UNIVERSITY OF EAST ANGLIA School of Mathematics Main Series UG Examination 2013 14 CALCULUS AND MULTIVARIABLE CALCULUS MTHA4005Y Time allowed: 3 Hours Attempt QUESTIONS 1 and 2, and THREE other questions. Notes are not permitted in this examination. Do not turn over until you are told to do so by the Invigilator. MTHA4005Y Module Contact: Dr Hayder Salman, MTH Copyright of the University of East Anglia Version: 1
- 2-1. (i) Given the vectors, a = i + 2j k, b = 4i + 2j 3k, c = 2j k, calculate a.b, a b, (a b).c, (a c).a. (ii) Given the complex numbers z 1 = 1 + i, z 2 = 3 i, calculate the real and imaginary parts of z 1 + z 2, z 1 z 2, z 1 z 2, z1. For the final part you may find it easier to use the complex exponential form of z 1. (iii) Evaluate the following integrals, showing all your working, 3 1 (2x + 3) 1 2 dx, x e 3x dx. (iv) By using the substitution y = e kx, or otherwise, find the general solution of the differential equation d 2 y dx + 4dy + 5y = 0. 2 dx
- 3-2. (i) Given the function f(x, y) = 4xy x 2 y 4, show that f(x, y) has three stationary points with one located at the origin. Hence, determine whether they are a minimum, a maximum or a saddle. (ii) Find the Taylor expansion of f(x, y) = x sin(y), at (π/4, 0) upto and including quadratic terms. [4 marks] (iii) Using the divergence theorem, or otherwise, evaluate the surface integral v ds, S where S is the surface of the sphere centred at the origin and of radius 2 and v is the vector field v = (z 3 + x)i + (z 2 + e x )j + y 3 k. (iv) Classify the equilibrium point of the linear system dx dt = 2x 3y, dy dt = 5y MTHA4005Y PLEASE TURN OVER Version: 1
- 4-3. (i) Differentiate the following functions with respect to x, x 1 3, x tan(3x), sin 2 (x 2 ). (ii) Calculate all the stationary points of the function f(x) = x3 x + 1. Sketch the curve y = f(x). You should explain your reasoning. (iii) Calculate the Maclaurin series for cos x and log(1 + x), up to and including the x 4 terms. Hence or otherwise find the Maclaurin Series for cos(x) log(1 + x), sin x. [8 marks] 1 + x
- 5-4. (i) Using a suitable substitution show that 1 ( x ) a2 + x dx = 2 sinh 1 + C. a Hence calculate the integral x x2 + 2x + 5 dx. [7 marks] (ii) Find the general solution of the differential equation x dy dx 3y = x5 sin x. (iii) Find the solution of the differential equation d 2 x dt 2 + 7dx dt + 6x = 6t2 + 2t 6 subject to the conditions that at t = 0, x = 0 and dx/dt = 0. [7 marks] MTHA4005Y PLEASE TURN OVER Version: 1
- 6-5. (i) Evaluate the closed contour integral (P dx + Qdy) for the functions P = xy, Q = 2xy, where C is the contour consisting of the lines connecting (0, 0) to (π/2, 0), and (π/2, 0) to (π/2, 1), and the curve y = sin(x). (ii) State Green s theorem in the plane for any differentiable functions P (x, y) and Q(x, y). Obtain a statement of Green s theorem in the plane for the functions P and Q given in part (i). [4 marks] (iii) Hence, verify that Green s theorem holds by evaluating the double integral. [4 marks] (iv) A double integral is given by 1 1 0 x x sin(πy 3 )dydx. Write down the integral with the order of integration reversed and then evaluate the integral.
- 7-6. A surface S is defined by the union of two surfaces S 1 and S 2. The surface S 1 is given by x 2 +y 2 = 4 for 2 z 2 and the surface S 2 is given by (4 z) 2 = x 2 +y 2 for 2 z 4. A vector function on these two surfaces is given by F = yz 2 i. (i) Using cylindrical coordinates (r, θ, z) calculate the directed area element ds 1 the surface S 1. Hence evaluate ( F) ds 1. S 1 of (ii) Using a similarly suitable parameterisation of the surface S 2, evaluate S 2 ( F) ds 2. [8 marks] (iii) Hence verify that Stokes theorem applies for the function F defined on the surface S. END OF PAPER