18.01 EXERCISES
Unit 1. Dierentiation
1A. Graphing
1A-1 By completing the square, use translation and change of scale to sketch
a) y = x2 2x 1
b) y = 3x2 + 6x + 2
1A-2 Sketch, using translation and change of scale
a) y = 1 + |x + 2|
b) y =
2
(x 1)2
1A-3 I
Problem Set 6 Solutions, 18.100C, Fall 2012
October 25, 2012
1
k=1 xk
Let sk = n=1 xk . Then we say that
k
sn converges to s as n .
converges to s if the sequence
Now suppose xk = s converges to some s = 0. We wish to show
k=1
that limk xk = 1. Let yk = x
TOPICS FOR THE FIRST TEST FOR 18.102, SPRING 2009
TEST: THURSDAY 5 MARCH, 9:30-11:00.
RICHARD MELROSE
Completeness of answers will be taken strong into account on the test. You are
supposed to get things really right!
1. Question 1
Will be on proving some
SOLUTIONS TO PROBLEM SET 4 FOR 18.102, SPRING 2009
WAS DUE 11AM TUESDAY 10 MAR.
RICHARD MELROSE
Just to compensate for last week, I will make this problem set too short and
easy!
1. Problem 4.1
Let H be a normed space in which the norm satises the paralle
Solutions to Problem set 9
P9.1: Periodic functions
Let S be the circle of radius 1 in the complex plane, centered at the origin,
S = cfw_z ; |z | = 1.
(1) Show that there is a 1-1 correspondence
(21.40)
C 0 (S) = cfw_u : S C, continuous
cfw_u : R C; con
Solutions to Problem set 10
Problem P10.1 Let H be a separable, innite dimensional Hilbert space. Show
that the direct sum of two copies of H is a Hilbert space with the norm
(23.18)
1
2
H H (u1 , u2 ) (u1 2 + u2 H ) 2
H
either by constructing an isometr
TEST 2 FOR 18.102: 9:35 10:55, 9 APRIL, 2009.
WITH SOLUTIONS
For full marks, complete and precise answers should be given to each question
but you are not required to prove major results.
1. Problem 1
Let H be a separable (partly because that is mostly wh
PRELIMINARY PROBLEMS FOR TEST 2 FOR 18.102, SPRING
2009
TEST ON THURSDAY 9 APR.
RICHARD MELROSE
Let H b e a separable Hilbert space with an orthonormal basis e i , i E W with
inner product (., .) a nd norm I I . I I :
( 1 ) Give an example of a sequenc
SOLUTIONS TO PROBLEM SET 5 FOR 18.102, SPRING 2009
WAS DUE l l A M TUESDAY 17 MAR.
RICHARD MELROSE
You should be thinking about using Lebesgue's dominated convergence a t several
points below.
Let f : R -+ C be an element of L1( R ) .Define
f (x) x
E [-
LECTURE NOTES FOR 18.102, SPRING 2009
151
Apendix. Exam Preparation Problems
EP.1 Let H be a Hilbert space with inner product (, ) and suppose that
B : H H C
(27.1)
is a(nother) sesquilinear form so for all c1 , c2 C, u, u1 , u2 and v H,
(27.2)
B (c1 u1 +
Solutions to Problem set 8
Problem 8.1 Show that a continuous function K : [0, 1] L2 (0, 2 ) has the
property that the Fourier series of K (x) L2 (0, 2 ), for x [0, 1], converges
uniformly in the sense that if Kn (x) is the sum of the Fourier series over
Solutions to Problem set 7
Problem 7.1 Question:- Is it possible to show the completeness of the Fourier
basis
exp(ikx)/ 2
by computation? Maybe, see what you think. These questions are also intended to
get you to say things clearly.
(1) Work out the Four
PROBLEM SET 2 FOR 18.102, SPRING 2009
DUE 11AM TUESDAY 24 FEB.
RICHARD MELROSE
I was originally going to make this problem set longer, since there is a missing
Tuesday. However, I would prefer you to concentrate on getting all four of these
questions real
PREPARATORY QUESTIONS FOR TEST 1
FOR 18.102, SPRING 2009.
RICHARD MELROSE
These questions, in addition to those in Problems3, are intended to help you
study for the test on Thursday March 5. In fact the questions on the test will be
very similar to some o
PROBLEM SET 1 FOR 18.102, SPRING 2009
SOLUTIONS
RICHARD MELROSE
Full marks will be given to anyone who makes a good faith attempt to answer
each question. The rst four problems concern the little L p spaces lp . Note that
you have the choice of doing ever
PROBLEM SET 3 FOR 18.102, SPRING 2009
MY SOLUTIONS.
