GATE Mechanical Engineering Preparation

Welcome to GATE Mechanical Engineering Preparation. Follow Engineering mathematics, then practise in the learning system. Sign in and use the study header to open your authorised course. Select this published lesson. Start Course Tutor with Auto audio on: it teaches continuously, answers a typed question, then resumes. Numerical methods are almost guaranteed. The Newton-Raphson method replaces an estimate by itself minus the function divided by its derivative, converging rapidly from a good start but failing where the derivative is near zero. First, Newton-Raphson. Find a root of x cubed minus x minus 1, starting from an estimate of 1.5, and perform one iteration. The function at 1.5 is 3.375 minus 1.5 minus 1, which is 0.875. The derivative is three x squared minus one, which at 1.5 is 6.75 minus 1, that is 5.75. The iteration replaces 1.5 by 1.5 minus 0.875 divided by 5.75, which is 1.5 minus about 0.152, giving about 1.348. The true root is about 1.325, so one iteration has already brought the error from about 13 per cent to under 2 per cent. Second, a comparison of integration rules. Estimate the integral of x cubed from 0 to 2 using two intervals, by both rules. With two intervals the step is 1 and the function values at 0, 1 and 2 are 0, 1 and 8. The trapezoidal rule takes half the step times the first value plus twice the middle plus the last, which is 0.5 times 0 plus 2 plus 8, that is 5. Simpson's one third rule takes the step over three times the first value plus four times the middle plus the last, which is one third times 0 plus 4 plus 8, that is 4. The exact value is 4. So Simpson's rule is exact here while the trapezoidal estimate is 25 per cent high, because the integrand is a cubic and Simpson's rule integrates cubics exactly while the trapezoidal rule handles only linear integrands exactly. Third, a directional derivative. For the field x squared y, the gradient is the vector whose components are two x y and x squared. At the point where x is 1 and y is 2, that is the vector with components 4 and 1. The greatest rate of increase is the magnitude of that vector, which is the square root of 17, and it occurs along the gradient direction itself. Ask Course Tutor to explain this step: With two intervals the step is 1 and the function values at 0, 1 and 2 are 0, 1 and 8. The trapezoidal rule takes half the step times the first value plus twice the middle plus the last, which is 0.5 times 0 plus 2 plus 8, that is 5. Simpson's one third rule takes the step over three times the first value plus four times the middle plus the last, which is one third times 0 plus 4 plus 8, that is 4. The exact value is 4. So Simpson's rule is exact here while the trapezoidal estimate is 25 per cent high, because the integrand is a cubic and Simpson's rule integrates cubics exactly while the trapezoidal rule handles only linear integrands exactly. Say what you tried. Pause and try this source task. What is the greatest rate of increase of that field at that point, and in which direction? The source answer is: the magnitude is the square root of 81 plus 144, which is 15, and it occurs along the gradient direction. Compare your method and result, correct an error, then continue. Use Stop to end Course Tutor.