Introduction to basic Maths

Vibhor Saxena

vs57@st-andrews.ac.uk

F8, School of Economics and Finance

Office hours: Wednesday 2 – 4 PM

Phone: 2438

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Before we start:

1) This class is targeted for Financial Econometrics students.

2) A large portion of it will focus on Matrix Algebra.

3) If you find it too basic, remember it is not mandatory to attend.

4) I will try to leave 20 – 30 minutes at the end of each session for one-to-one discussion.

5) You need to pick any standard text and practice some questions.

6) I will upload these slides on www.vibhorsaxena.weebly.com

7) We have six hours1) Wednesday 2 – 4 P.M. (today)

2) Friday 11 – 1 P.M. (14th October)

3) Wednesday 2 – 4 P.M. (26th October)

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The first thing in mathematics is counting for which we need numbers.

• Natural number (0, 1, 2,….)

• Integers (…..-2, -1, 0, 1, 2…..)

• Fractions (Are all integers fractions?)

• Rational numbers/Irrational numbers

• Imaginary numbers (real numbers are physically analogous)

• Complex numbers

You can also read about (1st chapter of Alpha C. Chiang).

• The concept of sets

• Relations and Functions

• Types of Functions

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One- variable Differentiation

The derivative of a function tells you the impact (change) on the function when its variable (argument) changes.

Suppose we have just one variable (argument) function f(x). The impact of changes (∆x) of x on the function f(x) is f(x+∆x) – f(x). The rate of change is denoted by:

𝑟𝑎𝑡𝑒 𝑜𝑓 𝑐ℎ𝑎𝑛𝑔𝑒 ≡𝑓 𝑥+∆𝑥 −𝑓 𝑥

∆𝑥.

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One- variable Differentiation

This is slope of the line (or gradient) connecting the points (𝑥, 𝑓 𝑥 ) and ( 𝑥 + ∆𝑥 , 𝑓 𝑥 + ∆𝑥 ). It’s a right triangle!

If the base of the triangle ∆𝑥 gets smaller and smaller, then this gradient becomes the slope of the tangent line of f(x) at (x, f(x)).

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One- variable Differentiation

Now, employing limiting notations:

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Some rules of differentiation

These are important basic rules you need to remember! If you want to become familiar with these, pick a basic textbook.

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Some special functions

The exponential function: 𝒆𝒙𝒐𝒓 𝒆𝒙𝒑 𝒙 . e is an irrational number, approximately equal to 2.7182818.

This definition of e implies that for every one dollar you have now, after continuous compounding for T years, you would then have 𝒆𝒓𝑻. This concept is quite useful in economics and finance. (e.g., A firm has 1000 employees with an average continuous-time quit rate of β; after 2 years, the firm would have 𝟏𝟎𝟎𝟎𝒆−𝟐𝜷.

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Some special functions

The (natural) logarithm function: log(x) or ln(x).

(**It can be useful to revise what log is all about**)

(**log is only possible on positive values: ln x, x ∈ 𝑅+ (𝑥 > 0)**)

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Graphical form of these special functions:

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Some rules of differentiation of special functions

For exponential:

For natural log:

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Optimisation:

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Optimisation:

Note: We are not covering convexity and concavity. However, if you joint tow points on a curve and the line always remain above (below) the curve, then the curve will be strictly convex (strictly concave). Or simply convex (concave).

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Optimisation:

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Optimisation:

A short discussion of optimisation-

Lets focus on 𝑥0and its neighbourhood (domain)-

How about an example now?

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Before we proceed for more formal ways of optimisation, lets focus on a very important tool of maths in economics/finance – Taylor’s expansion/series

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Taylor’s Expansion/Series:

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Taylor’s Expansion/Series:

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Back to Optimisation:

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Multivariate Calculus

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Partial Differentiation:

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Partial Differentiation:

(A nice video – https://www.youtube.com/watch?v=GkB4vW16QHI)

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Total Differentiation:

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Total differentiation:

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Total Differentiation (Chain Rule 1):

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Total Differentiation (Chain Rule 2):

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Total Differentiation (Chain Rule 3):

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Total differentiation (Chain rule 4):

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There is a lot more in differentiation. However, these are the basics from where you can proceed. We will cover integration basics next

week and then proceed to Matrix Algebra!

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