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 reread 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 tutormarked assignment for this course.
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ENROLMENT NO: ……………………………………………
NAME: ……………………………………………
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…………………………………………… COURSE CODE: ……………………………. COURSE TITLE: ……………………………. ASSIGNMENT NO.: ………………………….… STUDY CENTRE: ………………………..….. DATE: ……………………….………………... PLEASE FOLLOW THE ABOVE FORMAT STRICTLY TO FACILITATE EVALUATION AND TO AVOID DELAY.
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ASSIGNMENT (To be done after studying the course material.)
Course Code : MTE01 Assignment Code : MTE01/TMA/2009 Total Marks : 100
1) Which of the following statements are true? Give reasons for your answers. (This means that if you think a statement is false, give a short proof or an example that shows it is false. If it is true, give a short proof for saying so. For instance, to show that ‘{1, padma, blue}is a set’ is true, you need to say that this is true because it is a welldefined collection of 3 objects.)
i) If f is a function defined on [0, 1] for which exists but , then f is discontinuous at .
ii) If f is an invertible function that is increasing in [a, b], then is also an increasing function.
iii) If x = a is a point of inflection for y = f(x), then (a) = 0.
iv) f, defined by f(x) = , is integrable in [?1, 1].
v) The domain of a surjective function must be an infinite set.
vi) .
vii) y = 1 is an asymptote of y2 = x(x – 1)2.
viii) If f is a function defined on [a, b] which is not continuous, then L(P, f) > U(P, f) for any partition P of [a, b].
ix) If f is derivable on [?a, a], then .
x) The area under to , is the same as the area under to . (20) 2) Trace the curve y2 = (x + 1)2 (x – 5). Clearly state all the properties you have used for tracing it (e.g., symmetry about the axes, symmetry about the origin, points of intersection with the coordinate axes, tangents at some chosen points, range of definition, its asymptotes (if any), extreme points, etc.) (10)
3) a) Write the curve x3 + y3 = 3axy in polar form. Hence find the area encircled by its loop. (4)
b) i) Show that the arc length from 0 to x of the graph of a function f, defined by , where (x) ? 1, is g(x).
ii) Find a function f whose arc length from 0 to x is 2x. (6)
4) a) A village doctor rides her cycle from her home to the health centre, a trip of about 7 km. She cycles at a constant speed till she reaches a big hill, 3 km from her home. She slows down as she goes to the top of the hill, which is a 1km ride. Then she goes down the hill quite fast, which is another kilometre. The last 2 kms of the ride to the health centre are on flat ground. Draw a graph which shows the doctor’s speed as a function of the distance from her home. Also check whether this function is one  one. (3)
b) Find the domain and range of the function f, defined by . (3)
c) If , where f is defined by , then check if f is continuous at x = 0. (4) 5) a) Given f(x) = 3x – 2x, find the quadratic polynomial p(x) = ax2 + bx + c such that p(0) = f(0), . [We say that p is the best fit quadratic polynomial for f.] (3)
b) An aeroplane is flying at a constant height of 5 km, and at a velocity of 450 km/hr. A camera on the ground is pointed towards the plane, at an angle ? from the horizontal. As the plane flies over the camera, how fast does the camera have to rotate in order to keep the plane in view, when ? = ? (4)
c) If g(2) = 3 and (2) = ?4, find (2), where f(x) = x2 ln (g(x)). Also find the equation of the normal to f at x = 2. (3)
6) a) Find the quantity q of production which maximises profits, if the total sales revenue and total cost (in thousands of rupees) of production are given by
R(q) = 5q – 0.003q2 and C(q) = 300 + 1.1q, where 0 ? q ? 800 units.
What production level gives the minimum profit?
[Hint: Maximum profit is when marginal revenue is the same as marginal cost.] (5)
b) One hallway 3 metres wide meets another hallway 5 metres wide in a right angle. Find the length of the longest ladder which can be carried horizontally around the corner.
[Hint: The longest possible ladder would need to be the smallest length ABC in Fig. 1.] (5)
3 Fig. 1
7) a) Evaluate
i)
ii)
iii) (3 + 3 + 2)
b) Use Simpson’s rule, taking 5 points in the required interval, to approximate the value of to two decimal places. (3)
c) The region bounded by and the xaxis between x = 0 and x = 1 is revolved around the xaxis. Find the volume and surface area of this solid of revolution. (4)
8) a) Let f be a continuous and differentiable function on [0, ?[, with f(0) = 0 and such that is an increasing function on [0, ?[. Show that the function g, defined on [0, ?[ by
is an increasing function. (6)
b) Show that for . (4)
c) Find the seconddegree Taylor polynomial for f(x) = ax2 + bx + c, a, b, c ? R about x = 0, and the thirddegree Taylor polynomial for g(x) = ?0 + ?1x + ?2x2 + ?3x3, ? R about . Based on your answers, make a conjecture about Taylor approximations of f when f is a polynomial.
School of Sciences Indira Gandhi National Open University Maidan Garhi, New Delhi110068 (2009)
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