1. (a) Use Taylor's theorem to show that, for r> 0, I-S tanzSz. Hence find the limit tan-r lim 40+ (b) For t> 0, the function J is given by J6) = [" 1/(21) sin (tz) dz. %3D Use the Leibniz rule to show that eJ'(t) k for some constant k that you should find. (c) Determine whether each of the integrals VI+-1 +-1 dz and (ii) VI+-1 dr (i) dr, (ii) is convergent or divergent.

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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.