Instructions for Formatting Your Assignments

Dear Student,

We hope you are familiar with the system of evaluation to be followed for the Bachelor’s Degree Programme. At this stage you may probably like to re-read the section on assignments in the Programme Guide for Elective Courses that we sent you after your enrolment. A weightage of 30 per cent, as you are aware, has been earmarked for continuous evaluation which would consist of one tutor-marked assignment for this course.

Instructions for Formatting Your Assignments

Before attempting the assignment please read the following instructions carefully.

1) On top of the first page of your TMA answer sheet, please write the details exactly in the following format:

ENROLMENT NO: ……………………………………………

NAME: ……………………………………………

ADDRESS: ……………………………………………

……………………………………………

COURSE CODE: …………………………….

COURSE TITLE: …………………………….

ASSIGNMENT NO.: ………………………….…

STUDY CENTRE: ………………………..….. DATE: ……………………….………………...

PLEASE FOLLOW THE ABOVE FORMAT STRICTLY TO FACILITATE EVALUATION AND TO AVOID DELAY.

2) Use only foolscap size writing paper (but not of very thin variety) for writing your answers.

3) Leave 4 cm margin on the left, top and bottom of your answer sheet.

4) Your answers should be precise.

5) While solving problems, clearly indicate the question number along with the part being solved. Be precise. Recheck your work before submitting it.

Answer sheets received after the due date shall not be accepted.

We strongly feel that you should retain a copy of your assignment response to avoid any unforeseen situation and append, if possible, a photocopy of this booklet with your response.

We wish you good luck.

Assignment

(To be done after studying the course material)

Course Code: MTE-09

Assignment Code: MTE-09/TMA/2009

Maximum Marks: 100

1. a) Let f be defined on R by if , and f(0) = 0. Show that is continuous on R and it is not derivable at x = 0. (5)

b) Prove that, if

(3)

c) Is every convergent series also absolutely convergent? Justify your conclusion. (2)

2. a) Test the following series for convergence

ii) (4)

b) Let denote the derived set of a set X. For any two sets A, B prove that . Is the converse of the implication true? Justify your answer. (3)

c) Show that there is no real number for which the equation has two distinct roots in [0, 1]. (3)

3. a) Find the following limits:

ii) (4)

b) Obtain the values of x (x>0), for which the following series is convergent:

(4)

c) For what value of a does

tend to a finite limit as ? Also find the limit. (2)

4. a) Determine the points of discontinuity of the following functions, and the type of discontinuity each is:

ii) (4)

b) Assuming the validity of expansion, prove that

(4)

c) Check whether the sequence where is a Cauchy sequence. (2)

5. a) Evaluate by taking a partition of which divides it into 2n equal sub- intervals. (5)

b) Check whether the collection

is an open cover of ]0, 1[. (3)

c) Show that the set of natural numbers and the set of integers are equivalent. (2)

6. a) Is the sequence given by

convergent? Justify your answer. Find also a convergent sub-sequence of . (4)

b) Show that the function defined by is a continuous

function on R. (4)

c) Find an interval over which the functions f(x) = sin x and g(x) = cos x, satisfy the hypothesis of the Cauchy’s Mean Value Theorem. Verify Cauchy’s Mean Value Theorem for f and g over this interval. (2)

7. a) Find the primitive of the function f defined on [0, 3] by

Hence, find the value of the integral (5)

b) Show that the series

is uniformly convergent on [2, 4]. (3)

c) Show that the series

is convergent (2)

8. a) Show that the series

is uniformly convergent for all real values of x. (5)

b) A function f is defined on R by

Check whether f is derivable at x = 0. (3)

c) Let be a strictly increasing function such that f(X) = Y. Show that exists and it is also strictly increasing. (2)

9. a) Check whether the following functions f, are uniformly continuous or not:

ii) (4)

b) Let g be an integrable function defined on [a, b], and let

Show that h is uniformly continuous on [a, b]. (4)

c) By applying the Cauchy General Principle of convergence, show that the sequence , given by , is convergent. (2)

10. Are the following statements true or false? Give reasons for your answers.

i) The function defined by is a continuous function at x = 50.

ii) If , then the series is convergent.

iii) The function f, define on R by

is increasing in every interval.

iv) The series converges absolutely.

v) An integrable function can have finitely many points of discontinuities. (10)