1.
2.
1. Given that
and
, show that
[4]
(nov2003)
2. Given that
show that
.
By finding the second derivative, or otherwise, show that the series expansion of
ascending powers of , up to and including the term in
is
.
3. The radius of sphere is increasing at rate of 3cms-1. Obtain, as a multiple of
increase of the volume of the sphere when the radius is 9 cm.
[4]
in
[5]
(nov2003)
the rate of
[3]
(june2004)
4. The total cost of producing
radio sets per day is $
The selling price per set is
.
.
(i)
Find, in simplified form, an expression for the total profit per day.
[2]
(ii)
Show that the maximum profit is achieved when
[2]
(iii)
Show that the cost per set has a stationary value.
.
Find this value and determine whether it is a maximum or minimum.
5. A curve is given by the equation
=
.
(i)
Show that
.
(ii)
Hence show that the gradient of the curve cannot be equal to 1.
6. Given that
[4]
(june2004)
[3]
[3]
and
at
=
.
(a)
find
(b)
show that the value of
[4]
is independent of
[3]
(june2005)
1. A gas law is given by the equation
Find
(i)
(ii)
in terms of
where is a constant.
and
[3]
the percentage change in given that increases by 1% and state
whether it is an increase or a decrease.
[3]
(june2007)
2. The parameter equations of a curve are
(i)
Show that
(ii)
Find the equations of the tangent and normal to the curve at the point Q
where = .
(iii)
= 2sin .
If the tangent and normal meet the
state the coordinates of A and B.
[3]
[5]
axis at points A and B respectively,
[1]
(june2007)
1. Given that
(i)
(ii)
(iii)
,
Find
of
and
.
and hence write down the Maclaurin series
up to the term in .
Find by integration the area in the first quadrant bounded by
the
axis and line
giving your answer correct to
3 decimal places.
[4]
Use Maclaurin series obtained in (i) to estimate the area and state the
absolute error.
[4]
Find the coordinates of each of the stationary points on the curve
Show that there is only one minimum point and state its coordinates.
1.
[6]
Given that
and
(i)
an expression for
(ii)
the values of
for
increases as
[5]
[3]
(nov2007)
find
in terms of .
for which
.
[3]
increases
[3]
(june2009)
1. Given that
(i)
(ii)
show that
=
[3]
[5]
1. Given that
2.
and
(i)
an expression for
(ii)
the values of
Given that
(i)
(ii)
for
find
in terms of .
for which
increases as
[3]
increases
[3]
show that
=
[3]
[5]
1. Differentiate with respect to ;
(i)
(ii)
2. Given that
[4]
and
, find
in its simplest form.
[4]
Hence or otherwise describe the shape of the graph of y against x.
Given that
show that
4
[1]
[6]
A curve has parametric equations given by
and
where
(i)
(ii)
(a)
Show that
=
Find the equation of the tangent to the curve at
in the form
.
[3]
giving your answer
[4]
Solve the equation
for
(b)
Given that
stationary points.
An open rectangular box has length
(i)
(ii)
[4]
find the
values of the
[4]
cm and width
cm.
Given that its volume is 128 cm3, find in terms of
the total surface area cm2, of the box.
, the depth,
cm, and
Find the dimensions of the box correct to 3 decimal places such that the
surface area is minimum. Justify that the surface area is a minimum.
[4]
[6]