Mathematical Tidbits and Snacks and BiggerAdventures

Srinivasan Nenmeli-K

My aim in this article is to introduce many methods and approaches ofmathematics at the level of middle-school and high school students.This will also introduce you to techniques you will learn at college levelmathematics later.Further some of the numerical methods used in computerprogramming are explained with simple examples.This will take you to 'numericalalgorithms'--a subject of practical importance.Towards the end, I will give a short intro to Chaos theory and 'FRACTALS'

I sincerely hope that math teachers pick up some of these and enliven their mathclasses with these tid-bits. They can serve as fillers for 10 to 15 minutes.I also add a few historical notes that might interest you.These are marked: 'H N.'You can skip these at first reading.All you need is a paper,pencil and a calculator for these exercises.

I would also encourage students to write simple computer programs for some ofthe computations using the steps here.I hope that at the end of this study, if you are a high school student,you will gain "mathematical maturity" to pursue further work in the college.Above all, share these mathematical tidbits with your friends.! You can study thevarious sections in any order you like.Here we go!------------------------------------------------------------------------------------------------------------- PART 1

How to find the square root of a number in a few steps ?

Suppose you want to find the square root of a number n.

Start with an approximate root,call it x.Then a better approximation is given by this simple equation: x' = (x + n/x)/2 ----------------FORMULA 1Now you plow back x' in the right side and get the next best approximation x''and so on.

This method is called a 'method of iteration" Let us see a simple example: Example 1 To find the square root of 2:

Let us start with x=1 x' = (1 + 2/1)/2 = 3/2 = 1.5

Plow back x=1.5 in the right side: x'' = (1.5 + 2/1.5)/2= (1.5 + 1.333)=2.8333/2=1.4166The answer is already correct to 2 decimal places.Let us repeat or iterate once more: x''' = (1.4166 + 2/1.4166)/2=1.414215573

Your calculator gives :

In two iterations we got an answer correct to 5 decimal places...Well, most probably your calculator uses the same procedure.

This numerical procedure we call a "Numerical Algorithm" -- a fancy nameindeed.!

This formula is called 'Heron's formula'---Heron of Alexandria, most probably thesame math genius who gave us the formula for the area of a triangle.

[H N :Heron could have found this formula by trial and error or empirically, but we can derive thisformula using a method called "Newton-Raphson method" which you will learn in a "Calculus"course.Incidentally, Alexandria was a centre of great learning and math knowledge.The Libraryat Alexandria was a treasure-house of great math manuscripts.This place was later destroyedtwice--once by Romans and later in the 7th century [around 641AD]by muslim invaders whoburnt the library.Now a modern library has been built at the same site in 1999.]

We have introduced an "iterative formula" --there are many such formulae orformulas.The word "iterare" means ,in Latin, " to plow back".We can use this formula for finding square roots of any positive number, not onlyjust integers.

Try this exercise:1 Find the square root of 1943 by Heron's formula2 Find the square root of pi= 3.1416[May be, you can write a small computer program for the formula and run theprograms]

Can we extend this method ,similar to Heron's formula, to find cube-roots? Theanswer is yes.But to get at that formula,you have to use Calculus and Newton's method.

Cube Root and higher roots

The iteration formula for cube root is this: x' = 1/3[ 2x +n/x^2] -------------------------Formula [2]

Example 2 Find the cube root for n= 27 Start with x =2 x' = 1/3[ 4 + 27/4] = 3.58

Iterate now: x'' = 1/3[ 7.1666 + 27/12.82] = 3.0909 The result is close to the actual value of 3.

The iteration formula for 4th root? ---well, you can guess : x' = 1/4[3x+n/x^3]

Try this for n=16. Start with x=1. x' = 1/4[ 3 + 16/1] =19/4 = 4.75 x'' = 1/4[ 14.1 + 16/91.125]=1/4[14.1+0.1756]=3.569 x''' = 1/4[10.707 + 16/45.46]=2.76 x''''= 1/4[ 8.294+16/20.873]=2.265The iteration slowly converges to the actual value of x=2.

In general, to find the nth root of m:

x' = 1/n [ (n-1)x + m/x^(n-1)] -------Formula 3

[May be, you can remember formula 1 and 2 for your future work in school.

Successive Approximation

We have used the iterative method to find square roots. This can also be called amethod of successive approximation.We start with a rough estimate of thesolution and refine it gradually by iteration.

To solve algebraic equations, you have to rewrite the equation in the form :

x= f(x) --------------------FORMULA 4

Then substitute x as an approximate solution or root in the right side;get x-thefirst approximation. Call this x'.Repeat with x =x' by substituting in the right side and get x".

Many times, engineers use this formula to get quick answers.Example 3Let us solve this equation: exp(x) -3x=0 Let us rewrite it: x= f(x) = exp(x)/3

Start with x= 0.5 x' = exp(.5)/3=0.549 x'' = exp(0.549)/3= 0.5775 x"' = exp(0.5775)/3= 0.59385 x"" = exp(0.59385)/3=0.60365 x""' = exp(0.60365)/3 = 0.60959

After you repeat a few times ,we get x= 0.61889This method is also called "Picard's iteration" .

Fixed Points

You may wonder whether successive numbers will converge to some value.This isa big question.Let us get some bearing on this problem.

Graphical method Suppose you want a graphical method for this problem:To solve x=f(x)Write two equations: y= x y1= f(x)y=x is just a straight line passing thro origin....easy to draw.Now draw the function y1= f(x)The point where the two curves intersect , gives the solution where x=f(x)But we start at some value x close to the solution x* , gradually approaching thepoint of intersection if the method yields the root.

