University of London The London School of Economics and Political Science
ECON MA100

Winter 2014
Name _
Class Teacher_ Class group _
201314
MA100
Mathematical Methods
Test
This test does not count towards your final grade.
Instructions to candidates
Time allowed: 1 hour.
Answers should be justified by showing work.
Use blue or black ink to write you
University of London The London School of Economics and Political Science
ECON MA100

Winter 2014
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 i
University of London The London School of Economics and Political Science
ECON MA100

Winter 2014
Calculus Lecture Notes, Week Seven
1 Vectors, Lines, Hyperplanes, Flats, Surfaces,
Contours and Stationary Points in R"
The geometrical ideas introduced in previous weeks are extended below to multi
dimensional spaces.
1.1 Vectors in R
131
02
An elem
University of London The London School of Economics and Political Science
ECON MA100

Winter 2014
Calculus Lecture Notes7 Week Three
1 Functions, Derivatives, Taylor Series and Sta
tionary Points
1 .1 Intervals
A subset I of R is called an interval if, Whenever it contains two real numbers, it
contains all the real numbers between them. An interval ca
University of London The London School of Economics and Political Science
ECON MA100

Spring 2014
Exercise Set 13
Solutions
1. (a) Set z = loga y. This means exactly that az = y. Now a = eln a , so y = az = ez ln a .
Taking natural logs of both sides gives ln y = z ln a. All of this is based on the exponential
and logarithmic function being inverse to
University of London The London School of Economics and Political Science
ECON MA100

Spring 2014
Exercise Set 11
Solutions
*1. The second derivative is given by:
2 2 0
f (x) = 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
University of London The London School of Economics and Political Science
ECON MA100

Spring 2014
Exercise Set 14
Solutions
1. See the corresponding Maple le for the graph of the function y = x4 6x2 + 12
which will indicate appropriate domains for the dierent local inverses.
To nd formulae for local inverses, solve for x in terms of y:
x4 6x2 + 12 y =
University of London The London School of Economics and Political Science
ECON MA100

Spring 2014
Exercise Set 18
1. *(a) xy
Solutions
dy
y 2 = 3x2 e2y/x
dx
The terms xy, y 2 and 3x2 e2y/x are all homogeneous of degree 2.
Let
dy
dv
=v+x
dx
dx
y = vx
Substituting into the equation,
)
(
dv
2
vx v + x
v 2 x2 = 3x2 e2vx/x
dx
dv
v 2 = 3e2v
dx
dx
ve2v dv
University of London The London School of Economics and Political Science
ECON MA100

Spring 2014
Exercise Set 17
Solutions
1. *(a) Rewrite (x2 + 6x) dy = y 2 dx 12 dy as (x2 + 6x + 12) dy = y 2 dx.
This equation is separable,
1
1
1
dy =
dx =
dx
2
2 + 6x + 12
y
x
(x + 3)2 + 3
(
)
y 1
1
x+3
= arctan
+C
1
3
3
)
(
1
1
x+3
= arctan
+ C, C R
y
3
3
(b) 2x
d
University of London The London School of Economics and Political Science
ECON MA100

Spring 2014
Exercise Set 19
Solutions
*1. (i) y(x) = a + bx + cex + dxex , a, b, c, d R. The term bx dominates, so as x :
y
if
y
b>0
if
ya
b<0
if
b=0
Note that dxex 0 as x .
(ii) y(x) = ex (a cos 3x + b sin 3x), a, b R.
solution oscillates innitely.
2. *(i)
yx+2 yx+
University of London The London School of Economics and Political Science
ECON MA100

Winter 2014
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:
University of London The London School of Economics and Political Science
ECON MA100

Winter 2014
Exercise Set 7
*1.
Solutions
f (x, y, z ) = 3x2 + 2y 2 z 2 .
f
= 6x
x
f
= 4y
y
f
= 2z
z
(a) For the tangent hyperplane to the hypersurface u = 3x2 + 2y 2 z 2
when x = y = z = 1,
u = 4,
f
= 6,
x
f
= 4,
y
(b) The hyperboloid
hypersurface,
f
= 2
z
=
3x2 + 2y
University of London The London School of Economics and Political Science
ECON MA100

Winter 2014
Calculus Lecture Notes, Week Eight
1
VectorValued Functions and Tangent Flats
We extend the geometrical ideas presented in previous weeks by considering functions whose domain and codomain are both multidimensional spaces. We will not
cover contours and
University of London The London School of Economics and Political Science
ECON MA100

Winter 2014
Calculus Lecture Notes, Week Nine
1
Chain Rule for VectorValued Functions and
Second Order Taylor Polynomial for f : Rn R
We extend the chain rule to compositions of vectorvalued functions. We also dene the second derivative and the second order Taylor
University of London The London School of Economics and Political Science
ECON MA100

Winter 2014
MA100 Mathematical Methods
Background You Should Know and Exercises
This is background material for MA100 and is intended to be a review of your Alevel
Mathematics course. Please work through it before term or in your spare time. (You do
not need and sho
University of London The London School of Economics and Political Science
ECON MA100

Winter 2014
Calculus Lecture Notes, Week Ten
1
Classication of the Stationary Points of a
Function f : Rn R
This week, we revisit the classication of stationary points for functions of a single
variable and then discuss functions of n variables.
1.1
Classication of t
University of London The London School of Economics and Political Science
ECON MA100

Winter 2014
Calculus Lecture Notes, Week One
1 Vectors
1.1 Visualising the set R2 using position vectors
a
b
)l e 6 Kb 6 a}. We call the
The set R2 is dened as the set of all 2 x 1 column matrices ( ) where e and b are
a
real numbers. In set notation, we write R2 = 6
University of London The London School of Economics and Political Science
ECON MA100

Winter 2014
Exercise Set 11
*1. Show that the function
f (x, y, z ) = x2 + 5y 2 + z 2 + 2xy + Ayz
is convex if the positive constant A is less than 4, but not convex if A > 4.
Identify one stationary point of f , and show that this is the global minimum of f if A < 4
University of London The London School of Economics and Political Science
ECON MA100

Winter 2014
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,
dy
= 3 cm/min.