Differentiation

1. Using the chain rule, differentiate .
2. Show that the gradient of is given by

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3. Show that the gradient of the curve at the point where is -3 .
4. A potter is making an open topped vessel shaped as a right circular cylinder of radius and height .
(a) Show that the volume of the vessel is given by
(b) Find the rate at which the volume is increasing when the radius is 2 cm and increasing at a rate of .
Given that the volume is increasing at a rate of when the radius is 5 cm , find the rate at which the surface area is increasing at this point.
5. In this question you must show detailed reasoning.

A curve has equation .
(a) Find the coordinates of the stationary points and determine their nature.
(b) Sketch the curve.
Find the values for for which the curve is concave upwards.
6. In this question you must show detailed reasoning.

A log of wood is modelled as a cylinder with radius 10 cm and height 25 cm . It is to be made into a cuboid with dimensions by by 25 cm by trimming the cylinder. The cross-section is shown in the diagram.

(a) Find the value of for which the area of the rectagle is maximum.
(b) Calculate the volume of the largest cuboid that can be cut from the log.