This is called "contraction mapping' from function y= f(x) to y=xThe root you get is called "the FIXED POINT"The value you got x= 0.61889 is called the fixed point for the equation: exp(x) - 3x=0Once you reached the fixed point ,you repeat the iteration ,you will get the samenumber: exp(x)=exp(0.61889)=1.856865 3x=1.85667

There is considerable theory behind these fixed points....This topic also leads tothe topic of 'Fractals' which was developed in 1950's.See Part 9 later.

------------------------------------------------------------------------------------------- PART 2

Golden Ratio

Now let us solve a famous,significant quadratic equation:

x^2 = 1 + x --------- equation 1or x^2 -x -1 =0

You can use the clumsy quadratic formula you learnt in middle school.

Taking the positive root , = 1.61803

This magic number is called the Golden Ratio which was used by Greek architectsto design 'Parthenon" :the height/width ratio is this number---aestheticallysatisfying.Renaissance writers called this ratio "The divine proportion".

The negative root gives :

You might have visited this ratio in your geometry book...Check this.

What this ratio is all about? Let us pick two numbers A and B such that

A/B = (A+B)/ASee the diagram below:

<--------A---------><----B-----> |--------------------|---------------| <----------------A+B ----------->

Then cross-multiplying A.A = AB + B.BSuppose we write : A =x , B=1 we get x.x = x +1 ---> same as equation 1 above.Now A = x = 1.61803 You can also see that" x = 1/x-1 , x(x -1) =1 ---> x.x = 1 +x

This ratio is usually denoted by the greek letter "phi" .

The designation of phi comes from the name of the Greek artist 'Phidias' whohad used this ratio frequently in his sculpture.![Draw one rectangle of length A=3.22 in and width B= 2 inches.Draw anotherrectangle with length =5.22 ih and width =3.22.Look at the figures now.Theyare in golden ratios]You can learn from geometry books how to draw rectangleswith the Golden ratio. Check for yourself: x = 1.61803 = 1/0.61803

Properties of phiThis 'phi' has many interesting properties. Several pages can be written on that.What about higher powers of phi?:

It is so easy to generate higher powers of phi.Continuing thus, we get:

Look at the numbers in these higher powers of Phi: 1,1,2,3,5, 8,13 What do you find? The third number is sum of the previous two.The numbers

are the sum of previous two numbers.This sequence is a fascinating one.

Start again with

We can rewrite this:

You can also show: In general:

You may say: "Leave aside the Greek architects and masons.What are the usesof this phi?"To see this let us move on to the next,related topic.

Fibonacci Sequence

Fibonacci [also known as Leonardo of Pisa,(1170-1250)] gave the followingproblem of rabbit breeding in 1202 AD.Suppose you start with a single pair ofrabbits; each pair produces a new pair of rabbits every month, and further eachpair becomes fertile after one month.The rabbits never die.After one month wewill find two pairs.After two months,there would be three pairs;In the nextmonth ,there would be 5 pairs and so on.

The number of pairs follow the sequence: 1,1,2,3,5,8,13,21,34,55,89 and so on.

You see the pattern: just keep adding the previous two numbers.

We will show the close link with the golden ratio "PHI" immediately.... Take the ratio of any two consecutive numbers: R = 21/13= 1.61538

R = 34/21=1.61905

R = 89/55= 1.61818 18 R = 144/89=1.617977

R = 233/144 = 1.618055The ratios go towards phi= 1.61803 ,the same Golden ratio.In mathematical language ,the limit of this sequence is the golden ratio,phi.

The sequence of Fibonacci numbers we will write with this notation:

F(0)= 0 F(1) =1 F(2) =1 F(3)=2 F(4) =3 F(5)=5 and so on.

[ HN:Historically ,the Fibo sequence was given by Indian mathematicians, following some musical notes.Gopala (about 1135) and Hemachandra (about 1150) mentioned Fibo sequence explicitly.Ref: Donald Knuth -The Art of Computer Programming].

Now the higher orders of phi given earlier follow the equation: Thus Knowing the Fibo sequence, we can easily work out any higher power of phi.

Some important identities for Fibo numbers are useful: F(n+1).F(n-1) - F(n)F(n)= (-1)^n

This is due to Johannes Kepler.Another tidbit for Fibo fans.!

Fibo sequence and the Pythogorus triples

You have definitely studied in Algebra and Geometry the Pythagorean formulaand triples: {3,4,5}which means: There are other triples too.How to discover them.?Take the Fibo sequence and choose the alternate numbers after 5: 0,1,1,2,3,[5], 8, [13] , 21 [34], 55, [89] ,144, [233], ....These numbers are the 'hypotenuses' of the triples!

Take 13: you can find the triple: The next triple with hypotenuse of 34: same as {8,15,17}The next triple----------------------of 89: The next triple ----------------------of 233 We found these triples: {3,4,5} , {5,12,13}, {16,30,34}, {39,80,89}, {105,208,233}

If the hypotenuse is , then one of the legs of the right triangle is

Thus 13= The other leg of the right triangle is 5.

Several Egyptologists surmise that the ratio of height:semibase : slant lengthmay follow one of these triples,as known to Egyptian mathematicians.!

Natural Products and Fibo Sequence

Many natural products exhibit the Fibo sequence such as petals offlowers,bee-hives pattern, the seeds in flowers ,the shape of shells and so on.The family trees of bees and cows have l been related to the Fibo sequence.

For example, male honey bees have only one parent, a female.Female honeybees hatch from fertilised eggs; therefore female bees have two parents-one maleand one female.The family tree then follows a Fibo sequence.There are excellent web sites devoted to Fibo sequence and even a dedicatedjournal:'Fibonacci Quarterly.'

Continued "staircase-like" Fraction

This is a fascinating subject by itself----Continued Fraction....Let us see onerelationship to phi.

F =1+ 1/ (1+1/(1+1/(1+ ........... and so on.

