Chain rule

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).

Find an equation of the tangent to the curve

A particle is moving such that its position vector, r metres, at time t seconds, is given by

Fully justify your answer.

Water is poured into an empty cone at a constant rate of 8 cm³/s

After t seconds the depth of the water in the inverted cone is h cm, as shown in the diagram below.

When the depth of the water in the inverted cone is h cm, the volume, V cm³, is given by

Show that when t 3

Hence, find the rate at which the depth is increasing when t 3

Give your answer to three significant figures.

The volume of a spherical bubble is increasing at a constant rate.

Show that the rate of increase of the radius, r, of the bubble is inversely proportional

A curve, C, has equation

Show that C has exactly one stationary point.

Fully justify your answer.

A company is designing a logo. The logo is a circle of radius 4 inches with an inscribed rectangle. The rectangle must be as large as possible.

The company models the logo on an x - y plane as shown in the diagram.

Use calculus to find the maximum area of the rectangle.

Fully justify your answer.

A function f is defined by f (x)

State the maximum possible domain of f.

Show that the graph of y f (x) has exactly one point of inflection.

Write down the values of x for which the graph of y f (x) is convex.