Question 1. Sand is leaking from the back of a dump truck and forming a conical pile on the ground. The sand is leaking at the rate of 0.2 m³ per hour. If the base radius of the pile is always 0.6 times the height, how fast is the base radius changing when the height is 1.5 m? Show all the working.

(Volume of cone of height h and base radius r is V = 1/3 πr²h.)

Question 2. Let f(x) = 5 + (x² - x - 3) ln(√x - 1)

(a) Find the derivative of f.

(b) Using the derivative and linear approximation, estimate f(4.08).

Question 3. Find the following integrals, showing all working.

(a) ∫x(3x² - 1)^1/2dx

(b) ∫3x(4x + 2)^-2dx

(c) ₀∫⁸ (t^2/3 + 4e^-t/2) dt

(d) ₀∫¹4x²e^2-x3dx

4. A dam is overflowing at the rate

V′(t) = 3t/(t² + 1)^2/3 thousand m³ per day,

where t is the time in days after the overflow begins. Find the amount of water that overflows from the dam during the second day of the overflow. Give the answer exactly first and then to 4 significant figures.

Question 5. Sketch the region bounded by f(x) = √x + 1, g(x) = e^-2x, x = -1 and x = 2.

Using calculus, find the area of the region, showing all the working. Express your answer in simplified exact form.