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 few 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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Integration:

Usage in English language – Let’s integrate the society which is divided in different races and religions.

Usage in mathematics – not very different .

A tad bit formal – we need integration to calculate (add up) the area under a function (this may be indefinite or definite)!

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Integration:

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Integration:

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Integration (another way):

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Rules of integration:

As we have seen in the case of differentiation, there are some rules of integration-

Time for some practice!

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Rules of integration:

Integration by substitution (look carefully)-

න𝑓 𝑥 𝑑𝑥 = න𝑔 𝑢 𝑢′ 𝑥 𝑑𝑥 = න𝑔 𝑢 𝑑𝑢

This means, if we can’t apply the rules we learned above to integrate f(x), we need to find g and u such that

𝑓 𝑥 = 𝑔 𝑢 𝑢′ 𝑥 .

Now, example.

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Rules of integration (a little complicated):

Integration by parts-

-In this case, if we cant integrate f(x) by the ways mentioned before, we find two functions by inspection, u(x) and v(x), such that f(x) = u(x) v’(x). Then

න𝑓 𝑥 𝑑𝑥 = න𝑢 𝑑𝑣 = 𝑢𝑣 −න𝑣 𝑑𝑢 .

Let’s see how did we write the equation above:

-if we can find u and v: f(x) = u(x) v’(x), then dv = v’(x)dx and

න𝑓 𝑥 𝑑𝑥 = න𝑢 𝑣′ 𝑥 𝑑𝑥 = න𝑢 𝑑𝑣.

-Also, 𝑢 𝑑𝑣. can be further expressed as 𝑢𝑣 − 𝑣 𝑑𝑢, and we can get the first equation above.

Examples!!

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Definite integrals:

-You integrate a function and find a solution F(x)+c

-Now choose a, b E X (X is a set of real numbers)such that a < b. Form the difference

[F(b) +c] – [F(a)+c] = F(b) – F(a).

-This difference F(b) - F(a) is called the definite integral of f from a to b. The point a is termed the lower limit of integration and the point b, the upper limit of integration.

Notation

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Examples

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Graphical illustration

The absolute value of the definite integral represents the area between f(x) and the x-axis between the points a and b.

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Properties of Definite integrals

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There is a lot more in integrals (which we don’t use frequently). They are also useful in differential equations which we haven’t covered.

Now you understand differentiation and integration basics, and it really depends up to you how you would practice and proceed for a few more

topics.

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