The marginal profit made by a manufacturer is the additional profit made when production is increased by a small amount. It is the rate of increase of profit with respect to the volume of production. A manufacturer of plastic pipes has determined that with its present equipment the profit function for the production of pipes is:

P(n) = 67500n + 600n^2 - n^3 - 1200000

In the first seconds after launch, the height of a rocket above the ground is given by the expression:

h(t) = 5t^3 + 30t^2 + 45t + 4

where h is in metres and t is in seconds. How long does the rocket take to reach a speed of 75 m/s

NOTE: Speed is the rate of change of distance with respect to time

A farmer has 140m of chicken wire which she is going to use against an existing boundary fence, to enclose a vegetable garden, which she wants in the shape of a rectangle.

Find the maximum area of her vegetable garden.

(You must justify, using calculus, that your result is the maximum area)

A drink manufacturer proposes a new style of can packaging. Given that the sum of the circumference and the height of the new can is to be 30cm, what is the maximum possible volume of the new can package?

(You must justify, using calculus, that your result is the maximum volume)

A particle moves in a straight line so that its displacement from a point O, at any time t, is

x = *square root* 2t^3 + 4 (x is in metres and t is in seconds)

Find the acceleration when t = 2

(NOTE: Acceleration is the rate of change of velocity with respect to time)

The sum of two numbers is 10. What are the two numbers if the sum of their squares is a minimum?

For the curve, y = ax^2 + bx + c, where a, b and c are constants, it is given that at the points (2, 12) and (-1, 0) the slope of the tangent is 7 and 1 respectively.

Find a, b and c.