ETF2700/ETF9700
MathematicsforBusiness
Week 10 Lecture 1
TOPIC 7
Functions of Several Variables
PARTIAL
DIFFERENTIATION
FunctionsofMoreThan1Variable
So far, all of our functions have contained only one variable on the righthand-side of the equation. Now w

ETF2700/ETF9700
MathematicsforBusiness
Week 4 Lecture 2
TOPIC 4
Non-Linear Functions
QUADRATICS
NonLinearFunctions
The linear models used up until now are good introductory models for
demand, supply and cost, but they are not necessarily realistic when it

ETF2700/ETF9700
MathematicsforBusiness
Week 5 Lecture 2
TOPIC 4
Non-Linear Functions
LOGARITHMIC
FUNCTIONS
Logarithms
Logarithms are used to solve equations where the unknown or variable is
in the power, for example
10 x 1000
We need a new technique to so

ETF2700/ETF9700
MathematicsforBusiness
Week 5 Lecture 1
TOPIC 4
Non-Linear Functions
CUBICS AND OTHER
POLYNOMIALS
CubicFunctions
We now look briefly at cubic functions then at polynomials in general.
Cubic functions have the general form
y = ax3 + bx2 + c

ETF2700/ETF9700
MathematicsforBusiness
Week 7 Lecture 2
TOPIC 5
Financial Mathematics
DEPRECIATION
Depreciation
Depreciation is when something decreases in value over time instead of
increasing as was the case with the interest calculations.
There are two

ETF2700/ETF9700
MathematicsforBusiness
Week 6 Lecture 2
TOPIC 5
Financial Mathematics
SEQUENCES AND
SERIES
Sequences&Series
Sequences
A list of numbers which follow a definite pattern or rule.
e.g. 2, 5, 8, 11, 14, 17,
Referred to as T5.
Series
The sum o

ETF2700/ETF9700
MathematicsforBusiness
Week 7 Lecture 1
TOPIC 5
Financial Mathematics
INTEREST
SimpleInterest
Simple interest is a fixed percentage of the principal, P0, that is paid to an
investor each year.
So the amount of interest earned each year doe

ETF2700/ETF9700
MathematicsforBusiness
Week 9 Lecture 1
TOPIC 6
Differentiation and Applications
OPTIMISATION
TurningPoints
We are able to use derivatives to find the turning points on non-linear
functions.
y
20
15
10
5
x
0
-2
-1
0
1
2
3
4
5
turning point

ETF2700/ETF9700
MathematicsforBusiness
Week 4 Lecture 1
TOPIC 3
Simultaneous Linear
Equations
EQUILIBRIUM
ApplicationofSimultaneousEquations
Goods Market Equilibrium
Market equilibrium occurs when the quantity demanded (Qd) by consumers and
the quantity s

ETF2700/ETF9700
MathematicsforBusiness
Week 3 Lecture 1
TOPIC 2
The Straight Line and
Applications
BUDGET & COST
CONSTRAINTS
BudgetConstraints
We consider a consumer who can only spend their income, M, on two goods
for simplicity, X and Y. The price of go

ETF2700/ETF9700
MathematicsforBusiness
Week 10 Lecture 2
TOPIC 7
Functions of Several Variables
APPLICATIONS OF
PARTIAL
DIFFERENTIATION
ApplicationsofPartialDifferentiation
In this lecture we see 4 different applications for partial differentiation in
the

ETF2700/ETF9700
MathematicsforBusiness
Week 11 Lecture 1
TOPIC 7
Functions of Several Variables
UNCONSTRAINED
OPTIMISATION
OptimisingFunctionsof1Variable
We are by now familiar with the process for optimising functions of just 1
variable, y = f(x).
dy d 2

ETF2700/ETF9700
MathematicsforBusiness
Week 9 Lecture 2
TOPIC 6
Differentiation and Applications
CURVATURE
Curvature
We now concern ourselves with the overall shape of a function. Graphs or
portions of graphs can be classified into two categories dependin

ETF2700/ETF9700
MathematicsforBusiness
Week 1 Lecture 2
TOPIC 1
Mathematical Preliminaries
INEQUALITIES
Inequalities
John Smiths market stall
We saw last lecture that his workers can only make a maximum of 225
tracksuit pants per week. We can therefore ex

ETF2700/ETF9700
MathematicsforBusiness
Week 2 Lecture 2
TOPIC 2
The Straight Line and
Applications
MORE MATHEMATICS
Findingtheslopegiven2points
Recall the equation for the gradient of a line is as follows
rise
run
change in y
change in x
gradient = m
We

ETF2700/ETF9700
MathematicsforBusiness
Week 2 Lecture 1
TOPIC 2
The Straight Line and
Applications
THE STRAIGHT LINE
FeaturesofaStraightLineGraph
The most common function used is the linear function, so we need to explore
all of the various facets of line

ETF2700/ETF9700
MathematicsforBusiness
Week 8 Lecture 2
TOPIC 6
Differentiation and Applications
GRADIENTS AND
DERIVATIVES
GradientsandTangents
The straight line graphs we studied earlier had a characteristic called the
gradient. This tells us how steep t