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Engineering Mathematics | CHEN30101 problem sheet 1 1. The equation 2x−1 = cos x has a root between x = 0 and x = 1. Compute the root to an accuracy of two digits. (Check your answer by trying both values as starting points.) 2. A tank is fed at a constant rate and also drains a second tank. In turn, this tank drains into a third tank. The volume of fluid in the tank varies like V = 10 − 9e−t − e−3t . Compute an accurate estimate (correct to 5 digits) of the time when the volume of fluid in the tank is equal to 8. 3. The coupled equations 2x + y + e− 5x − 2y − 1 10 xy 5 − 11 = 0 sin(x + y) − 14 = 0 have a solution near the point x = 4, y = 3. Construct a Newton iteration to solve these two equations∗ and use this strategy to compute the solution to an accuracy of 5 digits. 4. The proportion of two kinds of impurities in a liquid vary with height and temperature as follows: T A = e− 100 /100 + T × sin(πh)/11000 + h/10 T B =e− 60 /80 + T × sin(πh)/5000 + (1 − h)/5. Find a temperature near T = 600K and a position near h = 0.5 where A = 0.1 and B = 0.2.∗ Record your final answer to an accuracy of 3 digits. 5. Consider the coupled equation system x2 − 2xy + y 2 − 5 = 0 x2 + 4xy + 4y 2 − 7 = 0 Compute (on paper) the first Newton iteration for solving the system starting from from x0 = 1, y0 = 1. What happens on a computer? Repeat the above experiment, this time starting from the point x0 = 1, y0 = 3. Does the iteration converge? 6. Compute a solution to the system of equations x2 + y 2 + z − 40 = 0 x2 − y + 3z 2 − 22 = 0 x + y 2 − 2z 2 − 20 = 0 that is accurate to two decimal places by constructing a Newton iteration† and starting from the point x = 5, y = 4, z = 0. ∗ An efficient way of doing this is to modify the function fdef.m that is associated with Computational Exercise I. † An efficient way of doing this is to modify the functions in the M-file newtonit.m that are presented in Computational Exercise I.