Take the first few: F =1+ 1/ (1+ 1/(1+1)= 1+ 1/1+1/1+1/2=1+3/5=1.60 Continue further: F = 1.61803=phi

In general, we can write a continued fraction as follows:

x = a + 1 --- b + 1 --------- c + 1 -------- d + 1 ---------This is represented by the notation: x= [a; b,c,d...]For golden ratio: phi = [1;1,1,1,1,....] Very simple!

Every infinite continuous fraction is an irrational number. It follows that irrationalnumbers can be represented by an infinite continuous fraction.

Can we find a continued 'staircase fraction' for pi?

We give here one such continued fraction ,which can be used for approximatingpi. Let us see a few steps of this staircase CF.

For the next step down in the stair-case , we get:

Let us stop here: The approximate values of pi are: 3, 22/7= 3.1428 333/106=3.141509 355/113=3.14159292

[ H N :The subject of continued fraction was written down by the Hindu mathematician Aryabhattasometime in 5th century.He also made many other math work like the sine tables. He was anoted astronomer.He gave the circumference of the earth as 62832 miles.He gave a value ofpi=3.1416,taking earth's diameter as 20000 miles.!He used the 'arabic numerals' with decimalpoint system.He wrote the 'sutras' or formulas in the classical language of India--'Sanskrit' and used letters for numbers and vowels for decimal points.!He also introduced the word"approximate " ['asanna'] in math work.

His infinite series expansion for square root of 2 is as follows:

accurate to five decimal places.His work "Aryabhattiyam" was translated into Arabic by Al-Khwarizmi sometime around 820 ADand introduced into the Arab world. Thus Indian numerals became Arabic numerals later inEurope.These works spread to Spain which was ruled by the moors,then translated from Arabic to Latin

and then to other European languages.][H N:The text books give credit to John Wallis [1616-1703] for introducing continued fraction.The value of pi --we obtained by continued fraction as 355/113-- was given by Zu Chongzhi in400 AD and remained the accurate one with 7 decimal places for nearly 1000 years.]

--------------------------------------------------------------------------------------------------------------- PART 3

Simple Addition and Multiplication Problems

Carl Frederich Gauss, [1777-1855],one of the three greatest mathematicians ofall times [the other two being Archimedes and Isaac Newton] was posed with thisproblem by his elementary school teacher---hoping to have some respite from hiswork:

Problem: Find the sum of integers from 1 to 100;Gauss was ready with the answer in a minute.How did he do this?Well- he developed an 'algorithm ' or procedure to find the sum, as follows:He used vertical addition ,instead of adding one term at a time: S= 1+2+3+5 +6 ------------+96+97+98+99+100

S =100 +99 +98 +96 -------+5+ 4+ 3+ 2 +1--------------------------------------------------------------------Add vertically: 2S = 101 + 101 +101 -----------101+101+101+101Thus we have ,on the right side, 101 repeated 100 times. 2S= 10100 S= 5050Cool algorithm!There are many such tricks used by mathematicians for finding sums for series ofnumbers.

Triangular Numbers

These are number of units or dots you would need to build a triangle as shownbelow:

* * * * * * * * * * * * * * * * * * * *

Thus we have the numbers 1,3,6,10 and so on.The sequence is generated by the formula: n=k(k+1)/2

Can we find the sum of reciprocals of triangular numbers?: S = 1+ 1/3 + 1/6 + 1/10 + 1/15 +.....

The answer was provided by Leibniz,another great mathematician , by a clevermethod :

Take S/2 and rearrange: S/2= 1/2 + 1/6 + 1/12 + 1/20 +1/30 -------- S/2 = (1-1/2) + (1/2- 1/3) + (1/3 - 1/4) + (1/4 - 1/5) ------- Simplify: S/2 = 1 -1/2 + 1/2 -1/3 +1/3 -------- S/2 =1

or S =2This amazing result , the sum of an infinite series,made Leibniz well known.

[ H N :Leibniz (1646-1716) succeeded in developing Calculus at the same time as Isaac Newton andthere was bitter verbal battle between them for claiming the priority for "Invention ofCalculus."In those days,math results were communicated through letters to friends,professors andleading lights of the day.There were no research journals. Communication was slow and also thetranslation from one language to another and the problem of notation cropped up.So,there weremany reasons for misunderstanding over priority claims and bitter feelings betweenmathematicians in Newton's time.There was also the issue of national pride and rivalry betweenBritain and France ,France and Germany and so on.]

We shall now explore the sums of infinite and finite series by clevermethods,starting with the Geometric Series you have learnt in Algebra orPrecalculus.--------------------------------------------------------------------------------------------------------------Geometric Series

You recall that the sum of numbers :

going to infinte terms is a geometric series:

The ratio of consecutive numbers is 'r'.

If r is between -1 and +1 , that is -1< r < 1, the sum is: S = 1/ (1- r) --------------------------------Equation 1

We can get many useful results with this equation:

If r= 1/2, we get: S = 1+(1/2) + (1/4) + (1/8) + (1/16)------ = 1/(1-(1/2)=2Sounds remarkable!

Now ,if r= 1/3 , S = 1+ (1/3) + (1/9) + (1/27) --------= 1/(1-(1/3) = 3/2

We call this a 'converging series' because the sum leads to a final number.

What if r is greater than 1? : You know that the sum gets very large after a few terms are added.

There is no convergence at all.

We introduce a couple of interesting 'summing' problems here and also Mersennenumbers.

Let us recall a simple,delightful problem from folk-lore ,called St Ivesproblem,sometimes given inrhymes for children, but this is not a child's play:

St IVES PROBLEM

I was going to St Ives, I met a man on the way with seven wives;Each wife hadseven sacks; Each sack had seven cats;each cat had seven kittens; How manywere going to St Ives ?

