Download Advanced Higher Mathematics

Advanced Higher Mathematics
Homework 2.3A
1.
Calculate each of these integrals.
(a)
ò
3
ò
1
1
(d)
0
2.
(b)
7.
ò
p
6
tan 2x dx
0
dx
(f)
ò
(c)
 x  x  1 dx
4 - x2

x2
dx
x 1
(b)
x
2
x2
dx
 2x  8
1
dy
in terms of x and y.
dx
2x2  y 2  4x
(b)
x 2  xy  y 2  0
dy
in terms of x and y.
dx
Hence find the equation of the tangent at the point (1,1).
Find
Differentiate each of these expressions w.r.t. x, simplifying your answers as far as possible.
(a)
6.
(e)
(c)
A curve has equation xy + y 2 = 2.
(a)
5.
x
dx
x +2
2
For each of these, find
(a)
4.
(b)
Find each of these integrals:
(a)
3.
4
ò0 5 - 4 x dx
dx
ò 4 + x2
1
3
(4 x - 3)2 dx
(b)
sec x tan x
2
tan 1 x
(c)
- 4
e 2 x sec 2 x
Given that y = (x + 1) (x + 2) , and x > 0, use logarithmic differentiation to show that
dy
can be
dx
æ a
ö
b ÷
expressed in the form çç
+
y, stating the values of a and b.
÷
çè x + 1 x + 2 ÷
ø
A particle moves so that its position x at time t is given by x  6t 2  t 3 .
Find expressions for v and a, the velocity and acceleration at time t.
8.
Find the derivative of y w.r.t. x, where y is an implicit function of x defined by the equation
x 2  xy  y 2  1.
Find also the coordinates of each of the two stationary points on the graph of this function.
9.
(a)
(b)
1
1  cos 2 x .
2
Find the volume of the solid formed when the graph of y  sin x, from x  0 to x  , is
given a rotation of 360 about the x-axis.
Show that sin 2 x 
10.
A car manufacturer is planning future production patterns. Based on estimates of time, cost and
labour, he obtains a set of three equations for the numbers x, y and z of three new types of car.
These equations are
x  2 y  z  60
2 x  3 y  z  85
3x  y  (  2) z  105
The integer  is a parameter such that 0    10.
11.
(a)
(b)
(c)
Use Gaussian elimination to find an expression for z in terms of  .
Given that z must be a positive integer, what are the possible values for z?
Find the corresponding values of x and y for each value of z.
(a)
Find a real zero of the cubic polynomial c( x)  x3  x 2  x  2 and hence factorise it as a
product of a linear term (x ) and a quadratic term q ( x).
Show that c (x ) cannot be written as a product of three linear factors.
Use your factorisation to find values of A, B and C such that
(b)
(c)
5x  4
A
Bx  C


.
2
x  x  x2
( x)
q( x)
3
Hence obtain the indefinite integral
12.
x
3
5x  4
dx.
 x2  x  2
g (x)= x tan- 1 x, where x is a real number.
C
Show that g ¢¢(x ) =
, where C is a constant to be determined.
2
(1 + x 2 )
Explain why the graph of g (x) has no points of inflexion.
13.
ò sin
2
x cos 2 x dx =
ò cos
2
ò cos
4
(a)
Show that
(b)
By writing cos 4 x = cos x cos3 x and using integration by parts, show that
ò
p
4
cos 4 x dx =
0
(c)
Show that
ò
p
4
cos 2 x dx =
0
(d)
x dx -
p
1
4
+ 3ò sin 2 x cos 2 x dx.
0
4
p+ 2
.
8
Hence, using the above results, show that
ò
0
p
4
cos 4 x dx =
3p + 8
.
32
x dx.