Differentiating powers of x

A curve C has equation

where k is a positive integer.

Show that

where A is a constant to be found.

Hence show that the y-coordinate of the stationary point of C can be written as Fully justify your answer.

Given that the stationary point of C has coordinates – state the value of k

A block of ice is melting.

At time t minutes, the block is in the shape of a cuboid with dimensions of 4x, 2x and x, as shown in the diagram.

All measurements are in centimetres.

The volume, V cm³, of the block of ice decreases at a rate which is proportional to its surface area.

When x = 4 the volume of the block of ice is decreasing at a rate of 7 cm³ per minute.

Show that

Using the results from parts (a) and (b), find

Interpret, in context, your answer to part (c)(i).

Circle your answer.

A curve has equation

The point P with coordinates (4, 10) lies on the curve.

Find an equation of the tangent to the curve at the point P

Show that the curve has no stationary points.

A function f is defined for all real values of x as

The function has exactly two stationary points when x 0 and x

f x 9 (a) (i) Find ()

Determine the nature of the stationary points.

Fully justify your answer.

State the range of values of x for which

is an increasing function.

A second function g is defined for all real values of x as

State the single transformation which maps f onto g.

Circle your answer.

A curve has equation y x⁵ 4x³ 7x q where q is a positive constant.

Find the gradient of the curve at the point where x 0

Circle your answer.

0 4 7 q