Distances to planes and lines
1. Using vector methods, nd the distance from the point (1,0,0) to the plane 2x + y
Include a cartoon sketch illustrating your solution.
2z = 0.
Answer: The sketch shows
18.100B Problem Set 1
Due Friday September 15 2006 by 3 PM
Problems
1) (10 pts) Prove that there is no rational number whose square is 12.
2) (10 pts) Let S be a non-empty subset of the real numbers,
The Tangent approximation
4. Critique of the approximation formula.
First of all, the approximation formula for functions of two or three variables
(6)
(7)
w
x
w
w
w
x
w
y
x +
0
x +
0
w
y
y,
if
x 0,
Functions of two variables
Examples: Functions of several variables
f (x, y) = x2 + y 2 ) f (1, 2) = 5 etc.
f (x, y) = xy 2 ex+y
f (x, y, z) = xy log z
Ideal gas law: P = kT /V .
Dependent and indepen
18.100B Problem Set 2
Due Friday September 22 2006 by 3 PM
Problems
1) (10 pts) Prove that the empty set is a subset of every set.
2) (10 pts) If x, y are complex, prove that
|x|
|y| |x
y|
(Hint This
18.100B Problem Set 4
Due Friday October 6 2006 by 3 PM
Problems
1) Give an example of an open cover of the set E = cfw_(x , x2 ) 2 R2 : x2 + x2 < 1 R2 which
2
has no nite subcover. (As usual, R2 is e
SOLUTIONS TO PS2
Xiaoguang Ma
Problem 1
Proof. It is true that for any two sets A, B, the intersection A \ B is a subset of
A. Now consider = A \ Ac . So is a subset of A for any set A.
Problem 2
Proo
18.100B Problem Set 5
Due Friday October 20 2006 by 3 PM
Problems
1) Let M be a complete metric space, and let X M. Show that X is complete if and only if X
is closed.
2) a) Show that a sequence in an
18.100B Problem Set 3
Due Friday September 29 2006 by 3 PM
Problems
1) (10 pts) In vector spaces, metrics are usually de ned in terms of nor s which measure the length
of a vector. If V is a vector sp
18 100B Problem Set 9 Solutions
awyer Tabony
1) First we need to show that
1
nx + 1
converges pointwise but not uniformly on (0, 1). If we x some x 2 (0, 1), we have that
1
lim fn (x) = lim
= 0.
n!1
n
The Tangent Approximation
1. The tangent plane.
w
For a function of one variable, w = f (x), the tangent line to its graph
at a point (x0 , w0 ) is the line passing through (x0 , w0 ) and having slope
18 100B Problem Set 5 Solutions
awyer Tabony
1) We have X M, with M complete. X is complete if and only if every Cauchy sequence of X
converges to some x 2 X. Let (xi ) be Cauchy, with xi 2 X. M being
18.100B Problem Set 9
Due Friday December 1 2006 by 3 PM
Problems:
1) Let fn (x) = 1/(nx+1) and gn (x) = x/(nx+1) for x 2 (0, 1) and n 2 . Prove that fn converges
pointwise but not uniformly on (0, 1)
SOLUTIONS TO PS 8
Xiaoguang Ma
Solution/Proof of Problem 1 From f (x) = f (x2 ), we have
1
1
1
f (x) = f (x 2 ) = f (x 4 ) = = f (x 2 )
1
Now let yn = x 2 , and assume x = 0, so lim yn = 1. Since f is
18 100B Problem Set 3 Solutions
awyer Tabony
1) We begin by dening d : V V ! R such that d(x, y) = kx yk. Now to show that this
function satises the denition of a metric. d(x, y) = kx yk 0 and
d(x, y)
18.100B Problem Set 6
Due Friday October 27 2006 by 3 PM
Problems
P
P
1) Prove that if
|an | is converges, then
|an |2 also converges.
2) Prove that
1
X
n=1
1
1
= .
n(n + 1)(n + 2)
4
Hint: Use a teles
18 100B Problem Set 1 Solutions
awyer Tabony
1) The proof is by contradiction. Assume 9r 2 Q such that r2 = 12. Then we may write r as a
b
with a, b 2 Z and we can assume that a and b have no common f
18 100B Problem Set 7 Solutions
awyer Tabony
1) We have ai > 0 and ai
convergence of
1
ai for all i = 0, 1, 2, ., and lim ai = 0, and we want to show the
i!1
1
X
( 1)i ai = a0
a1 + a2
.
i=0
So we de
18.100B Problem Set 8
Due Thursday November 9 2006 by 3 PM
Problems
1) Let f : [0, 1) !
be continuous, and suppose
f (x2 ) = f (x)
holds for every x 0. Prove that f has to be a constant function.
Hint
SOLUTIONS TO PS6
Xiaoguang Ma
Solution/Proof of Problem 1. Since
P
|an | converges, lim |an | = 0. So 9N 2
n!1
such that for n > N , |an | < 1. Thus for n P N we have |a2 | |an | and
n
P 2
by the comp
StokesI Theorem
Our text states and proves Stokes' Theorem in 12.11, but ituses
the scalar form for writing both the line integral and the surface integral
involved. In the applications, it is the vec
18.100B Problem Set 7
Due Friday November 3 2006 by 3 PM
Problems
1) Consider an in nite series with alternating signs
a0
a1 + a2
a3 + a4
. =
X
( 1)n an .
Prove the Leibnitz criteria for convergence:
SOLUTIONS TO PS4
Xiaoguang Ma
Solution/Proof of Problem 1 Consider the open set
Bn =
(x1 , x2 ) 2 R2 : x2 + x2 < 1
1
2
1
n
Then we can see that E [Bn because for any point (x, y) 2 E, x2 + y 2 < 1, we
18.100B Problem Set 10
Due Friday December 8 2006 by 3 PM
Problems:
1) Let (fn ) be the sequence of functions on R dened as follows.
2
f0 (t) = sin t and fn 1 (t) = fn (t) + 1 for n 2 .
3
Show that fn