Reference no: EM132178021

1. Differentiate the following functions with respect to x.

a) Use the quotient rule to differentiate y = 2x⁴ - 3x/4x - 1
b) Use the chain rule to differentiate y = 6 cos(x³ + 3)
c) Select an appropriate rule to differentiate y = (4x²- e^2x) sin(3x)

2. The angular displacement, 0 radians, of the spoke of a wheel is given by the expression 0 = 0.5t⁴ - t³, where t is time in seconds. Find;
a) The angular velocity after 2 seconds.
b) The angular acceleration after 3 seconds.
c) The time when the angular acceleration is zero.

3. From a rectangular sheet of metal measuring 120 mm by 75 mm, equal squares of side x are cut from each of the 4 corners. The remaining flaps are then folded upwards to form an open box.

a) Draw a neat diagram of the rectangular sheet of metal and show the dimensions given including the squares of side x.

b) Show that the volume of the box is given by : V= 9000x - 390x² + 4x³

c) Find the value of x such that the volume is maximum.