Csv Exam with Solutions 2009
Student ID No:
Pages: 5 Questions: 10
UNIVERSITY OF TASMANIA [Australian Maritime College]
EXAMINATIONS FOR DEGREES AND DIPLOMAS June 2009
JEE235 Calculus of Several Variables
First and Only Paper
Examiner: Dr Irene Penesis
Time Allowed: THREE (3) hours. Instructions: - Candidates should attempt ALL TEN (10) questions. - Total marks: 100. - In any question, full marks will not be awarded unless sufficient working is shown.
JEE235 Calculus of Several Variables Exam 2009
-2 -
Question 1 Use the Tables of Laplace transforms, along with the operational theorems, to find the Laplace transform of the following functions: (a) t(t + cos t)e−3t (b) (c) t 0
e−u sinh 2u du
4 − 4e2t t [4+2+2=8 marks]
Question 2 (a) Write f (t) = 2t + 3, 0 ≤ t < 9 in terms of Heaviside functions. Find the Laplace −2, t≥9 transform of the function.
(b) Use Laplace transforms to determine the solution y(t) of the following initial-value problem y − 2y = 3 + H(t − 5); y(0) = 1, y (0) = 0 . [4+6=10 marks]
Question 3 Find the inverse Laplace transform of 10s2 (s + 1)(s2 + 6s + 10) and hence write down L
−1
10s2 e−s . 2 + 6s + 10) (s + 1)(s [6 marks]
continued. . .
JEE235 Calculus of Several Variables Exam 2009
-3 -
Question 4 Use the convolution theorem to solve the following integral equation for f (t): f (t) = tet − 2 t 0
f (τ ) cos(t − τ ) dτ . [7 marks]
Question 5 A uniformly metal bar of length π, having its lateral edges insulated is initially at a temperature of x2 degrees Celsius throughout. The ends of the bar at x = 0 and x = π are placed in contact with heat reservoirs which are kept at zero temperature, so that the bar begins to cool. The boundary value problem governing the temperature u(x, t) of the bar at time t is given by ∂u ∂2u = for 0 < x < π, t > 0 ∂t ∂x2 with ∂u u(0, t) = (π, t) = 0 ∂x and u(x, 0) = x2 . Use the method of separation of variables to obtain (a) the eigenvalues and corresponding eigenfunctions
Pages: 5 Questions: 10
UNIVERSITY OF TASMANIA [Australian Maritime College]
EXAMINATIONS FOR DEGREES AND DIPLOMAS June 2009
JEE235 Calculus of Several Variables
First and Only Paper
Examiner: Dr Irene Penesis
Time Allowed: THREE (3) hours. Instructions: - Candidates should attempt ALL TEN (10) questions. - Total marks: 100. - In any question, full marks will not be awarded unless sufficient working is shown.
JEE235 Calculus of Several Variables Exam 2009
-2 -
Question 1 Use the Tables of Laplace transforms, along with the operational theorems, to find the Laplace transform of the following functions: (a) t(t + cos t)e−3t (b) (c) t 0
e−u sinh 2u du
4 − 4e2t t [4+2+2=8 marks]
Question 2 (a) Write f (t) = 2t + 3, 0 ≤ t < 9 in terms of Heaviside functions. Find the Laplace −2, t≥9 transform of the function.
(b) Use Laplace transforms to determine the solution y(t) of the following initial-value problem y − 2y = 3 + H(t − 5); y(0) = 1, y (0) = 0 . [4+6=10 marks]
Question 3 Find the inverse Laplace transform of 10s2 (s + 1)(s2 + 6s + 10) and hence write down L
−1
10s2 e−s . 2 + 6s + 10) (s + 1)(s [6 marks]
continued. . .
JEE235 Calculus of Several Variables Exam 2009
-3 -
Question 4 Use the convolution theorem to solve the following integral equation for f (t): f (t) = tet − 2 t 0
f (τ ) cos(t − τ ) dτ . [7 marks]
Question 5 A uniformly metal bar of length π, having its lateral edges insulated is initially at a temperature of x2 degrees Celsius throughout. The ends of the bar at x = 0 and x = π are placed in contact with heat reservoirs which are kept at zero temperature, so that the bar begins to cool. The boundary value problem governing the temperature u(x, t) of the bar at time t is given by ∂u ∂2u = for 0 < x < π, t > 0 ∂t ∂x2 with ∂u u(0, t) = (π, t) = 0 ∂x and u(x, 0) = x2 . Use the method of separation of variables to obtain (a) the eigenvalues and corresponding eigenfunctions