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My stage 2 math paper exam The paper involves calculus, and linear algebra. I am given 2 hours from when the exam goes online, I will need 10 mins to download the exam paper and send it to the tutor. This means I will need to have a tutor ready on standby waiting for me to give the questions. Then the tutor will complete the paper. Needs to be done in 85 mins max (1.25 hours) so that I can have time to receive the results from the tutor and submit it myself. Attached is a past paper so the new exam will be similar to this. The tutor can send the worked answers as a pdf file of the handwritten answers within the exam timeframe (usually 2 hours from when the exam time begins) I would prefer the tutor to send each question as they complete it if possible Is this all possible?
MATHS 208 THE UNIVERSITY OF AUCKLAND SUMMER SESSION, 2021 Campus: City MATHEMATICS General Mathematics 2 (Time allowed: TWO HOURS) NOTE: • This exam is open book with unrestricted calculators. • The exam contains SIX questions on pages 2 to 6. Answer ALL SIX questions. • Each question is worth 20 marks, giving a total of 120 marks. • You must give full working and, where appropriate, reasons for your answers to obtain full marks. • Please begin each question on a new page in your answer book. Page 1 of 6 MATHS 208 1. Consider the function f given by f(x, y) = x2 + y2 − 2y + 1. (a) (i) Find the gradient of the function f at the point (0, 0). (ii) Find a unit vector u such that Duf(0, 0) = 1 2 . (iii) Find a unit vector u such that Duf(0, 0) = 0. (iv) The level curve f(x, y) = 1 passes through the origin (0, 0). Find the equation of the tangent line to the level curve at the origin. (12 marks) (b) Consider the constrained optimisation problem: Optimise f(x, y, z) = x2 + y2 + z2 subject to the constraint: x + 2y + 3z = 14. Use the method of Lagrange multipliers to find the critical point of this constrained optimisation problem (without classifying the type of critical point). (8 marks) Page 2 of 6 MATHS 208 2. (a) Only one of the following sequences converges. Find this sequence and its limit. (i) bn = (−1)n ( 1 n + 1 ) (ii) cn = lnn + sin(n) n Do not attempt to say anything about the divergent sequence. (6 marks) (b) When we apply the ratio test to the following three series, it will only tell us that one of the series converges. Which series converges by the ratio test, and what is its sum? (i) ∞∑ n=1 ( 1− 1 n! ) (ii) ∞∑ n=1 1 n2 (iii) ∞∑ n=1 2n 3n+1 Do not attempt to say anything about the other two series. (6 marks) (c) Consider the function f(x) which has a MacLaurin series, f(x) = ∞∑ n=0 (−1)nxn n! for all points x on the interval of convergence. (i) Find the interval of convergence of ∞∑ n=0 (−1)nxn n! . (ii) Find the MacLaurin series of f ′(x) by differentiating the MacLaurin series of f(x) given above. (iii) Calculate the value of f(1). (8 marks) Page 3 of 6 MATHS 208 3. (a) Consider the matrix A = 1 2 3 −11 1 2 −1 0 1 −1 0  . (i) Find a basis for the nullspace of A. (5 marks) (ii) Find a vector that is orthogonal to every vector in the nullspace of A. (2 marks) (iii) What is the rank of A? (1 mark) (iv) Does the equation Ax = b have a solution for all b ∈ R3? Give a reason for your answer. (3 marks) (v) Let v =  1 1 1 1 . Suppose b ∈ R3 and that Av = b. Determine all solutions x to the equation Ax = b. (3 marks) (b) Compute the least squares solution and error to the following equation:1 00 1 1 −1 [x1 x2 ] = 23 2  (6 marks) Page 4 of 6 MATHS 208 4. (a) The population of a country is divided into urban and rural residents. Suppose that each year, 5% of the urban population moves to rural areas and 15% of the rural population moves to urban areas. In 2020 there were 2,500,000 residents living in rural areas and 3,500,000 residents living in urban areas. Let xn = [ rn un ] be the state vector denoting the number of rural and urban residents (rn and un respectively), where n is measured in years after 2020. Assume that the population of the country remains constant. The state vector changes according to the formula xn+1 = Sxn. (i) Find the stochastic matrix S. (2 marks) (ii) What will be the eventual populations of rural and urban residents in the long term? (4 marks) (b) Consider the following three matrices: A1 = [ 2 3 3 2 ] , A2 = [ 2 0 −1 2 ] , A3 = [ 1 3 1 −1 ] . (i) Find the characteristic equation of each matrix Aj, j = 1, 2, 3. Hence compute the eigenvalues of each matrix. (6 marks) (ii) Determine which matrix Aj is NOT diagonalisable. Explain why it cannot be diagonalised. (4 marks) (iii) Which matrix Aj has an orthogonal diagonalisation V TAjV = D? Find V and D in this case. (4 marks) Page 5 of 6 MATHS 208 5. (a) Determine whether each first order differential equation below is • linear or nonlinear • separable or nonseparable (i) dy dx = x− y x (ii) dy dx + xy = 0 (iii) (ex + 1) dy dx = x y (6 marks) (b) Find the general solution y(x) of y′ = xny2, where n is a positive integer. Give your answer in explicit form: y = f(x). (7 marks) (c) Solve the initial-value problem y′ − 3y = 6ex, y(0) = 1 for y(x). Give your answer in explicit form. (7 marks) 6. (a) Given the initial-value problem y′ = t2 − ty, y(0) = −1 use Euler’s method with stepsize h = 0.5 to estimate the value of y(1). (6 marks) (b) Solve the linear system x′ = 2x + 3y y′ = 2x + y for x(t) and y(t) with initial conditions given by x(0) = 1 and y(0) = 1. (8 marks) (c) Solve the initial-value problem y′′ + 6y′ + 9y = 0, y(0) = 1, y′(0) = b for y(t) where b is a given real number. (6 marks) Page 6 of 6
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