Differentiation

The mass, m grams, of a radioactive substance t years after it was first measured is modelled by m = 50e^-kt. The half-life of the substance is 8 years. (a) Show that k = (ln 2)/8. (b) Find the mass after 20 years. (c) Find the time taken for the mass to fall to 1 gram. (d) Find the rate at which the mass is decreasing when t = 0.

(a) Prove, from first principles, that the derivative of x³ is 3x². (b) The curve C has equation y = x³ - 3x + 1. Find the equation of the tangent to C at the point where x = 2, in the form y = mx + c.

An open box is made from a square sheet of card of side 30 cm by cutting a square of side x cm from each corner and folding up the sides. (a) Show that the volume of the box is V = x(30 - 2x)² cm³ and state the possible values of x. (b) Use calculus to find the value of x that maximises V. (c) Justify that this value gives a maximum. (d) Find the maximum volume.

A curve has parametric equations x = 3cos t, y = 2sin t, for 0 ≤ t < 2π. (a) Find dy/dx in terms of t. (b) Find the equation of the tangent to the curve at the point where t = π/4, in the form ax + by = c√2. (c) Find a Cartesian equation of the curve.

Water is poured into an inverted cone at a constant rate of 20 cm³ s^-1. When the depth of water is h cm, the radius of the surface is h/2 cm. (a) Show that the volume of water is V = (π h³)/12. (b) Find the rate at which the depth is increasing when h = 10. (c) The cone is initially empty. Find the time taken for the depth to reach 10 cm.