. [10 marks] Find the derivatives of the following, showing the working and simplifying. (a) x) : (cosx + xsinx)"3 (b) 30) = 511101! + 8"") 3 (c) y 2 x), where x siny2 + cos(5xy} = 7 . [4 marks] Find the tangent to the line 1,! = (2 + cosr)"2 at x = 7r/4. You may use that cos (g) = sin (E) 2 21/2. Tip; Complicated arithmetic expressions are easier to work with if you call them something else. In this problem let ,6 = (2 + 21/2)"2 and the algebra will look a lot easier! . [8 marks] A circular water tank with radius 4m and depth 2m is situated in a tropical village. In order to keep the mosquito population under control, a pesticide is put in the tank so that the concentration is 103/ m3. Water is withdrawn from the tank for town and farm supply at a rate of 51713 lday, and the level is regulated to remain at exactly 2m by an artesian bore (inflow), so the inow is also 53113 / day. (a) Let y (t) be the amountof insecticide (in grams) in the pond at time t (days). Write down and solve an appropriate differential equation AND initial condition for t). (b) After a long time, find the concentration of insecticide in the pond if none is added? (c) In order to kill the mosquito larvae it is necessary that the level of pesticide be kept above 4g/m3. When will more pesticide need to be added to the tank? . [8 marks] Find the following integrals exactly. Show all working. (a) fuxcosw gm: (b) f2x2(1 x6)-1/2dx . [8 marks] Waves passing the end of Busselton jetty are measured to have height (metres) above the ocean oor of h(t} = 8 + 0.3 cos((t 3075/3} where t Z 0 is time (secs). (a) Using the formula for Mt), find its period. Sketch the graph of ME) for D g t g 4. (b) At what time does the wave reache its maximum height for the 20th time (show reasoning). (c) Find h' (t) and evaluate h'(l) exactly. Is the wave moving up or down at t = 1 and why? . [7 marks] Electricity for the village in Q3 is provided by an engine. Suppose the instantaneous power required over the day is given by the equation Pft) = 10 5 sin [0]:E] where t in hours is measured from midnight. (a) At what time of day are power consumption maximal and minimal? (b) What is the total power consumption over a 24 hour period