The trapezium rule, with 2 intervals of equal width, is to be used to find an approximate value for fe"dx . i) Explain with the aid of a sketch, why the approximation will be greater than the exact value of the integral. ii) Calculate the approximate value and the exact value, giving each answer correct to 3 decimal places. ii1) Another approximation to 'e*dx is to be calculated by using two trapezium of unequal width. The first trapezium has width h and the second trapezium has width (1– k), so that the three ordinates are at x= 0, x=h and x=1. Show that the total area T of these two trapezium is given by 7 =(r! + h(1- e*1) + e* ). iv) Show that the value of h for which T is a minimum is given by h = In'

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Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.