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MATH102 Examination, Mathematics Foundations III, May 2009, Exams of Mathematics

Aliah UniversityMathematics

The may 2009 examination paper for the mathematics foundations iii course (math102) at year 1 level. Instructions for candidates, 12 questions in two sections (a and b), and covers topics such as taylor polynomials, differential equations, limits, partial derivatives, and double integrals.

Typology: Exams

2012/2013

Uploaded on 02/26/2013

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PAPER CODE NO.
MATH102
MAY 2009 EXAMINATIONS
Year 1
MATHEMATICS FOUNDATIONS III
TIME ALLOWED : Two Hours and a Half
INSTRUCTIONS TO CANDIDATES
Answer ALL questions of Section A and THREE questions from Section B.
Section A carries 55 % of the available marks.
Paper Code MATH102 Page 1 of 5 CONTINUED/
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PAPER CODE NO. MATH

MAY 2009 EXAMINATIONS

Year 1

MATHEMATICS FOUNDATIONS III

TIME ALLOWED : Two Hours and a Half

INSTRUCTIONS TO CANDIDATES

Answer ALL questions of Section A and THREE questions from Section B. Section A carries 55 % of the available marks.

SECTION A

  1. (a) Write down the Taylor polynomial of order n = 4 generated by the function f (x) = cos^2 (x) − sin^2 (x) at x = 0. [3 marks] (b) Write down the Taylor series about x = 0 for the function

f (x) = ln(1 + x). Does this series converge to f (x) for x = 3? [3 marks]

  1. (a) Find the solution of the differential equation

(1 + ex)ydy − exdx = 0,

which satisfies the condition y(0) = 1. [3 marks] (b) Find the solution of the differential equation dy dx

  • 2xy = 3x

with y(0) = −^32. [4 marks]

  1. A flat circular plate occupies the region x^2 +y^2 ≤ 4. The plate is heated so that the temperature T oC at the point (x, y) is given by

T (x, y) = x^2 + 3y^2 − 2 x.

Find the temperatures, and the coordinates, at the hottest and coldest points of the plate. [8 marks]

  1. Evaluate the double integral ∫ ∫ D x cos(x^2 + y^2 )dxdy,

where D = {(x, y) : x^2 + y^2 ≤ 1 , y ≥ 0 }. [8 marks]

SECTION B

  1. Let f (x) be a function defined in the interval (a−h, a+h), where h > 0. Assume that f (x) is differentiable n + 1 times for any x ∈ (a − h, a + h). Prove that for any x ∈ (a − h, a + h) there exists c ∈ (a, x) (with the interval of positive or negative orientation) such that

f (x) =

∑^ n k=

k!

dkf dxk^ (a)(x − a)k^ +

(n + 1)!

dn+1f dxn+^ (c)(x − a)n+1.

Hint: You can use the Cauchy Theorem: If f (x) and g(x) are functions defined on [a, b], differentiable in (a, b), continuous at x = a, x = b, and g′(x) ̸= 0 for all x ∈ (a, b), then there exists c ∈ (a, b) such that

f (b) − f (a) g(b) − g(a)

f ′(c) g′(c)

[15 marks]

  1. Solve the following differential equations with the given boundary con- ditions:

(a) x^2 y′′^ + xy′^ = 1, with y(1) = 0, y(2) = 1. (b) y′′^ + 9y = cos(3x), with y(0) = 0, y′(0) = 1. [15 marks]

  1. Use Taylor’s formula for f (x, y) at the origin to find a quadratic poly- nomial approximation of f near the origin: (a) f (x, y) = sin(x) ln(1 + y), (b) f (x, y) =

1 + x + y − xy

[15 marks]

  1. Let D be the plate, which occupies the region bounded by y = x, y = 1, x = 0.

(a) Find the total mass of the plate D, provided the mass density is ρ(x, y) = y/(1 + x). (b) Find the centre of mass (x, y) of the plate D. [15 marks]

PAPER CODE MATH102 Page 5 of 5 END.