Question 1. A river flows at the rate of 60,000 m3/day into a full reservoir of volume 400,000 m3, and the water in the reservoir is pumped out at the same rate for a town's water supply. The river water contains certain bacteria at concentration 0.02 g/m3, and the initial concentration of the bacteria in the reservoir is 0.005 g/m3.
(a) Let y(t) be the amount of bacteria (in grams) in the reservoir at time t (days). Write down and solve an appropriate differential equation for y(t) along with the appropriate initial condition.
(b) Using the solution in part (a), determine what happens to the concentration of bacteria in the reservoir as t → ∞.
(c) The water from the reservoir is not safe to drink if the bacteria concentration reaches 0.01 g/m3. How many days does it take for the bacteria concentration to reach this level? Express the answer in exact form and to 3 decimal places.
Question 2. Find the derivatives of each of the following, showing the working and simplifying.
(a) f(x) = (4 + 3 sin2 x)5/3
(b) g(t) = tan(3t + 4e-2t)
(c) y = y(x), where x2sin y + cos(x + 3y) = 1.
Question 3. Find the following integrals exactly. Use calculus and show all working.
(a) ∫(4 cos 2x - sin πx) dx
(b)0∫√πx cos(x2 - π/4) dx
(c) ∫2x3(1 + x8)-1dx
Question 4. The top of a buoy moves vertically (due to waves) such that its height (in metres) above sea level is h(t) = 0.8 sin(π(t - 1)) + 2 where t ≥ 0 is time in seconds.
(a) Using the formula for h(t), find its period. Sketch the graph of h(t) for 0 ≤ t ≤ 4.
(b) Find the time at which the top of the buoy reaches its maximum height for the 100th time, showing the reasoning.
(c) Find h′(t) and evaluate h′(4/3) exactly. Is the top of the buoy moving up or down at t = 4/3 and why?
(d) Find the smallest value of t ≥ 0 for which h′(t) is a maximum, showing the working.