text consider the oscillatory function yfraccos 2 xx text i describe w

Question

\text { Consider the oscillatory function } y=\frac{\cos (2 x)}{x} \text { (i) Describe what happens to the amplitude of oscillation as } x \text { increases in value. } \text { (ii) Show that } \frac{d y}{d x} \text { is given by } \frac{d y}{d x}=-\frac{2 x \sin (2 x)+\cos (2 x)}{x^{2}} \text { ii) Determine the equation of the tangent line to } y=\frac{\cos (2 x)}{x} \text { at the point }\left(\frac{\pi}{4}, 0\right) \text {. } The function f (t) = te¯", where a > 0, rises to a peak value then decays towards zero for t > 0.-at It can be used to model the transient response of an electrical circuit to the sudden application of a voltage. For the case where a =0.23, find the following. (i) f'(t). (ii) The maximum value of f (t) for t > 0. \text { Use integration by parts to evaluate the definite integral } \int_{0}^{3 \pi} x \sin x d x \text {. } \text { Expand the rational function } \frac{x+1}{x^{2}+7 x+12} in terms of its partial fractions and hence evaluate the indefinite integral \int \frac{x+1}{x^{2}+7 x+12} d x