Exercise Set 16
1. (i)
Find
(a)
(ii)
x3 x2 + x + 1
dx
(x2 + 1)2
(b)
cos8 x sin3 x dx
Find, using appropriate changes of variable
(a)
(x2 + 2)6 x3 dx
sin x cos x
dx
(1 + sin x)3
(b)
1
2. Use integration by parts to evaluate
arcsin x dx
0
*3. Find the gener
Exercise Set 14
Solutions
1. *(a) If re is the eective annual rate of interest, after t years a principal of 1000 will
have value
1000(1 + re )t
(i) The value of 1000 after t years compounded quarterly at 8% is
1000 1 +
0.08
4
4t
.
If these values are the
Exercise Set 15
*1. (a)
u
v
f : R2 R2 is dened by
Solutions
2x y
.
x+y
=
The derivative of the function is the matrix,
du
=
dx
=
ux
vx
uy
vy
2
1
=
1
1
.
2
1
1
=3=0
1
1
3
Since
11
1 2
local inverses exist everywhere.
The derivative of the local inverse is
Exercise Set 18
1. *(i)
x dy + y dx = x3 y 6 dx
y 6
Solutions
Let u = y 5 ,
dy
1
+ y 5 = x2
dx x
=
du = 5y 6 dy ,
5y 6
=
dy
1
+ (5) y 5 = 5x2
dx
x
du 5
u = 5x2
dx x
The equation is now linear in u with integrating factor, e
d
(ux5 ) = 5
dx
y 5 x5 =
5 2
x
Exercise Set 19
*1. (i) y (x) = a + bx + cex + dxex
y
if
b>0
Solutions
The term bx dominates, so as x :
y
if
b<0
ya
if
b=0
Note that dxex 0 as x .
(ii) y (x) = ex (a cos 3x + b sin 3x)
oscillates innitely.
As x , ex and the general solution
(iii) y (x) =
Exercise Set 20
Solutions
Solutions c(x) for (all but one of) the corresponding homogeneous equations for questions 1
and 2 were found in Exercise Set 18 and 19 respectively, so only particular solutions need be
found here. The general solution to the non
Exercise Set 8
1. The function f : R3 R2 with component functions f1 and f2 is dened by
u = f1 (x, y, z ) = x2 + y 2 + z 2 ;
v = f2 (x, y, z ) = x y.
Find all the points x = (x, y, z )T such that f (x) = (8, 0)T , and describe the curve consisting
of thes
Exercise Set 9
1. The length and width of a rectangle decrease at the rate of 2 cm per minute and 3 cm per
minute respectively. When the length is 6m and the width is 3m how fast are the following
changing:
(a) the area
and
*(b) the diagonal?
2. Suppose t
Exercise Set 11
Solutions
1. The second derivative is given by:
2
f (x) = 2
0
2
10
A
0
A.
2
Note that this matrix is independent of x. To test whether f is convex, we need to look
at the principal minors. The rst of these is the upper left entry, which is
Exercise Set 9
Solutions
1. Let x = x(t) and y = y (t) represent the length and width of the rectangle respectively.
If t0 is the time at which the length is 6m and the width 3m, at t0 ,
x(t0 ) = 600 cm,
dx
= 2 cm/min,
dt
y (t0 ) = 300 cm,
(a) the area,
A
Exercise Set 17
1. Classify each of the dierential equations below in one or more of the following categories:
(i) Linear equations (ii) Homogeneous equations (iii) Exact equations
Find the general solutions of the equations by any appropriate method.
(a)
Exercise Set 19
*1. Describe the behaviour as x of the general solutions of the equations in question 3
of Exercise Set 18.
2. Find the general solutions of the following dierence equations; describing their behaviour
as x
(i) yx+2 yx+1 2yx = 0
(ii) yx+3
Exercise Set 20
*1. Find the general solutions of the following dierential equations where x R+ :
(i)
d4 y
d3 y
d2 y
+ 2 3 + 2 = sin x
dx4
dx
dx
(ii)
d3 y
d2 y
dy
3 2 +9
+ 13y = x + ex
3
dx
dx
dx
2. Find the general solutions of the following dierence equ
Exercise Set 4
Solutions
The solutions below are only an outline. When writing out solutions show all your work.
*1. To nd the local maxima and minima of the cost function
C (x) = x3 24x2 + 117x + 784
=
C (x) = 3x2 48x + 117
The stationary points occur wh
Exercise Set 5
1.
(a)
z=
Solutions
x2 + y 2
x+y
z
2x(x + y ) (x2 + y 2 ) 1
2x2 + 2xy x2 y 2
x2 + 2xy y 2
=
=
=
x
(x + y )2
(x + y )2
(x + y )2
z
y 2 + 2xy x2
=
y
(x + y )2
by symmetry
To check that the function z = f (x, y ) is homogeneous:
f (tx, ty ) =
Exercise Set 6
*1.
Solutions
h : R2 R
h(x, y ) = 4000 0.001x2 0.004y 2
The direction of maximum increase of h is given by the gradient vector,
h=
h/x
h/y
=
0.002x
0.008y
=
h(500, 300) =
1
2.4
The rate of ascent in this direction is
h=
(1)2 + (2.4)2 = 2.6
Exercise Set 8
Solutions
1. The equations you have to solve are:
x2 + y 2 + z 2 = 8;
x y = 0.
Eliminating y , we see that the points x = (x, y, z )T we want are those points of the form
(x, x, z ) with 2x2 + z 2 = 8. There are other ways to express this: