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Fall 2014
140
Solutions
Solution to Exercise 5.29. (From the calculation done in the given hint, we guess that the derivative
of g at c is the map Rn h 7 2f (c)f (c)h, and we prove our claim below.)
We have for x Rn that
(f (x)2 (f (c)2 2f (c)f (c)(x c)
(f (x) + f
University of London The London School of Economics and Political Science
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Fall 2014
Solutions to the exercises from Chapter 4
129
Solution to Exercise 4.33. Let X = (0, 1], Y = R with the usual Euclidean metrics. Define f : (0, 1]
R by
1
(x (0, 1]).
f (x) =
x
Then f is continuous. Take (xn )nN to be the Cauchy sequence n1 nN . Then f (x
University of London The London School of Economics and Political Science
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Fall 2014
Solutions to the exercises from Chapter 4
121
Solutions to the exercises from Chapter 4
Solution to Exercise 4.3. That (1) implies (2 ) is immediate since (1) implies (2), and (2) implies (2 ).
Now suppose that (2 ) holds. Let (xn )nN be a sequence contai
University of London The London School of Economics and Political Science
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Fall 2014
Solutions
134
Solutions to the exercises from Chapter 5
Solution to Exercise 5.9. We have
f (a)
(a) = det f (a)
f (b)
g(a)
g(a)
g(b)
1
1 =0
1
since the first and second rows of the matrix are linearly dependent. Similarly,
f (b) g(b) 1
(b) = det f (a) g(a
University of London The London School of Economics and Political Science
ABSTRACT MATHEMATICS
MATHEMATIC MA203

Fall 2014
Solutions
Solutions to the exercises from Chapter 1
Solution to Exercise 1.7.
(D1) As d(x, y) is either 0 or 1 for all x, y X, clearly d(x, y) 0.
Also, for all x X, d(x, x) = 0 by definition.
If x, y X and x 6= y, then d(x, y) = 1 6= 0. So if d(x, y) = 0,
University of London The London School of Economics and Political Science
ABSTRACT MATHEMATICS
MATHEMATIC MA203

Fall 2014
Solutions to the exercises from Chapter 3
109
Solutions to the exercises from Chapter 3
Solution to Exercise 3.3. We have
(2n + 1) (2n 1)
1
1
1
= tan1
tan1
,
tan1 2 = tan1
2n
1 + (2n + 1)(2n 1)
2n 1
2n + 1
and so the partial sums telescope to give
X
tan1
University of London The London School of Economics and Political Science
ABSTRACT MATHEMATICS
MATHEMATIC MA203

Fall 2014
Solutions
104
Solution to Exercise 2.34. (If part) Suppose that (an )nN converges to 0. Let > 0. Then there
exists an N N such that whenever n > N , an 0 < . As an 0, we have that an < for all n > N .
But by the definition of an , this means that for an
University of London The London School of Economics and Political Science
ABSTRACT MATHEMATICS
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Fall 2014
Solutions
90
Solution to Exercise 1.25. Consider the open ball B(x, r) = cfw_y X : d(x, y) < r in X. If y B(x, r),
then d(x, y) < r. Define r = r d(x, y) > 0. We claim that B(y, r ) B(x, r). Let z B(y, r ). Then
d(z, y) < r = r d(x, y) and so d(x, z) d(x,
University of London The London School of Economics and Political Science
ABSTRACT MATHEMATICS
MATHEMATIC MA203

Fall 2014
Solutions to the exercises from Chapter 2
97
Solutions to the exercises from Chapter 2
Solution to Exercise 2.5. Let > 0. Since (an )nN is a Cauchy sequence, there exists an N N
such that for all n, m > N , an am  < . In particular for n > N , we have m
University of London The London School of Economics and Political Science
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Fall 2014
Solutions
116
Solution to Exercise 3.29.
(1) True.
Since the series
X
n=1
an  converges, we have necessarily that
lim an  = 0.
n
Thus there is an N N such that for all n > N , an  < 1. But now for n > N ,
a2n  = an an  an ,
and so it follows
University of London The London School of Economics and Political Science
Mathematical mathods
MATHEMATIC ma100

Fall 2014
Mathematical Methods, Lecture ThirtySix
36. Dierential and Dierence Equations, 3 of 5
36.1 Dierence equations
Dierence equations are recurrence relations satised by a sequence cfw_yx . The
sequence cfw_yx is the unknown of the problem, and x N is an ind
University of London The London School of Economics and Political Science
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Fall 2014
Mathematical Methods, Lecture ThirtyTwo
32. Multivariate Calculus, 4 of 5
32.1 The second derivative of a function
d2 f (x)
of a function f : Rn R is dened by taking the
dx2
derivative of the (transpose of the) derivative of f (x); that is, the transpose
University of London The London School of Economics and Political Science
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Fall 2014
Mathematical Methods, Lecture TwentySeven
27. Linear Transformations, 5 of 6
27.1 Diagonalisation
In the last lecture we saw examples of linear transformations T : R2 R2 where a
basis B of R2 can be found such that the matrix ABB
representing T is diagon
University of London The London School of Economics and Political Science
Mathematical mathods
MATHEMATIC ma100