Using your result from Geometric series:

or simply: S = 1+7(1+7(1+7(1+7)))= 1+7(1+7(1+8))= 1+7(1+7x57)=1+7x400=2801

Nested Multiplication-Horner's rule

This method of finding polynomials and higher powers of x by using repeatedbrackets is called Horner's rule or 'nested multiplication' and this is the widelyused method in computer programs!This procedure or algorithm reduced number of multiplications,compared tofinding each term in a polynomial and adding them up:Take the polynomial :We can find x^2, x^3,x^4 .....and multiply with the coecefficients and add themup....This involves too many multiplications.Here is the efficient 'nested'multiplication or Horner's rule:

Take a small example: For Take sum Next step: sum = previous sum .X + next coefficient of lower degreeRepeat these steps:

We are done with two multiplications and two additions.Check by reversing the order:

For many polynomial work,Horner's rule is handy,as we used for the St Ivesproblem.

Example:Evaluate the polynomial: y = 2 + 3x +4 x^2 + 6x^3 + 7x^4 at x= 2

Start with sum = 7 Next step sum = 7 x + 6 = 20Next step: sum = 20 x + 4 = 44Next step: sum = 44 x +3 = 91Next step: sum = 91 x +2 = 176 ---> y =184

Check: y = 2 + 6 + 16 +48 + 112= 184

---------------------------------------------------------------------------------

A problem similar to St Ives was given by an unknown Egyptian mathematician

,sometime in 1650 BC , found in the Rhind papyrus document:

There were seven houses,each had seven cats, each cat caught seven mices,each mouse had seven spelt and each spelt seven hekat. How many total werethere? The answer is : S = 1+ 7 + 7^2 + 7^3 +7^4 + 7^5 = 28043Using Horner's rule, S0 = 1 S1 = 1.7 +1 = 8 S2 = 8.7 +1 = 58 S3 = 58.7 +1 =407 S4 = 406.7 +1 =28043

[H N:'Rhind Mathematical Papyrus' is a Egyptian mathematical document purchased by AlexanderHenry Rhind in 1858 in Luxor ,Egypt and now kept in the British Museum.It contains 64problems, algebra and even trig included ,and is the main source of Egyptian mathematicsavailable.There is another similar document in Russian museum as well]

Now to Mercenne's numbers:

M1=1; M2=3; M3= 7 and so onMn's are: 1,3,7,15,31,63,127,255......

Soon enough the numbers become huge: The number of digits in Mercenne's number is

In digital computers, with base or radix 2, these numbers are just a string ofones: M1=1, M2=11, M3=111, M4= 1111 and so on.It is easy to write this way and convert to decimal numbers!M5=11111= 15 +16=31 M6=111111=31+2^5=31+32=63 and so on.

[ H NMercenne's numbers have been used to hunt for large prime numbers,using a computerprogram,since 1970's.A major project was undertaken at UCLA [Univ of California,at LosAngeles] and they found a million digit prime with Mercenne's number.If Mn is a prime,then nis a prime too.]

Checker Board ProblemHere is an ancient problem with modern algorithms.Take a checker-board with8x8=64 squares.Place one grain of wheat in one square .Then two grains in thenext square.Then place 4 grains in the third square and so on.How many grainswould be required for all the 64 squares.?

The answer would be : Sum S

This can be easily found with a small computer program you can write foryourself.

Sum of cubes of integers

Here is a math -tidbit:

Find

A theorem due to Nichomachus [about 100 AD] gives this interestingrelationship:

This theorem can be generalized and you can show that:

Example: John has a set of cubical boxes of width: 1in,2in,3in,4in and so on.Thebiggest box is of 8 inches.Find the total volume of all the boxes: S = (1+2+3+4+5+6+7+8)^2= [8(9)/2]^2= 36^2=1296 cubic inches

Harmonic Numbers Harmonic numbers are sums of reciprocals of integers: Hn= 1+1/2 + 1/3 + 1/4 + 1/5 -------+1/n

What could be simpler than this series!But does this converge as we go to further terms.?The answer is definitely "NO",even though we are adding smaller and smaller quantities...This was proved byJohann Bernoulli, one of the two brothers -- a team of mathematicians,havingtheir own quarrels about "who found first this theorem?"

[ H N:They came from Switzerland and studied with Leibniz in Berlin .Jakob Bernoulli,[1654-1705],theelder one,did much work in infinite series and Calculus.

His early work leads us to another topic: the Catenary curve, the curve you find for a cablesuspended between two poles .Galileo thought that this curve is a parabola,but it is not.It isdescribed by 'hyperbolic functions' which you might have studied in Precalc.The great arch in St Louis ,Missouri is designed,upside down, with the same catenerycurve.Jakob took one year to try to solve this catenary problem ,but could not succeed; but theyounger fellow,Johann [1667-1748] did that by 'burning mid-night oil' over a single night-time.Johann was justifiably proud of his achievement.[Jakob Bernoulli's tombstone wasdecorated with a logarithmic spiral ,shown later in this article!] What is the equation for a catenary or hanging cable?

where a is the minimum distance of the cable from the ground and y is the height.You can consider thisfunction as intersection of two curves: y1=(a/2)exp(-x/a) andy2=(a/2)exp(x/a)The two curves meet when y1=y2 at x=0.. Then y(0)=a/2 + a/2=a

There was a brilliant genius studying under Johann Bernoulli who cracked many problems of"infinite sums"--sums of infinite series--and he did have a productive life for 76 years beforeGauss came over the scene.You will meet his name repeatedly in many branches ofmathematics: Leonhard Euler.(1707-1783) ]

Leonhard Euler & Infinite sums

Euler [pronounced "oiler"] tackled the infinite sum for this series:

Find the sum : S = 1+ (1/4) + (1/9) + (1/16) + (1/25) ------

---------------------equation (1)This is a remarkable result because the sum leads to a term involving pi.We canuse such a series to find the value of pi, as we shall see later.It is also strange that while harmonic series does not converge,this series is

converging.[Sorry, I am not giving the proof ,which is available in many books.]

