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Indian Institute of Technology Madras
Department of Mathematics
MA 1010 - Calculus - l : Functions of One Variable
End Semester Examination
Time: 9 AM - 12 NOON Date: November 23, 2010 Max Marks: 60
Instructions:
(i) Answer ALL questions.
(ii) Answer PA

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Indian Institute of Technology Madras
Department. of Mathematim
MA 1010 - CALCULUS I : FUNCTIONS 0? ONE VARIABLE 6/
End Semester Examination '
Date: November 24. 309 Class: B. Tech.
Time: 09:00 - 12:“) Maximum Marks: 60
Instructions:
(i) Answer A

' ' lin'u’ J6,
3,
Indian Institute of Technology Madras
Department of Mathematics
' IA. 101 — CALCULUS I: FUNCTIONS or ONE VARIABLE
End Semester Examination
Class: B. Tech.
Maximum Marks: 60
l'!\:
-r -\LL questions.
PART A and PART B in w answer books.

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Department of Mathematics, I.I.T. Madras
MA 101 - Elements of Calculus
End Semester Examination
Date. :. 23, 2007 '7 ,- Class: [8. Tech.
Time: Uni)“ 12:00 Maximum Marks: 60
Instructions: Write your roll number. teacher‘s name and batch

DEPARTMENT OF MATHEMATICS
IIT MADRAS
MA1010: Calculus I : Functions of one variable1
Problem Sheet III
|x|
|x|
and lim
.
x0+ x
x0 x
1. Find lim
2. Let f : R R be such that lim f (x) = `. Prove that f (n) ` as n .
x
3. Let [x] denote the greatest integer n

DEPARTMENT OF MATHEMATICS
IIT MADRAS
MA1010: Calculus I : Functions of one variable
Problem Sheet II
an
1. Let (an ) and (bn ) be two sequences
P of positive
Psuch that bn l as n , where
l 6= 0. Prove that the series n=1 an and n=1 bn either both converge

Practice Problems -I
(Ref: S. Ponnusamy, Foundations of Mathematical Analysis, Birk
auser, 2012)
MA1010: Calculus 1: Functions of One Variable
(1) Which
of the following sequences
2 converge.
6
1
n 2n + 3
n + 3n4 2
(a)
(p > 0)
(b)
(c)
p
3
n
5n
n6 + 2n

h'nu ) c— {n r‘ "7. (
\ (N; W}: xx“? ,( n n
I: .
, . (_ ’ ."
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f. I,
n] (1 It"? 11
/
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n a N ,8)? “4
2 > ,L
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W 2 ,1“ W3 (3°)
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EXANIPLE 1.17 Suppose (tn > O for al 71. E N such that lim 0TH : E < 1. we
-n—>oo (In
show that an e O.
Since lim
n—mo an
a. T
“+1 g q for all n 2 IV. Hence,
an.
: E < 1, there exists q such that E < q < 1 and N E N such
that
o < a.” g (En-Nat v n >

DEPARTMENT OF MATHEMATICS
IIT MADRAS
MA1010 - Calculus I: Functions of one variable
Problem Sheet I: Sequences
1. Suppose that xn l as n . Prove that
(a) |xn l| 0 as n .
(b) |xn | |l| 0 as n .
2. Establish convergence or divergence of the following sequen

DEPARTMENT OF MATHEMATICS
IIT MADRAS
MA1010 - Calculus I: Functions of one variable
Problem sheet I: Sequences
1. Suppose that xn l as n . Prove that
(a) |xn l| 0 as n .
(b) |xn | |l| 0 as n .
2. Let xn > 0, n N. Prove that xn 0 as n if and only if
1
as

Indian Institute of Technology Madras
Department of Mathematics
Problem Sheet on Applications of Definite Integrals
1. Find the arc length of each curve between the points indicated
(a) x = et cos(t),
(b) (y + 1)2 = 4x3
(c) y = log sec(x),
2
2(e 1) ]
2
[

Solutions of Some Problems from Assignment Sheet-II
M.T.Nair
Department of Mathematics, IIT Madras
August 30, 2013
P
1. To test the convergence of the series n=1 an :
P
n n1
(g) an = log(n+1)
: The series n=1 an diverges since
1
n n1
1
,
=
an =
log(n +

Solutions of some Problems from Assignment Sheet-4
M.T.Nair
Department of Mathematics, IIT Madras
October 3, 2013
(
1. (Modified form) For c R, let f (x) =
x3 1
x1 ,
x 6= 1,
Then f is continuous at every
c,
x = 1.
x0 =
6 1, and f is continuous at x0 = 1 i

Department of Mathematics, IIT Madras
MA 1010 Calculus - I
Assignment - 5
1. Suppose f : R R and g : R R. If f and g are differentiable at a point x0 R,
then prove that
(a) f + g, f g and f g are differentiable at x0 .
(b)
f
g
is differentiable at x0 , pr

Department of Mathematics, IIT Madras
MA1010 Calculus-I
Assignment 3 (Limits)
1. Use definition of the limits of functions to prove:
(a) lim
x0
(b) lim3
x 2
x
2x2
1
4
x3 (x3)
= 2.
(e) lim
= 32 .
(f) lim sin x = sin .
x
(g) lim x2 + 3 = 4.
= .
(c) lim
(d)

Department of Mathematics, IIT Madras
MA1010 Calculus-I
Assignment 4 (Continuity of Functions)
1. Let f : R R be defined by
f (x) =
x3 1
x1
if x 6= 1;
0
if x = 1;
Prove that f is continuous at each x R, x 6= 1. Is f continuous at
x = 1? Justify your ans

INDIAN INSTITUTE OF TECHNOLOGY MADRAS
Department of Mathematics
Problem Sheet 2 - MA1010 (Calculus 1)
P
(1) Test the convergence of the infinite series ( an ) where an is given by:
n
(e)
( 2)n
n1 n
(f)
sin(1/n)
n1 n
(g)
log(n + 1)
(h) ( n n 1)n
(n!)2
(2n