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 ch
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
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 thi
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
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.
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
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
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
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
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 f
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 sens
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
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
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 t
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
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 pro
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
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 funct
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.
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;
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 term
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)
Unit 5. Integration techniques
5A. Inverse trigonometric functions; Hyperbolic functions
5A-1 Evaluate
a) tan1 3
b) sin1 ( 3/2)
c) If = tan1 5, then evaluate sin , cos , cot , csc , and sec .
d) sin1
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 si
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 + + + . . . +
+ .
2
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 mult
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
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 ) +
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 u
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 t