y) at the point (x, y) = (1, 1).
x and
y produce a sm
18.02 Practice Exam 2 A Solutions
Problem 1.
a) rf = (y
b)
w'
3
4x3 ) + x; at P , rf = h 3, 1i.
x+
y.
Problem 2.
dh
h
'
' .2.
a) By measuring, h = 100 for s ' 500, so
ds u
s
b) Q is the northernmost point on the curve h = 2200; the vertical distance betwe
Vector problems
1. a) A river ows at 3 mph and a rower rows at 6 mph. What heading should the rower
take to go straight across a river?
b) Answer the same question if the river ows at 6 mph and the rower rows at 3 mph.
Answer:
river = 3
_
/O
net velocity
MODEL ANSWERS TO HWK #7
(18.022 FALL 2010)
2
2
(1) (a) F is a gradient eld given by the potential f (x, y) = x + y2 + C.
2
(b) The ow line r(t) = (x(t), y(t) satises x0 (t) = x(t), y 0 (t) = y(t) and x(0) = a, y(0) = b.
The solution is x(t) = A1 et and y(
Cusp on the cycloid
The graph of the cycloid has point where the graph touches the x-axis. These points are
usually called cusps.
What you saw in the previous video was an analysis of the behavior of the trajectory near
the cusps. We will go through that
Matrix inverses
0
1
1 2 1
1. a) Find the inverse of A = @ 1 4 0 A.
2 1 5
b) Use part (a) to solve the system of equations
x
+ 2y + z
= 1
x
+ 4y
= 0
2x + y
+ 5z = 3
Answer: a) We compute in sequence: the determinant, the matrix of minors, the matrix of
cof
Velocity and acceleration
Now we will see one of the benets of using the position vector. Lets assume we have a
moving point with position vector
r(t) = x(t)i + y(t)j.
(We assume the point moves in the plane. The extension to a point moving in space is
tr
Geometry of linear systems of equations
Very often in math, science and engineering we need to solve a linear system of equations.
A simple example of such a system is given by
6x + 5y = 6
x + 2y = 4.
You have probably already learned algebraic techniques
18.02 Exam 2 Solutions
Problem 1. a) rf = h2xy 2
1, 2x2 yi = h3, 8i = 3 + 8
i
j.
c)
y = 1.1 1 = 1/10. So z ' 2+3 x+8 y = 2 3/10+8/10 = 2.5
b) z
d)
2 = 3(x
2) + 8(y
x = 1.9 2 =
df
ds
u
1)
1/10 and
= rf u = h3, 8i
or
h 1, 1i
p
=
2
z = 3x + 8y
12.
3+8
5
p
=
MODEL ANSWERS TO HWK
(18.022 FALL 2010)
6
(1) The curve C is given in rectangular coordinates by ~( ) = (f ( ) cos( ), f ( ) sin( ). Then
r
~0 ( ) = (f 0 ( ) cos( )
r
f ( ) sin( ), f 0 ( ) sin( ) + f ( ) cos( ),
and the arc length of C is given by
Z
s( )
18 014 Problem Set 11 Solutions
Total: 32 points
Problem
limsup.
: Prove that a sequence converges if, and only if its liminf equals its
Solution (4 points) Suppose cfw_an is a sequence that converges to a limit L. Then
given > 0, there exists an integer
Meaning of Matrix Multiplication
1. In this problem we will show that multiplication by the matrix
!
1
1
p
2
1
p
2
A=
p
2
1
p
2
acts by rotating vectors 45 counterclockwise. As usual, we write the vector v = xi + yj as
x
a column vector
.
y
a) Show that
Volumes and determinants
1. a) Find the volume of the parallelepiped with edges given by the origin vectors h1, 2, 4i,
h2, 0, 0i, h1, 5, 2i
Answer: The gure below shows the box.
0
1
1 2 4
The volume is | det(A, B, C)| = det @ 2 0 0 A = |
1 5 2
2 ( 16)| =
Cross product
1. a) Compute h1, 3, 1i h2, 1, 5i.
b) Compute (i + 2j) (2i
3j).
Answer: a) We use the determinant method:
h1, 3, 1i h2, 1, 5i =
i
1
2
j k
3 1
1 5
j(3) + k( 7) = h16, 3, 7i
= i(16)
b) Using determinants we get
(i + 2j) (2i
3j) =
i
1
2
j k
2 0
A.l
Integers and exponents
Definition. A set of real numbers is called an inductive
(a) The number 1 is in the set.
(b) For every x in the set, the number x + 1 is in
the set also.
The set R+ of positive real numbers is an example of an
inductive set. [
18.02 Exam 2
Problem 1. Let f (x, y) = x2 y 2
x.
a) (5) Find rf 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 the direct
Parametric Curves
General parametric equations
We have seen parametric equations for lines. Now we will look at parametric equations of
more general trajectories. Repeating what was said earlier, a parametric curve is simply
the idea that a point moving i
B.1
Square roots, and the existence of irratiOnal numbers.
Denition. If b2 = a, then we say that b is a sguare root of a.
A negative number has no square root (see Theorem I .20), and the number 0 has
V only one square root,'namely 0. We shall show that a
18 014 Problem Set 10 Solutions
Total: 12 points
Problem : Evaluate
log(a + bex )
a. lim p
b. lim log(x) log(1
x!1
x!1
a + bx2
x).
x)
Solution (4 points) For part (a), we rst evaluate the limits limx!1 log(a+be
x
x
limx!1 pa+bx2 . The second limit can be
Distances to planes and lines
In this note we will look at distances to planes and lines. Our approach is geometric. Very
broadly, we will draw a sketch and use vector techniques.
Please note is that our sketches are not oriented, drawn to scale or drawn
Parametric equations of lines
General parametric equations
In this part of the unit we are going to look at parametric curves. This is simply the idea
that a point moving in space traces out a path over time. Thus there are four variables
to consider, the
MODEL ANSWERS TO HWK
9
1. There are a number of ways to proceed; probably the most straightforward is to view the region D as something of type 2:
ZZ
Z 2 Z 2 y
x + y dx dy =
x + y dx dy
D
=
=
=
=
Z
Z
Z
1
2
1
2
y 2 2y
x
+ yx
2
y)2
(2
dy
y 2 2y
+ y(2
2
1
2
Matrix multiplication
1. Let A =
1 3
4 5
0
1
1 4
1 1 1
@ 1 5 A, D = 3 0 , E = 5 .
, B=
, C=
4 5 6
0 3
3
1 6
For each of the following say whether it makes sense to compute it. If it makes sense then
do the computation.
(i) AA (ii) AB (iii) AC (iv) AE (v