1. [16 marks] Solve the following differential equations. Note that the solution may be either a general solution or a particular solution.

(a) dy/dx= (y + 1)2 e −3x with y(0) = 2

(b) x^2 dy /dx − 4xy + 1 x = 0 with y(1) = 2

(c) 3 dy/dx + x 3 y − 4x 3 = 0

2. [13 marks] A stream, which is polluted with insecticide at concentration 9 g/m3 , flows at a rate of 25 m3 /day into a pond of volume 2000 m3 . At the same time, water from the pond is flowing into the sea at rate 25 m3 /day. The initial insecticide concentration in the pond is 2.5 g/m3 .

(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(t) 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 g/m3 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 simplifying.

(a) f(x) = (2 cos^2 x + 3)^5/2

(b) g(t) = tan(t^ 2 − 2e ^−4t )

(c) y = y(x), where x ^2 cos y + sin(3x − 4y) = 3

(d) h(x) = cos^−1 (x^ 5/3 )

4. [8 marks] Find the following integrals exactly. Show all working.

(a) ∫^ π/4 cos(2x) ^e sin(2x) dx

0

(b) ∫ 3x(1 + x^ 4 )^ −1 dx

5. [11 marks] In a suburban area, hourly internet usage is measured and recorded from 0:00AM on October 1, 2017. It is observed that the hourly usage f(t), measured t hours from this time, is a sinusoidal function (a sine or cosine curve). The maximum usage is 5TB (5000GB). This maximum/peak usage occurs at 2AM, 10AM and 6PM each day. The minimum usage is 1TB.

(a) What is the period of f(t)?

(b) Find the formula for f(t) and sketch the graph of f(t) for 0 ≤ t ≤ 24.

(c) Find the time(s) at which the minimum usage occurs each day, showing the reasoning.

(d) Find f ′ (t) and evaluate f ′ (3) exactly. Is the internet usage increasing or decreasing at t = 3 and why?

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