Determine the maximum value of C and the temperature at which it occurs. (5)

6. Determine for the curve y = 2x2 − 3x at the point (2, 2): (a) the equation of the tangent (b) the equation of the normal (6)

7. A rectangular block of metal with a square crosssection has a total surface area of 250 cm2. Find the maximum volume of the block of metal. (7)

8. A cycloid has parametric equations given by: x = 5(θ − sin θ) and y = 5(1 − cos θ). Evaluate (a) dy

dx (b) d

when θ = 1.5 radians. Give answers correct to 3 decimal places. (8)

9. Determine the equation of (a) the tangent, and (b) the normal, drawn to an ellipse x = 4 cos θ, y = sin θ at θ = π

3 (8)

10. Determine expressions for dz dy

for each of the following functions:

(a) z = 5y2 cos x (b) z = x2 + 4xy − y2 (5)

11. If x2 + y2 + 6x + 8y + 1 = 0, find dy dx

in terms of x and y. (3)

12. Determine the gradient of the tangents drawn to the hyperbola x2 − y2 = 8 at x = 3. (3)

13. Use logarithmic differentiation to differentiate

y = (x + 1) 2√(x − 2)

(2x − 1) 3√(x − 3)4 with respect to x. (6)

14. Differentiate y = 3e θ sin 2θ√

θ5 and hence evaluate

dy dθ

, correct to 2 decimal places, when θ = π 3 (9)

15. Evaluate d dt

[ t √(2t + 1)] when t = 2, correct to

4 significant figures. (5)