Differentiation

1. Given that , write down .
2. A curve has equation .
   (a) Sketch the gradient function for the curve.
   (b) State the range of values of for which is decreasing.
3. Show that the only stationary point on the curve is a minimum point.
4. A curve has equation .

Find the equations of the two tangents to the curve which have gradient 4 .
5. (a) Find the equation of the normal to the graph at the point where .[3]
(b) Show that the normal does not meet the curve again.
6. Sabah is using a spreadsheet to investigate the gradient of the curve . She uses the spreadsheet to calculate the gradient of the chord from the point where to the point where for different values of . Part of her spreadsheet is shown below.

	A	B	C	D	E	F
1	x	f(x)	h	x+h	f(x+h)	Gradient
2	2	16	1	3	81	65
3	2	16	0.1	2.1		34.481
4	2	16	0.01	2.01	16.32240801
5	2	16	0.001		16.03202401	32.024008
6	2	16	0.0001	2.0001	16.00320024	32.00240008

(a) Fill in the missing values in cells D5, E3 and F4.
(b) Use differentiation from first principles to show that the gradient of the curve at the point where is 32 .
7. In this question you must show detailed reasoning.

Find the stationary points of the graph .
Determine the nature of each.
8. Two real numbers and are such that , where is a positive constant. Find, in terms of , the maximum value of the product of the two numbers.
9. A cuboid has a square base of length and height . The surface area of the cuboid is .
(a) Show that .
(b) Show that the maximum volume of the cuboid occurs when , and find this maximum volume.