Instructions for Formating Your Assignments

Dear Student,

Please 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 two tutor-marked assignments for this course. The assignments are in this booklet.

A sample exam paper is also given at the end of this booklet for your information.

Instructions for Formating Your Assignments

Before attempting the assignments please read the following instructions carefully.

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

ROLL 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 a 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 which part of which question is being solved.

6) The assignments are to be submitted to your Study Centre.

Your answer sheets will only be accepted within 2003, not later.

Please keep a copy of your answer sheets.

Wish you good luck.

ASSIGNMENT 1

(To be done after studying Blocks 1 and 2.)

Course Code : MTE-01

Assignment Code : MTE-01/AST-1/2003

Maximum Marks : 100

1. Which of the following statements are true? Give reasons for your answer. The reasons should be presented as a short proof or an example to make your point. (For example, if you say (i) below is true, then give a short proof for showing it is true. If you say (i) is false, then given an example for which it is not true.)

i) If y = f(x) such that exists at every point of [a, b], then y must have a critical point in [a, b].

ii) A function that is continuous everywhere has no point of inflection.

iii) If f : [a, b] ® R is differentiable over [a, b], then f(a) = f(b).

iv) The domain of the function f given by f(x) = is R  {–5}.

v) Any proper subset of R has an infimum.

vi) If f : [a, b] ® R is a function such that x = y Þ f(x) = f(y) " x, y Î [a, b], then f is 1–1.

vii) " x Î ] a – h, a + h [ , where h > 0.

viii) If y = f (x) is such that yn = yn – 1 for some n Î N, then f(x) = c, a constant.

ix) y2 – x2 = 3x has oblique asymptotes.

x) If f is concave on [a, b], then it is convex on [ – b, – a ] . (20)

2. (a) Express the following without using the absolute value signs.

i) (2)

(b) Express the following as an interval or union of intervals.

(5)

3. (a) Examine the continuity of the following function f. (This means that you must give all the points at which the function is continuous, and all the points at which it is discontinuous, along with the justification.)

(4)

(b) Using the e-d definition, show that .

Further, define a function f : [ 0, 1 ] ® R such that . (6)

4. (a) Find the slope of the tangent at each point of the curve . (2)

(b) If y is a function of x such that xy . yx = 1 for x ³ 0, find y1. (2)

(c) Is f given by f(x) = x5 + 2x + 1 invertible? If so, check if f –1 is differentiable. If f is not invertible, define another function g which is invertible and for which g–1 is derivable.

Also find , whichever exists. (5)

(d) A spherical balloon is leaking air so that its surface area is decreasing at the rate of 1 sq. cm./sec. At the moment that the radius is 10 cm., how fast is the volume decreasing? (3)

(e) Find . (3)

5. (a) Sketch the curve r = 1 – 2cos q. Clearly state the properties you use for doing so. (7)

(b) Draw the graph of in the same diagram, giving the features you used for doing so. (6)

Find the intervals over which F is concave upwards. Also find all the inflection points of F. (5)

(d) Give an example of a function with two inflection points. Justify your choice of example. (2)

6. (a) A woman on an island wishes to go to a small town on the mainland, the coast of which is 1 km. away from the island. From the point on the coast closest to the island it is 4 km. along the straight shore line to the town. The woman can row her boat at a speed of 2 km. per hour and walk at a speed of 4 km. per hour along the beach. Where should she land the boat in order to get to town as quickly as possible?

(6)

(b) Let f(x) = x2 + 4x + 3. Find c so that (c) equals the average slope of f(x) on [ a, b ]. (2)

(d) Find all the critical points of the function f defined by f(x) = x3(x2 – 1)2 " x Î[ –2, 2 ]. Classify these critical points as local extrema, absolute extrema and those which are not extrema. (7)

7. (a) Find the Taylor polynomial of degree 4 around zero for f(x) = sinh x. Hence estimate sinh . (3)

(b) Find a and b if we know that the Maclaurin’s expansion of degree at most 4 for

f(x) = ax + b sin x is x – (2)

ASSIGNMENT 2

(To be done after studying Blocks 3 and 4.)

Course Code : MTE-01

Assignment Code : MTE-01/AST-2/2003

Maximum Marks : 100

1. Which of the following statements are true, and which are false? Give reasons for your answers.

i) The function f, defined by f(x) = sin x + cos 2x, is monotonic in .

ii)

iii) For a function f defined on [a,b] and a partition P ={x0, x1, ¼, xn} of [a,b], L(P,f) £ U(P,f) only if x1 – x0 = x2 –x1 = ¼ = xn – xn – 1.

iv) If f is integrable on [a,b], then it is continuous on [a,b].

v) If f is defined and differentiable on [a,b], then it is integrable on [a,b].

vi) , where f and g are two integrable functions and g(x) ¹ 0 " x.

vii) If f(x) = f(a – x) " x Î [0,a], then

viii) The area under the curve y = f(x) for xÎ[a,b] is U(P,f) – L(P,f), where P is a partition of [a,b].

ix) The Maclaurin’s series for a function is a reduction formula for the integral of the function.

x) We can find the integral of any function by Simpson’s rule. (20)

2. (a) Find the upper and lower integrals of the function g, defined on [0,1] by g(x) = .

