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
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
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 a
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 t
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
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
rat
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.
Wh
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
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 th
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 p
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
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
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 illuminate
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 an
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 conditi