Solutions to Exercises 3
by Biplab Basak
1. Extra problem: If |a| < 1 then limn xn = 0 where xn = an .
Proof: If a = 0, then there is nothing to prove. So let 0 < a < 1 then
xn+1 = a xn < xn . So, the sequence cfw_xn is monotonically decreasing
sequence
Solutions to Exercises 2
by Sayani Bera
1. (a) |x + y| < 1, i.e. 1 < x + y < 1.
Thus it is the region in between the straight lines x + y = 1 and
x + y = 1.
(b) x + 2y is integer, i.e. x + 2y = n, n Z.
They are the straight lines in R2 given by x+2y = n w
Solutions to Exercises 1
by Sumit Kumar
1. Let a and b be two rational numbers. Without loss of generality we
assume that a < b. Take a1 := a+b
2 . Clearly a1 is a rational number
and lies between a and b (prove it!). Repeat this process to get the
result
Solution to Exercises 4
by Bidyut Sanki
(1) Consider the following function f : R R defined by,
cfw_
1 , x Q
f (x) =
+1 , x R \ Q
where Q is the set of all rational numbers. Then it is easy to check that
f is continuous nowhere and |f |(x) = 1 for all x R
Solution to Exercise 6
by Sumit Kumar
1. We know that f (x) = [x] is a continuous function except at integers. So f is not
dierentiable at integers. For non-integers the derivative is zero. Since for any a R
which is not an integer, there exists a unique
UM 101 : Analysis and Linear Algebra I
August - December 2012
Indian Institute of Science
Exercises 1
13 August, 2012
1. Show that there exist infinitely many rational numbers between two distinct
rational numbers.
2. Let Ai , i = 1, 2, . . ., be countabl
UM 101 : Analysis and Linear Algebra I
August - December 2012
Indian Institute of Science
Exercises 5
7 September, 2012
1. Suppose that f : [a, b] R is continuous and that f (x) is always rational.
What can be said about f ?
2. Prove that there exists a r
Solution to Exercises 5
by Mizanur Rahaman
1. The function f : [a, b] R is continuous that takes only rational values. Claim: f
is constant. If not suppose f takes two distinct rationals x and y. But in between
two rationals we can always get an irrationa
MA 101 : Analysis and Linear Algebra I
August - December 2012
Department of Mathematics, Indian Institute of Science
Exercises 3
24 August, 2012
1. In each of the following cases, determine whether the given sequence converges
or diverges. Find the limit
UM 101 : Analysis and Linear Algebra I
August - December 2012
Indian Institute of Science
Exercises 6
14 September, 2012
1. Find f if f (x) = [x].
2. Suppose that f is dierentiable and periodic with period a, i.e., f (x+a) = f (x)
for all x. Prove that f
UM 101 : Analysis and Linear Algebra I
August - December 2012
Indian Institute of Science
Exercises 4
31 August, 2012
1. Give an example of a function f such that f is continuous nowhere, but |f | is
continuous everywhere.
2. Suppose that f and g are func
ECE 440
HW Assignment 4
Spring 2011
ECE 440 Homework Assignment 4
Due: Friday February 25th, 2011
(
).
1. An intrinsic Si sample is doped with donors from one side such that
(a) Find an expression for the built-in electric field at equilibrium over the ra
ECE 440
Homework VIII
Fall 2009
Print your name and netid legibly. Show all work leading to your answer clearly
and neatly. Staple multiple pages.
Problem # 1: A semi-infinite p-type bar is illuminated with light which generates
GL electron-hole pairs per
Date : 19 Feb 2008
Due by: 28 Feb 2008
PH 352 : Semiconductor Physics and Technology
Instructor: Prof. V. Venkataraman
Assignment # 3
1. Consider a direct gap semiconductor with one conduction band and one valence band, both
having the same eective mass a
EEE 352A, Properties of Electronic Materials, Spring 2007
Homework 12
Due: Friday, April 27, in class
1. (20 points) Consider an n-type semiconductor of length L. Show that, under
steady-state conditions
pn (x) = pn0 (1 x/L),
0xL
is the special-case solut