[5]
[4]
[2]
(i) No. f (x) = x2 .
(ii) Does every continuous function f : R R have a xed point?
[2]
(ii) No. f (x) = x + 1.
(iii) Does every function f : [0, 1] [0, 1] (not necessarily continuous) have a xed
point?
(iii) No.
f (x) =
[2]
1 x=0
0 x > 0.
(iv)
Problem 2) Matching problem (10 points) No justications are needed.
Problem 3) Matching problem (10 points) No justications are needed.
a) (6 points) Match the volumes of solids.
a) (6 points) Match the integrals with the pictures.
Integral
1
1(11 x)
11 |
8)
T
F
If we blow up a balloon so that the volume V changes with constant rate,
then the radius r(t) changes with constant rate.
15)
Solution:
This is a related rates problem. The balloon radius grows slower for large volumes.
d
9)
T
F
The identity
f.
dx
BUS/ST 350 Fall '07
Exam 3 Version 1
page 2
1. A survey of 296 recent deliveries by Fumble Express shows that 131 were delayed. The 95%
p Fumble
Express is
confidence interval for the proportion of delayed deliveries by
a. (109, 153) b. (.386, .499) c. (.
3B Dynamics and Relativity
The motion of a planet in the gravitational eld of a star of mass M obeys
2
d2r
GM
= 2 ,
h
2
3
dt
r
r
r2
d
=h,
dt
where r(t) and (t) are polar coordinates in a plane and h is a constant. Explain one of
Keplers Laws by giving a g
(b) By evaluating the Wronskian, or otherwise, nd functions p(x) and q(x) such
that () has solutions y1(x) = 1 + cos x and y2(x) = sin x.
[3]
W (x) = (1 + cos x) cos x sin x( sin x) = 1 + cos x.
So p(x) = W (x)/W (x) = sin x/(1 + cos x).
sin x
cos x + q(x
Problem 1) True/False questions (20 points). No justications are needed.
1)
T
F
The functions ex2+y3yand x2+ y3 y have the same critical points.
2)
T
F
The line r(t) = t2, t2, t2hits the plane x + y + z = 100 at a right angle.
3)
T
F
4)
T
F
The relation |
15)
Solution:
Use Hopital
T
F
Hopitals rule assures
0
that cos(x)/ sin(x) has a limit as x .
Solution:
0
The nominator does not go to zero for x .
8)
T
F
The derivative of f(f (x) is f (f (x) for any dierentiable function f .
16)
Solution:
This is not the
Problem 1) TF questions (20 points) No justications are needed.
1)
T
F
1 is the only root of the log function on the interval (0, ).
Solution:
Yes, log is monotone and has no other root.
2)
T
F
exp(log(5) = 5, if log is the natural log and exp(x) = exis t
Problem 1) TF questions (20 points) No justications are needed.
1)
T
F
1 is the only root of the log function on the interval (0, ).
2)
T
F
exp(log(5) = 5, if log is the natural log and exp(x) = ex is the exponential
function.
3)
T
F
The function cos(x) +
Problem 2) Matching problem (10 points) Only short answers are needed.
Problem 3) Matching or short answer problem (10 points).
We name some important concepts in this course. To do so, please complete the sentences with
one or two words. Each question is
By the pigeonhole principle, there exist distinct i, j cfw_0, 1, . . . , N + 1
with gi= gj. WLOG i < j. Then gji = e so j i cfw_n > 0 : gn= e
and were done.
Suppose g = khk1. Then for n Z we have gn= khnk1 so
gn= e khnk1 = e hn(= k1k) = e.
[5]
Thus cfw_n
Glaeser Question
Fall 2013 Micro General Exam
It has been suggested that the ability to deter crime with large penalties may be limited by the
willingness of police to accept bribes that are below the penalty amount.
(1) (1) Craft a simple model of crime
2
Paper 1, Section I
3D Analysis I
What does it mean to say that a sequence of real numbers (x n) converges to x?
Suppose that (xn) converges to x. Show that the sequence (yn) given by
n
yn= 1
n
xi
i=1
also converges to x.
Paper 1, Section I
4F Analysis I
MATHEMATICAL TRIPOS
Part II
2016
List of Courses
Algebraic Geometry
Algebraic Topology
Applications of Quantum Mechanics
Applied Probability
Asymptotic Methods
Automata and Formal Languages
Classical Dynamics
Coding and Cryptography
Cosmology
Dierential G
2
Paper 1, Section I
3F Analysis I
Find the following limits:
(a) lim
sin x
x
0
x
(
(b) lim 1 + x)1/x
0
x
x
(1 + x) 1+x cos4x
(c) lim
ex
x
Carefully justify your answers.
[You may use standard results provided that they are clearly stated.]
Paper 1, Secti
MATHEMATICAL TRIPOS
Part IB
2015
List of Courses
Analysis II
Complex Analysis
Complex Analysis or Complex Methods
Complex Methods
Electromagnetism
Fluid Dynamics
Geometry
Groups, Rings and Modules
Linear Algebra
Markov Chains
Methods
Metric and Topologica
MATHEMATICAL TRIPOS
Part II
2015
List of Courses
Algebraic Geometry
Algebraic Topology
Applications of Quantum Mechanics
Applied Probability
Asymptotic Methods
Classical Dynamics
Coding and Cryptography
Cosmology
Dierential Geometry
Dynamical Systems
Elec
HARVARD UNIVERSITY
DEPARTMENT OF ECONOMICS
General Examination in Macroeconomic Theory
SPRING 2013
You have FOUR hours. Answer all questions
Part A (Prof. Laibson): 48 minutes
Part B (Prof. Aghion): 48 minutes
Part C (Prof. Farhi): 72 minutes
Part D (Prof
Art 113
26 September 2011
Duality and Moravian Roots
Rudy S. Ackermans 1984 sculpture Duality is an abstract piece on a grand scale.
The sculpture, approximately 15 feet high, is a rectangular vertical- geometric in shapeconsisting of five rectangular bro
Art History 113
9 December 2011
Question One: Architecture
The development of art over time is easily traced through the architectural
wonders that still exist today. Beginning with the prehistoric Stonehenge and working up
to the late Gothic Chartres Cat
Art History 113
9 December 2011
Question 2: Sculpture
The art form known as sculpture has evolved immensely since its beginning.
From the prehistoric era through the Gothic period, sculptures have taken numerous
different shapes and sizes utilizing a vari
Art 113
18 November 2011
New York Kouros and Virgin and Child
The New York Kouros from Attica is a sculpture from Ancient Greece, dating back
to the Archaic period (c. 600-480 B.C.E.) and is an accurate embodiment of the art of that
time. Little is known