Differentiation

1. It is given that x=(t^2)+t and y= sq.root (3t^2+2t-4)
(a) Find dy|dx in terms of t.
(b) Find dy|dx when x=2

2. The length of a diagonal of a cube increases at a rate of 0.1 cm/s, find the rate of change of the volume of the cube when the diagonal is 30cm long.

3. A rectangle is inscribed in a circle of radius a cm. Find the dimensions of the rectangle if it has a maximum area.

4. A particle moves along a straight line so that its displacement s m, from the origin O, after t seconds is given by s = 4t + 2t^2 -t^3, where t>0.
(a) Find the displacement of the particle from O when its velocity is zero.
(b) Find the acceleration of the particle when its velocity is 5 m/s.

5. A right circular cone has a volume of 18'pie' cm3. Let l and h be the slant height and the height of the cone respectively.
(a) show that l^2=h^2 + 54/h
(b) hence, find the minimum value of l as h varies.

6. A solid triangular prism has a given volume of V cm3. Its base is an equilateral triangle. Find the length of each side of the equilateral triangle so that the total surface area of the prism is a minimum.