RICHARD MELROSE
This problem set is also intended to be a guide to what will be on the in-class
test on March 5. In particular I will ask you to prove some of the properties of the
Lebesgue integral, as
Problem Set 4 Solutions, 18.100C, Fall 2012
October 11, 2012
1
Let X be a complete metric space with metric d, and let f : X X be a con
traction, meaning that there exists < 1 such that d(f (x), f (y ) d(x, y )
for all x, y X . Then there is a unique poin
Problem Set 10 Solutions, 18.100C, Fall 2012
December 5, 2012
1
We have a continuous K : [0, 1] [0, 1] R which is continuous. For each
xed x [0, 1], the function y K (x, y ) is thus a continuous function from
[0, 1] R, hence is Riemann-integrable. Since f
Problem Set 7 Solutions, 18.100C, Fall 2012
October 31, 2012
1
We wish to dene /2 as the smallest zero of cos(x), i.e. as the positive real
number such that cos(/2) = 0 and for 0 < x < /2, cos(x) = 0. Clearly
such a number, if it exists, is unique; if b a
2. Applications of Dierentiation
2A. Approximation
2A-1 Find the linearization of a + bx at 0, by using [(2), Notes A], and also by using the
basic approximation formulas. (Here a and b are constants; assume a > 0. Do not confuse
this a with the one in (2
Unit 3. Integration
3A. Dierentials, indenite integration
3A-1 Compute the dierentials df (x) of the following functions.
c) d(x10 8x + 6)
a) d(x7 + sin 1)
b) d x
d) d(e3x sin x)
e) Express dy in terms of x and dx if x + y = 1
3A-2 Compute the following i
Unit 4. Applications of integration
4A. Areas between curves.
4A-1 Find the area between the following curves
a) y = 2x2 and y = 3x 1
c) y = x + 1/x and y = 5/2.
b) y = x3 and y = ax; assume a > 0
d) x = y 2 y and the y axis.
4A-2 Find the area under the
Unit 6. Additional Topics
6A. Indeterminate forms; LHospitals rule
6A-1 Find the following limits
sin 3x
a) lim
x0
x
x2 3x 4
d) lim
x0
x+1
xa 1
g) lim b
x1 x 1
ln sin(x/2)
j) lim
x (x )2
ln x
x x
x sin x
f) lim
x0
x3
ln sin(x/2)
i) lim
x
x
cos(x/2) 1
x0
x
7. Innite Series
7A. Basic Denitions
7A-1 Do the following series converge or diverge? Give reason. If the series converges, nd
its sum.
1
1
1
1
+
+
+ . + n + .
4 16 64
4
12
n
c) 1 + + + . . . +
+ .
23
n+1
2n1
e)
3n
1
b) 1 1 + 1 1 + . . . + (1)n + . . .
a
September 20, 2012
1
We deﬁne
F = Q = {a +
√
2b|a, b ∈ Q} ⊂ R
We wish to show that F is a subﬁeld of R. In order to show this, we need
to show that a) 0, 1 ∈ F ; b) F is closed under addition and multiplication;
and c) if x ∈ F and x = 0, then −x ∈ F and
Problem Set 5 Solutions, 18.100C, Fall 2012
October 18, 2012
1
n
Let sn = n xi and n = i=1 yi be the partial sums. Then we claim
i=1
that n is the average of the rst n sn , i.e.
n =
s1 + s2 + sn
n
To see this, we will look at the coecients of the xk in th
Problem Set 9 Solutions, 18.100C, Fall 2012
November 30, 2012
1
Write B 1 := B 1 ([a, b]), d(f, g ) = supx[a,b] |f (x) g (x)| the uniform met
ric on real bounded functions, and d1 (f, g ) = d(f, g ) + d(f ' , g ' ) the given
metric on B 1 . Note that if f
Problem Set 8 Solutions, 18.100C, Fall 2012
November 14, 2012
For bounded functions f , g : [a, b] R, we use the notation |f | =
supcfw_|f (x)|x [a, b] and d(f, g ) = |f g |.
1
We have fn f and gn g uniformally. We wish to show that fb gn f g
uniformally
Problem Set 3 Solutions, 18.100C, Fall 2012
September 26, 2012
1
We have a metric space (X, d), and dene the function d' (x, y ) = d(x, y ).
We wish to show that (X, d' ) is also a metric space with the same open sets
as (X, d). We rst check that d' is a