Hence conclude whether g is integrable on [0, 1] or not. (6)

(b) Prove Th.6, Sec. 10.4, by proving the following steps.

(i) Let m = min {f(x)½x Î [a,b]} and M = max {f(x)½x Î [a,b]}. Show that m(b – a) £ £ M (b – a).

(ii) Next, use the intermediate value theorem (Block 1) to find [a,b] so that f() = (4)

3. (a) Find constants b, c, d so that the function g defined by g(x) = bx2 + cx + d satisfies

(4)

(b) Find the intervals in which G is increasing or decreasing, where

G(x) = (3)

4. Evaluate the following integrals :

i) (2+4+4)

5. (a) Find a reduction formula for (4)

(b) Find a reduction formula for 0. Use this to find

(6)

6. (a) Prove that

(7)

(b) Find an approximate value of ln (0.9) upto 3 decimal places. (3)

7. (a) The velocity of a train which starts from rest is given by the following table. The time is recorded in minutes and the speed in kilometres per hour.

Minute / 2 / 4 / 6 / 8 / 10 / 12 / 14
Km./hr. / 15 / 20 / 27 / 33 / 26 / 24 / 18

Use the Trapezoidal rule and Simpson’s rule to estimate approximately (upto 2 decimal places) the total distance travelled in 12 minutes. (4)

(b) Find the area of the region enclosed by the curves x2 = y and y = (x4 + x). (2½)

(c) Give an example to show that the region between the curves f(x) and g(x) over [a,b] may not have the same area as the region between (2)

(d) Let y = and x2 + y2 = 1 intersect in A in the 1st quadrant. Find the area of the surface obtained by revolving the portion OAB of the curves about the x-axis, where O is the origin and B is the point (1,0). (2½)

(e) Consider two single-cell organisms. One is spherical, and the other has the shape of the solid of revolution obtained by revolving x2 + 2y2 = 1 around the x-axis. If both have the same volume, do they have the same requirements for nutrients? Note that the amount of nutrients absorbed is proportional to the surface area. (4)

8. (a) Find the length of the arc of the curve y = ex between x = ln and x = ln . (2)

(b) The half-life of uranium 238U is 4.5 billion years. If the quantity of this isotope in a rock sample has diminished to 90% of its original amount, how old is the rock sample? (1 billion = 109.) (4)

(c) Find the volume of the solid obtained by revolving about the y-axis the planar region bounded by the y-axis, y = and y = 2x2 – 1. (4)

9. (a) Find the arc length of the curve r(t) = (1 + t2, 3 – 2t3) for tÎ [0,1]. (2½)

(b) Use Simpson’s rule with n = 4 to estimate the arc length of r(t) = (cos2t, sin3t), tÎ [0,p]. (2½)

S A M P L E P A P E R

MTE-1 : CALCULUS

Time : 2 hours Maximum Marks : 50

(Weightage 70%)

______

Note : Q. No. 1 is compulsory. Do any four questions from Q.Nos. 2 to 7.

1. Which of the following statements are true? Justify your answers with a short proof or a counterexample. 10

(i) f(x) = x3 has no critical point.

(ii) The function f(x) = is differentiable in

(iii) The function f(x) = sin x + cos 2x is not monotonic in

(iv)

(v) The domain of the function f given by f(x) =

2. (a) Examine the continuity of the function f defined by

f(x) =

over R. 4

(b) The velocity of a train which starts from rest is given by the following table, the time being recorded in minutes from the start and the speed in kilometres per hour.

Minute / 2 / 4 / 6 / 8 / 10 / 12 / 14
Km./hr. / 15 / 20 / 27 / 33 / 26 / 24 / 18

Estimate approximately (upto 2 decimal places) the total distance travelled in 12 minutes. 3

P = be a partition of [0, 1]. Show that L(P, f) £ U(P, f). 3

3. (a) If y =

(1 – x2)yn+2 – (2n + 1)xyn+1 – (n2 + m2)yn = 0. 4

(b) Use the e-d definition to show that .

Further, for what value of a is ? 6

4. (a) Using Lagrange’s mean value theorem, prove that

(b) Evaluate the following integrals : 4

(i) , (ii) .

5. (a) Find an approximate value of (1.01)5/2 using Maclaurin’s series, upto 3

decimal places. 3

(b) Show that (2, 0) is a singular point of the curve y2 = (x – 1) (x – 2)2.

Also determine the type of singular point it is. 3

6. Trace the curve x(x2 + y2) = a(x2 – y2), clearly stating the necessary properties required for tracing it. 10

7. (a) If

Hence find the value of 5

(b) Show that 3