Armed with this result, Euler reaped a rich harvest of many results;here are afew:

-----equation(2)

What about the series:

Euler had a quick answer:

--------equation (3)With these three infinite sums, S,S' and S'', Euler opened up a new area ininfinite sums....

Carl Frederich Gauss and geometric series

C F Gauss, one of the greatest mathematicians ,also a physicist andastronomer of first magnitude,whom we mentioned before, made many newdiscoveries in geometric series .

[H N : Gauss (1777-1855) is credited with least-square method of curve fitting ,at the age of 15 orso, to aid his astronomical observations.Gauss's work on magnetism was significant that themagnetic flux is now measured in "Gauss" units--like we use 'Newton' as the unit for force.Thefundamental theorem of algebra was also proved by Gauss...read about this inalgebra/precalculus book. He also found methods for numerical integration,known asQuadrature methods.Perhaps many people remember Gauss for the bell-shaped curve-'Normaldistirbution 'in statistics--the Gaussian distribution.]

You are familiar with the geometric series:

Substitute -x for x: --------------------------equation 4

Add the two series: 1/(1-x) + 1/(1+x) = 2/(1-x^2) ---------------------------equation 5

Here are other sums: ------------------------------equation 6 ------------------------------equation 7With geometric series alone,we can generate many infinite series.[Can you prove equations 6 and 7 ? ]

Power series / Taylor series and Approximations

Taylor series gives an easy and powerful method for generating power series formany functions.But you must learn basic 'differential calculus' for that.

Suppose you want to find a function f(x) at some value close to x ,say x+a.Knowing the function at x, that is f(x),

--------------equation 8The denominator terms are : factorials: 1!, 2!,3! and so on.The coefficients are the derivatives of f(x).

Taylor series can be treated as an approximation to f(x+a) ;We find f(x) and addsmall terms to it and arrive at f(x+a).

Take the first two terms only: This is indeed an approximation: without taking the limits.

The third term adds a quadratic term as a further approximation: + f''(x).a.a/2and so on.For most approximations, the first three terms may be adequate: -------------------------------equation 9

Taylor series is helpful in generating infinite series for functions.

Exponential function expansion

Take f(x) = exp(x)Now, f'(x) =exp(x) f''(x) = exp(x) and so on.

Using equation 9, we get:exp(x+a) = exp(x) + exp(x).a + exp(x).a.a/2 + exp(x)a.a.a/6 + ---Set x=0

----------equation (10)

This series is useful for many, many applications.

[This series ,expanded about x=0 ,is called MacLaurin Series.]

Suppose you substitute a= 1:You get e = 1 + 1+1/2+1/6+1/24+1/120+--- Keep adding the terms ,you will get e=2.718281828{here is another tidbit: the numbers get repeating .....x.abc2818 2818.... }

Suppose you want to expand :Substitute 2x for 'a' in the above series,equation (10):

------------equation 11

Example: Find the value of exp(.2) taking only the first three terms in eqn(10):

Your calculator would give: exp(.2) = 1.2214027

At this point , you may be wondering why we should use these expressions orapproximations when we can find these values easily with a calculator orcomputer.

The reason is simple: You should know how your calculator or computercomputes these expressions. In fact they use the same or similar expressions intheir software built in;this is how computer algorthms are developed in real life.Suppose you write a gaming software.You have to compute such numbersquickly.You may use these approximate relations.!

Logarithmic function expansion

Log fn is the inverse of exponential fn: y=exp(x) --> lny = x

But this function is a tricky one.Remember ln (1) is taken as zero. Let us expandthe function: ln(1+x) around 1:If you know how to differentiate, it becomes easy:

y= f(x)= ln (1+x) f'(x) = 1/(1+x) ----> f'(0) =1

f''(x) = -1/(1+x)^2 -----> f"(0)= -1/4

f"'(x) = 2/(1+x)^3 ------> f"'(0) = 1/4So the Taylor series for this function, is an alternating series ,with + and - termsas follows:

-------------equation(12)

It was a great exercise to find the ln(x) values and many mathematicians spentyears in constructing this table of log values for several decimal places.You can doall that in a few minutes now ,using a computer and the above equation (11) ofinfinite series.

It is enough if we calculate ln(x) for x>1 and x<2. Other values we cancompute easily with the formula: ln x^n = n lnx Suppose you want to findln(8) -->ln(8)= 3ln(2).

The above series converges slowly;so take many terms in the series.

Example: Find ln(1.5) with three terms of Taylor Series.

ln(1.5) = ln(1+.5) = 0.5 - .25/2 +.125/3 -.0625/4 = 0.40071Your calculator would give: ln(1.5) = 0.405465 -----> an error of 5/400 =1.25%

[H NThis series is attributed to Nicolaus Mercator [1620-1687] who also did navigation lines calledMercator lines which are used in long-distance flights,using Spherical geometry.]

Equation (12) can be modified to obtain other series:

You can also write the series for ln[ (1+x)/(1-x)]= ln(1+x) - ln(1 - x)

It is easy now to expand which you will find useful for error function andGaussian distribution in later work.

Trigonometric Functions

Computing trig fns [sin x ,cosx,and tanx ] had been tough for ancientmathematicians.They needed them for many astronomical [and also astrological]calculations.Even for calculating sun-rise and sun-set times at a given latitude,they needed trig function tables. They developed several empirical formulas andalso rapid means of computation.

[H N :The Hindu mathematicians had their own formulae for trig functions , which some astrologersstill use in India. The early formulae for sine x function was developed by Bhaskara II(1114-1185) in 11th century and filtered through the Arab writers and entered Europe.]

