2 100% + Due by: 4:00pin Monday 21 November, 2022 1. [16 marks] Solve the following differential equations. Note that the solution may be either a general solution or a particular solution. dy a) 20- d with (0) du b) dy (c) 25- 2. [13 marks] A stream, which is polluted with insecticide at concentration 8 g/m, flows at a rate of 24 m' /day into a pond of volume 1800 m . At the same time, water from the pond is flowing into the sea at rate 24 m /day. The initial insecticide concentration in the pond is 3 g/mi (a) Let y(t) be the amount of insecticide (in grams) in the pond at time ? (days). Write down and solve an appropriate differential equation for y({) along with the appropriate initial condition. (b) After a long time, what happens to the concentration of insecticide in the pond? (c) It is known that if the insecticide concentration in the pond reaches 7.5 g/m the water beetles in the pond will die. How many days does it take for the insecticide concentration to reach this threshold? 3. [12 marks] Find the derivatives of each of the following, showing the working and simpli- fying. (a) f(x) = (4cosix - 52)3/2 (b) g(t) = tant _ (c) y = y(r), where r cosy + sin(2 - 5y) 4 (d) h(z) - cos (27/2) (Note: cos hu = arceosu) 1. [8 marks] Find the following integrals exactly. Show all working. (a) 627 sin(mr (b) / 20(1 + 210)-1 da 5. [11 marks] A horse on a merry-go-round moves in such a way that its height (in metres) above the floor is (2) - 0.ssin . (f# 1)) + 1.8 where / 2 0 is time in seconds. (a) Using the formula for h(2), find its period. Sketch the graph of h(t) for 0 S t = 12. (b) Find the time at which the horse reaches its maximum height for the 15th time, showing the reasoning(c) Find h'(f) and evaluate h'(3.5) exactly. Is the horse moving up or down at t - 3.5 and why? (d) Find the smallest value of / 2 0 for which h'({) is a maximum, showing the working Notes: 5 marks are reserved for presentation in this assignment. There are penalties for late assignments. You must contact your tutor before the due date if you have difficulties making the deadline