introduction to basic maths vibhor saxena vs57@st-andrews ... · 4) i will try to leave 20 –30...
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Introduction to basic Maths
Vibhor Saxena
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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