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

Winter 2014
4 OneVariable Calculus, Part 2 of 7
Todays plan is to develop onevariable calculus further by introducing differentiation
rules, Taylor polynomials and Taylor series.
4.1 Derivatives of elementary f
University of London The London School of Economics and Political Science
ECON MA100

Winter 2014
5 OneVariable Calculus, Part 3 of 7
Todays plan is to introduce several important concepts: increasing and decreas
ing functions, local extrema and strict local extrema, stationary points and their
University of London The London School of Economics and Political Science
ECON MA100

Winter 2014
As discussed in the orientation lecture, the role of the lecture slides produced during
lectures is to complement your lecture notes; they are not a substitute for them.
During lectures, I will be mak
University of London The London School of Economics and Political Science
ECON MA100

Winter 2014
6 OneVariable Calculus, Part 4 of 7
, IomL W AMHJF
: [66ft *' iiif I"
II . a ' 0.: ." '1 ' '8?
6.1 Asymptotes and graph sketching
$29. 6.l \ (For mdkdoloy7 /I1;w./c3
m
6.2 Conics in sta
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 th
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 m
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 loc
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 +
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
= arcta
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),
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 mi
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 m
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
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 befo
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