P3_2022.pdf 1 2 100% + Due by: 4:00pm 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. (a) 247 - (9 - 3) (4r - ) with g(0) dy dy _ Ay + 52" =0 with y(1) 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/m. (a) Let y(t) be the amount of insecticide (in grams) in the pond at time t (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(z) - (4cos3 x - 5.2)3/2 (b) g(t) = tan(t - 2e34) (c) y = y(z), where a' cosy + sin(2x - 5y) = 4 (d) h(z) - cos" (27/2) (Note: cos ] u = arccos u) 4. [8 marks] Find the following integrals exactly. Show all working. (2) 617 sin(723 (b) / 224 (1 4 240)-1 dx 5. [11 marks] A horse on a merry-go-round moves in such a way that its height (in metres) above the floor is h(t) - 0.8 sin[ (t - 1)] + 1.8 where t 2 0 is time in seconds. (a) Using the formula for h(t), find its period. Sketch the graph of h(t) for 0 S t s 12. (b) Find the time at which the horse reaches its maximum height for the 15th time, showing the reasoning.(c) Find h (t) 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 t 2 0 for which h' (t) 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