Mst209 Mathematical Methods And Modelling Definition
How can the answer be improved? MST207 Mathematical Methods, Models & Modelling. MST209 Mathematical Methods and. A degree containing physics from the Open University is accredited provided it.
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TMA MST209 01 Part 1 Cut-off date 7 November 2012 Question 1 below, on Unit 1, forms the first part of TMA MST209 01. The remainder of the TMA (Part 2, on Units 2, 3 and 4 ) can be found immediately following Question 1 in this booklet. Question 1 is marked out of 25. When you use Mathcad, it is important that you make your tutor aware of the results that you wish to be considered: you should not leave your tutor to interpret Mathcad printouts.
(a) Real variables x and y are related by the equation ln(3y + 2) = 5 ln(x − 1) − ln(2 − x) − x. (i) Determine the range of values of x and y for which the expressions on each side of this equation are defined.
[2] (ii) Find y explicitly as a function of x, that is, express the equation in the form y = f(x), simplifying your answer as far as possible. [3] (b) (i) Show that the expressions 2 √ 3 sin(3t + π/3) and√ 3 sin(3t) + 3 cos(3t) are equivalent. [2] (ii) Use Mathcad to graph the function f(t) = √ 3 sin(3t) + 3 cos(3t) PCon the interval −π ≤ t ≤ π. On the same graph, plot the function g(t) = 0.8t2.
How many solutions does the equation f(t) = g(t) have on the interval −π ≤ t ≤ π? Briefly justify your answer. [3] (iii) Use Mathcad to find, correct to four decimal places, all of the PCsolutions of the equation f(t) = g(t) on the interval −3 ≤ t ≤ −1. (You can use the Mathcad worksheet 20901-02 Solving numerically.xmcd, where you will need to follow the instructions concerning solve blocks in the Mathcad tips pop-up.) Submit to your tutor just one page of Mathcad printout, on which you should clearly identify the input and your solutions. [3] (c) If xy2 + 3x2y2 − x3y = 5, use implicit differentiation to determine dy/dx, expressing your answer in the form dy dx = f(x, y), that is, an expression that involves terms in both x and y.
Igi 1 mission 12. [4] (d) (i) Without using Mathcad, determine the indefinite integral of the function f(x) = 1 (x + 3)(x − 2) (−3. TMA MST209 01 Part 2 Cut-off date 28 November 2012 Questions 2 to 6 below, on Units 2, 3 and 4, form the second part of TMA 01. Your overall grade on TMA 01 will be based on the sum of your marks on these questions and on the question in Part 1. Find the general solution of the differential equation, expressing y explicitly as a function of t.
Hence find the particular solution of the differential equation that satisfies the initial condition y(0) = 1. [6] Question 3 (Unit 2 ) – 9 marks This question is concerned with the use of Euler’s method to find a numerical solution to the initial-value problem dy dx = 2x2 − 3y2, y(0) = 0. In part (a) you may use a computer or calculator only to perform numerical calculations. In part (b), on the other hand, you are expected to use one of the MST209 Mathcad worksheets. You may find it helpful to use the same worksheet in part (c).
(a) Use Euler’s method with a step size of 0.1 to find an approximation to the value of y(0.3), where y(x) is the solution to the given initial-value problem. Carry out your calculations using at least five decimal places. Show all your working, and quote your final answer to four decimal places. [3] (b) Use the Mathcad worksheet 20902-02 Euler’s method.xmcd PCassociated with Unit 2, Activity 2.3, to calculate approximations to six-decimal-place accuracy to the value of y(1), where y(x) is the solution to the given initial-value problem, with step sizes h = 0.01, 0.001 and 0.0001. (You may have to edit the worksheet, by entering the appropriate right-hand side for the differential equation, the appropriate initial values and the number (three) of step sizes.) Submit to your tutor the Mathcad printouts that show your edited inputs and the output. [3] (c) The value 0.526 709 of the solution y(1), which is correct to six decimal places, has been obtained using a different numerical method. Using the three approximate values for y(1) that you have obtained PCusing Euler’s method in part (b), confirm that ‘absolute error is approximately proportional to step size’ (page 80 of Unit 2 ) when Euler’s method is used for this initial-value problem with step sizes h = 0.01, 0.001 and 0.0001.