Returning to Taylor series expansion: y=f(x)= sin x f(0) =0 f'(x) = cosx f'(0)=1 f''(x) = -sinx f"(0) =0 f"'(x) = -cos x f"'(0) = -1

Taylor series becomes:

-------------equation 13

Example: Calculate sin (30deg ) using the infinite series.To calculate sin 30 degrees, x= pi/6 = 0.524 radian sin (x) =sin(0.524) = 0.524 - (.524)^3/6+(.524)^5/120 = = 0.524 - .143604/6=0.524 - 0.02393=0.50007The answer is pretty close to sin(30 deg)=0.5We can also find the expansion for cos(x) :

------------equation 14

The case of tan(x) is tricky, since tan 90 deg is infinity.

So we expand arc tan (x): ------------equation 15

Equation 15 is useful to find another infinite series formula for pi:

--------------equation 16

It is easy to use this series for finding the value of pi.It is called Madhava-Gregoryseries.

[H N :This series is often credited to James Gregory,[1638-1675] ---this series is not very usefulsince it converges slowly and you may have to add several hundred terms.This is also called Gregory-Leibniz series ,since Leibniz also found this.Historians tell us thatone Hindu mathematician 'Madhava of Sangamagrama' [Kerala,India] found this seriessometime in 14th century and it is now called Madhava-Leibniz Series.also!][See the historicalnote at the end.]

Srinivasa Ramanujan [1887-1920], an Indian mathematical prodigy who was mostly self-taught, was invited by Prof G H Hardy to Cambridge Univ, UK..Ramanujan colloborated with G HHardy for about 5 years [during World War I years] .He gave many useful results and one wasan infinte series for 'pi' which converges very fast and is used in many computer algorithms.Healso did work on primes,continued fraction and elliptic integrals[See William Dunham for a briefbiography of Ramanujan and Donald Knuth for some of Ramanujan's work.]

Infinte series and integrals

This topic takes us to slightly advanced methods.I just indicate the steps;For many rational expressions, it is difficult to integrate.The clever method is asfollows:

Expand the rational function as an infinite series using Taylor series or geometricseries as given earlier, and then integrate term by term.[A similar method is usedfor solving many differential equations by "series method".]

Example

A simple example is given here: f(x) = 4/(x+2) Find

Expand this function using geometric series and equation (4) given earlier: f(x) = 4 / (2+x)=4/2(1+x/2)= 2 /(1+x/2)

f(x) =2[ 1-(x/2) +(x/2)^2 -(x/2)^3 +---]

= 2 (1-x/2 +x^2/4 -x^3/8---------)

Taking integrals term by term, we get:

Practice Problems:1 Expand using Taylor series and integrate term by term fromx=0 to x=0.5 and compare with table values for cumulative normal [Gaussian]distribution.2 Expand arctan(x) where x= 1/sqrt(3)3 Find the series expansion for cosh(x) and sinh(x)

----------------------------------------------------------------------------------------------------------------

PART 4

Binomial Theorem

This is ,perhaps, the last topic in most algebra text books.[H N:This theorem ,which has many applications ,is again credited to Isaac Newton.(1642-1727) Buthistorians add that one , Shih-Chieh Chu had mentioned binomial coefficients in a 13th centuryChinese treatise.Again,though Newton did invent Binomial theorem, he gave only intuitivederivation.This was partly proved by Euler and the complete,rigorous proof was left to theinimitable Carl Gauss much later.]

Let us see a few binomial expansions:

It is easy to build these expansions,but what about the coefficients,called

'binomial coefficients': [1 ,2,1], [1,3,3,1] [1,4,6,4,1]

You might have studied Pascal's triangle :

There is a still easier way: take the various powers of 11 and write out the digitsfor first four powers::

Here is a problem for 11^5---think about this!

Pascal triangle would give the numbers if we add the two numbers above in thepyramid. 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 PASCAL'S TRIANGLE

The Binomial theorem states:

------------ equation 1The interesting thing about this theorem is that n need not be an integer;it couldbe a fraction or negative too.For instance, find the expansion for: with n=1/2

This works, but we end up with an infinite series.

Newton [ around 1669] was delighted by this.Now he can expand many functionsinto infintie series by his theorem. Newton found that n can be negative too.Thisleads to negative binomial series:

---------------equation 2

For a=1, we get: ---------------- equation 3

Newton used this theorem to find square roots and other roots too:

For instance: to find Expanding by Binomial theorem ,we get:

Binomial theorem can be got by using Taylor series:Let f(x) = (1+x)^n

We get the same result as the binomial theorem.But Brook Taylor[1685-1731]came after Newton and published his series sometime in1715,nearly 40 years after Newton's work.

How to get the value of 'e'

There are many uses of the Bino theorem.Here is one:Expand f(n) = (1 + 1/n)^n by using the Bino thereom and simplify theexpression:

What happens when n is large? that is---- when n tends to infinity?

Well ,the function y tends to 'exponential number' "e"=2.718288.....You might have been introduced to 'e' through "compound interest" formula---ifyour principal amount earns interest every minute and the interest is added tothe principal every minute by your bank.------If the interest is added everysecond to the principal...and so on!!

------------------------------------------------------------------------------------------- PART 5

Polar - Cylindrical Coordinates & DNA Spiral helix & spiral stair cases

You have learnt to locate points and draw lines and curves in the x-y coordinates,invented by Rene Descartes [ a famous mathematician and philosopher ];hencethe name Cartesian system.But there are other ways of locating points anddoing calculations in the X_Y plane.One method is to use polar coordinates. Takea point P(x,y) and join this with the Origin O(0,0).The line OP is called the radius line or polar line.Now form a right triangle bydrawing a perpendicular line PQ from P to the x -axis.Now OPQ forms a righttriangle and the angle ^qop is called . Let OP =r. You can see readily:

We can write the polar coordinates for any point as follows:

For example, point P(2,3) in X-Y coordinates becomes:

, If you are chasing a rabbit, you would be interested in the distance between youand the rabbit and the angle this distance line makes with some reference line orchanges in the angle.Then use polar coordinates.Astronomers often used the polar coordinates to denote the position of stars andplanets.If r remains the same,the distance moved by the planet when the anglechanges over say 18 degrees, will be : Arc distance S = r (theta) with the anglein radians.

