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

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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

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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 which satisfy the partial differential equation and the boundary conditions at x = 0 and x = π; Note: You must solve all three cases for λ. (b) the complete solution u(x, t) of the boundary-value problem in the form of an inﬁnite series. [5+15=20 marks]

continued. . .

JEE235 Calculus of Several Variables Exam 2009

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Question 6 (a) The depth of a pond at the point with coordinates (x, y) is given by h(x, y) = 2x2 + 3y 2 where x and y are measured in metres. (i) If a boat at the point (−1, 2) is sailing in the direction of the vector 4i + j, is the water getting deeper or shallower? At what rate? (ii) In which direction should the boat at (−1, 2) move for the depth to remain constant? (b) Determine the tangent plane to z = 3ex−y ln x at the point (1, 1, 0). [(3+1)+3= 7 marks]

Question 7 Evaluate the following double integral by reversing the order of integration. 1 0 0 x

2xy dy dx 1 − y4

You must include a sketch of the region of integration. [5 marks]

Question 8 Use Green’s theorem in the plane to evaluate

C

xy dx + (e

√ y

+ x) dy

where C is the closed curve consisting of the straight line segment from (−1, 0) to (1, 0), followed by the semi-circle x2 + y 2 = 1, y ≥ 0 from (1, 0) to (−1, 0). [5 marks]

continued. . .

JEE235 Calculus of Several Variables Exam 2009

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Question 9 (a) Find the work done by F = xyi + zj + 3k from (1, 0, 0) to (0, 1, 0) along the arc of the unit circle, centre (0, 0) in the xy-plane. (b) Consider the vector ﬁeld F deﬁned by F = 2xe2y i + (2x2 e2y − z 2 sin y)j + 2z cos yk . (i) Show that F is a conservative vector ﬁeld. (ii) Determine a scalar potential φ for F; that is, determine a scalar ﬁeld φ such that F = φ. (iii) Hence, evaluate the scalar line integral x = 2 cos t, y = 2 sin t, z = t for 0 ≤ t ≤ 2π. [4+(2+3+3)=12 marks] C

F· dr where C is that part of the helix

Question 10 (a) Evaluate using cylindrical polar coordinates 2y dV where V is the region that

lies below the plane...