Fall 2014
Mathematical Methods, Lecture TwentyEight
28. Linear Transformations, 6 of 6
We now focus on symmetric matrices and a special form of diagonalisation applicable
to symmetric matrices, known as orthogonal diagonalisation. We begin by introducing orthogona
University of London The London School of Economics and Political Science
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Fall 2014
Mathematical Methods, Lecture Twenty
20. Vector Spaces, 4 of 4
In this section, we develop the concepts of angle and length, so that these concepts
can be applied to a general vector space V where angles and lengths do not arise
in a natural way. For exam
University of London The London School of Economics and Political Science
Further Quantitative Methods
MATHEMATIC MA207

Spring 2016
FQM Solutions to Exercises 7
1. The gradient vector for the function h(x, y, z) = x2 + y 2 + z 2 is given by
2x
1
1 1 1
h = 2y
At
, ,
,
h =
1
2 2
2
2z
2
so this is the normal vector to the surface at that point. The equation of the tangent plane
is
x
University of London The London School of Economics and Political Science
Further Quantitative Methods
MATHEMATIC MA207

Spring 2016
FQM Solutions to Exercises 10
1. (a) u(x, y) = x3 y 3 2xy + 1
ux (x, y) = 3x2 2y
The partial derivatives are
3x2 = 2y
uy (x, y) = 3y 2x
2
2x = 3y
2
27y 4 8y = y(27y 3 8) = 0
Solve the second equation for x, substitute into the rst, and obtain the last equ
University of London The London School of Economics and Political Science
Further Quantitative Methods
MATHEMATIC MA207

Spring 2016
FQM Solutions to Exercises 9
1. (a) f (x, y) = x2 + 4xy + 2y 2 + 6x + 6y + 1 has stationary points when
)
fx = 2x + 4y + 6 = 0
3
x=0 y=
2
fy = 4x + 4y + 6 = 0
The quadratic part of the function is the quadratic form
T
2
x Ax = x + 4xy + 2y
2
where
A=
1 2
University of London The London School of Economics and Political Science
Further Quantitative Methods
MATHEMATIC MA207

Spring 2016
FQM Solutions to Exercises 6
1
a = 2
2
1. We have
1
b= 1
4
2
c = b a = 1
2
The cosines of the three angles are given by
ab
1 + 2 + 8
1
=
= ;
a b
9 18
2
2+24
ac
=
= 0;
a c
9 9
21+8
bc
1
= =
b c
18 9
2
Thus the triangle has a rightangle, and two angl
University of London The London School of Economics and Political Science
Further Quantitative Methods
MATHEMATIC MA207

Spring 2016
FQM Solutions to Exercises 1
1.
Y R = ( 500
10000
1.05 0.95
1000 ) 1.05 1.05 = ( 12395
1.37 1.42
12395 )
so it is riskless since the guaranteed return is 12395 in each state.
Z = ( 1000
2000
cost(Z) = 1000 2000 + 1000 = 0
1000 )
1.05 0.95
1000 ) 1.05 1.05
University of London The London School of Economics and Political Science
Further Quantitative Methods
MATHEMATIC MA207

Spring 2016
FQM Solutions to Exercises 5
1. The auxiliary equation of yt 4yt1 + 4yt2 = 5
with two equal roots, = 2.
is 2 4 + 4 = ( 2)2 = 0,
The general solution of the associated homogeneous equation is yt = (C + Dt)(2)t .
A particular (constant) solution is y = 5/(1
University of London The London School of Economics and Political Science
Further Quantitative Methods
MATHEMATIC MA207

Spring 2016
FQM Solutions to Exercises 4
1. In matrix form, the equations are
ft
rt
.4
.4
=
.3
1.2
ft1
rt1
= Axt
We will give two methods which can be used to solve this system. The solutions will be linear
combinations of (1 )t and (2 )t where 1 and 2 are the eigenv
University of London The London School of Economics and Political Science
Mathematical mathods
MATHEMATIC ma100

Fall 2015
Exercise Set 14
1. = Use Maple to plot the function f : R R dened by
y = x4 6x2 + 12 .
Find a formula for a local inverse to the function at each of the points x = 1, 1, 2. Sketch
the graph of each local inverse function.
Calculate the derivative of the l
University of London The London School of Economics and Political Science
Mathematical mathods
MATHEMATIC ma100

Fall 2015
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
University of London The London School of Economics and Political Science
Mathematical mathods
MATHEMATIC ma100

Fall 2015
Exercise Set 17
1. Classify each of the dierential equations below in one or more of the following categories:
(i) linear equation (ii) separable equation (iii) exact equation
Find the general solutions of the equations by any appropriate method.
*(a) (x2
University of London The London School of Economics and Political Science
Mathematical mathods
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Fall 2015
Exercise Set 19
*1. Describe the behaviour as x of the general solutions of the equations in question 4
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
University of London The London School of Economics and Political Science
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Fall 2015
Exercise Set 16
*1. Given the inverse demand function yd = 116 + 2x x2 and inverse supply function
ys = (x + 2)2 , where x represents quantity and yd and ys each represents price, nd the
consumers surplus and the producers surplus.
2. Use partial fraction