Such calculations will be tedious in X_Y [Cartesian] system.]Let us write the equation of a circle in polar coordinates:

No surprise here.

Drawing spirals

Drawing a spiral is easy with polar coordinates....Let the value of r increase withincreasing theta.You will get a spiral line:

Now let this function be a linear relation: r = a + b (theta)Take a specifc example: r= 2 +0.1 (theta)

As the line OP sweeps a circle--counter-clockwise, theta increases from zero to360 degrees.This means the radial length r increases from, say ,2 cm to 2+36=38 cms....So you get a spiral...Well, you can have another revolution,from 360 to720 degrees and so on.This simple spiral is calledArchimedes Spiral. Archimedes made a spiral-like screw to lift water from a canalat a lower level.He definitely needed an equation for that.!

We can also make a widely increasing spiral: The length 'r' keeps increasing rapidly,exponentially, as we sweep a rotation.

This is called a Logarithmic spiral. See the figure below.

Log-Spiral --This was engraved in the tombstone of Jakob Bernoulli.

For an example of Log-spiral in nature,see the Fractal-Food in part 9 later{recurrence relation}

If you are designing a spiral stair case in your building [like the GuggenheimMuseum of Art in New York] you will use these spiral equations.Can you model a spiral galaxy in the sky.!

What about other functions for a polar equation?.We can generate closed curveswith a sine function: If a=b, we get an interesting 'heart-shaped' cardioid.

A polar equation,relating r to theta can describe an elliptical orbit of a planet orasteroid.A Halley's comet may have this polar equation: where d is a constant [some distance] and e is called the eccentricity of theellipse.When e =1, you get a circle.For ellipse,e <1.For this Halley's comet, e=0.97 and d = 1.102; We get :

Cylindrical Coordinates

Having learnt polar coordinates, the next step is to move to three dimensionsand the cylindrical coordinate system.Take the cartesian X-Y-Z system; retain theZ axis as before---say the vertical axis as the Z axis.The horizontal plane of X-Yaxes,convert to polar coordinates as r,theta.So any point in 3-D space (x1,y1,z1) can be expressed as Take the previous example: x=2 y= 3 and add z=4.In cylindrical coordinate system, P[r,theta,z] =

Spiral Helix--DNA structure You want to represent the peptide groups stringed

in a spiral [double helix].Take the structure of the molecule as a cylinder,find thepitch of the spiral,use cylindrical coordinate to represent the position of anygroup.You can also write the equation for horizontal rungs of the spiral joining thepeptide groups.{see the figure above}.One author contends that the diameter ofthis DNA is about 20 angstrom unit ,while the axial length is 34 AU,the ratio of20/34 is close to Golden ratio--should be 21/34---we have not reached perfectionyet.![ Angstrom = 10^(-10)meter]

Parametric Equations We have been writing y = f(x) for our representation of x and yrelationship.Suppose we find a common parameter ,say 't' ,through which we canrepresent x and y.Then we get parametric equations.For instance, x= g(t) andy=h(t) are the two other functions.

Let us take a simple example.We want to represent a rotating wheelmathematically.The wheel has a radius of r.It is logical to use time as thecommon parameter,"t".Let P be any point P (x,y) on the rim of the wheel.We canwrite:

Note that ---- the equation of a circle. is theangular velocity: where f is the frequency [no of revolutions per unittime.]

Practice problems

1 Write the parametric equations for volume and surface area of a right cylinderkeeping height as constant. Write also the equation for the ratio of volume tosurface area.

2 A cycloid is a 2D curve formed by taking a point on the rim of a coin as it rollson its edge on a level surface.The parametric equations are:x= a(t - sin t)y= a(1-cos t)Draw a acycloid for a = 3cm and t =5

--------------------------------------------------------------------------------------------------- PART 6

Strange numbers-united Euler Identity You may find both pi ( and e as strange numbers.Both are irrational:pi=3.14159..... e=2.7218218....Pi arises from geometry, as the ratio of cicumference of a circle to itsdiameter.When mathematicians and astronomers learnt that our earth is nearlyspherical ,they went ahead to find its circumference and its radius.You got the

value of pi.

The exponential [base of the natural logarithm] came from the sequences andlimits and might have been inspired from compound -interest formula.Is there is any connection between the two? Are there expressions which useboth?

We shall have to make a slight diversion into complex numbers.Much of theserealtions you must have studied already.

You know that :

A complex number is composed of two terms, a real part and an imaginary part: Example: x = 2 + 3iThe real part is 2 and the imaginary part is 3.It is like the thief and his shadow under the street lamp!;both have meaning andcan reveal something about the thief.How do we do algebraic operations with this complex number?You have learnt the following: Take the 'complex conjugate' of x:

Then multiply the two and expand:

We take the magnitude of x as the square root of this number:

What is the meaning of the ratio of 3/2?:In a graph paper, locate the point (2,3).Join with the origin.[This plane of x-y axiswill be called "complex plane".What is the the tan of the angle of the line drawnin the graph.?

Euler gave a simple ,but profound relation which connects this complex numberto 'e': What is the meaning of ?Euler expanded this: ----------------------(1)

This is the well-known Euler Formula or Identity.[ Bear with me---We shall derive this a little later in this part.] Now --------------------(2)[As a short form, you can write thus: For the example: x=2+3iThe real part=

The imaginary part=

Note that the Euler equation combines sine and cosine in a single expression. Let |x| = r =1

Now we return to our main adventure: In the Euler equation (1), put

We get

This is a remarkable relation: we have united 'e' and pi with the help of imaginaryquantity 'i.'Note that e is a number (=2.72.....) and pi is also a number (3.14....).They areunited here by 'i'..

Since sin (pi) =0, cos(pi) = -1, we have :

The Euler equation has many other uses.Suppose you want to multiply two complex numbers: z1=2 + 2i z2= 3+4iIt becomes easy after you convert them with exponentials [much as you do forlarge numbers in algebra.]

z1= 2 +2 i =

z2= 3 +4i =

Now

In the same way, you can divide complex numbers easily:

We alredy know how to expand exp(x).

Let us expand into an infinte series, using Part-3 equation (10)given earlier:

You can separate the real and imaginary parts in the expansion:

What do you find in the right side?:Comparing with the trig function expansions given earlier in Part 3:

Thus, we have got the result --"Euler Formula" using exponential functionexpansion.!

One more extension of this leads to another famous formula used in precalcbooks:

as for any exponent!

This important relation is called De Moivre's thereom and has manyapplications.It is easy to see from Euler's formula.!

Example: Using De Moivre theorem,find z^3 where z= 3[cos 30 +isin30]=3cis30

A Math Tidbit We have seen the expansion of exp(x) and the Euler's formula.Here isanother interesting number:What is ?You can expand this with the formula: We get This is called 'Gelfond's constant after Alexsand Gelfond.

[HN:De Moivre was a contemporary of the English and scottish mathematicians of Newton'stime.There is an interesting anedote.He found that he slept 15 minutes longer at certainage.Then he made a sum with arithmetic progression of 15 mins and predicted the day of hisdeath,which was true!.]

PART 7

Interpolating Polynomials

Certain polynomials are used for 'interpolating' between data points ,usuallyfound from experiments.For instance, you measure the solubility of a chemical in water at threetemperatures.You write the function: y=f(x) where y is the solubility and x is thetemperature.We wish to find solubility at some temperature between the measured ones.Weneed to know f(x) ,but we can approximate f(x) by simple polynomials.

[Again this topic is not discussed in many math text books,except as additionalproblems.It is of major interest to all science/engineering students!]

Linear InterpolationThe simplest interpolation is linear interpolation where we join neighbouringpoints by a straight line or linear variation of y with x.

Let three data points be P(xo,y0) , Q(x1,y1) and R(x2,y2).Then linear interpolation between P and Q is this: y= mx+c y1=mx1+cSubtract: y - y1= m(x-x1) [slope-point formula] or y= y1+ m(x-x1) = y1 + [(y1-yo)/(x1 - xo)] (x-x1) ---------------------equation (1)A similar one you can write for the two points Q and R.

Example: Prof Cavendish of Smart University measured the solubility of hisfavourite sugar-substitute sweetener in water at three temperatures, as follows:

Temperature: {x} [deg C] 25 35 45 Solubility {y}[mg/100cc of water] 5.5 6.3 7.5

Linear interpolation: For temp: [25,35] : y = 6.3 + [(6.3-5.5)/10 ] (x-35) =6.3 + 0.08 (x-35) y= 0.08 x + 6.3-2.8=0.08x +3.5Using this , solubility at 28 deg C: y= 0.08x28 + 3.5 = 5.74 mg/100ccFor temperature in the interval[35 ,45] y = 6.3 +0.12 (x-35))

Quadratic or parabolic interpolation:

We take the three data points and find a parabola which passes through these

points ---and then employ the quadratic equation of the parabola forinterpolation.First we derive the formula for the quadratic equation.Note that we have taken the simpler case,when x values are equally spaced:x2-x1=x1-xo=10 in this case.Let the spacing between x values be h. Then write : y = ax^2 + bx + c

Our aim is to find the values of the coeffcients,a,b and c from the given threedata points;To reduce our work, we shift the x values such that xo=-h , x1=0 & x2=h{This trick of shifting the x values we often do in algebra/calculus to reduce ourwork}

The three points are: (-h,yo), (0,y1) and (h,y2)

Now we can solve for a,b and c; c= y1

Add eqn(1) and eqn(3):

Subtract eqn(3) from eqn(1):

Therefore the interpolating quadratic polynomial is this: ---------------- equation (2)

Let us write this polynomial for the solubility data of Prof.Cavendish and find thesolubility at 28deg: y = [(5.5-12.6+7.5)/200] x^2 + [(7.5 -5.5)/20] x + 6.3 = 0.002 x^2 + 0.1 x + 6.3

Now , 28deg C correspond to x=-7; y(28) = 0.002(49) + 0.1(-7) + 6.3 y = 0.098 - 0.7 + 6.3 = 5.698 mg/100cc

Prof Cavendish should be happy with this increased precision in ourinterpolation,compared to linear interpolation which gave y(28) =5.74mg/100cc.Here we have used all the three data points for interpolation.

One can develop interpolating polynomial for four data points using a cubicpolynomial.If we have n+1 data points, we use a polynomial of nth degree.We can also have polynomials for data not equally spaced in x values.These aretreated in books on Numerical methods.

Average values of functions

I mention this topic in passing because this has been treated in another tutorialon integrals. Prof Cavendish is interested in finding the average solubility in the temp range 25to 45 deg C. This is really easy using the interpolating polynomial:Integrate the polynomial

for the temperature range and divide by 20:

between the limits x=10 and x=-10

The average solubility of Prof Cavendish's sweetener is 6.47 mg /100 cc of waterin the temperature range of 25 to 45 deg C.He can use this data to dissolve therequired amount of sweetener in his brew!.Prof Cavendish could have used the simple average of three data values:averagesolubility = 19.3/3=6.43,slightly less than the value of integration method.

[Note: The integration between the limits 25 to 45 degrees ,gives the area underthis curve,between x=25 and x=45.Dividing by the interval 20 gives the averageheight of a rectangle of same area;hence we get the average value of y.] This method of finding average of a function is widely used in science andengineering.!

Numerical Differentiation Prof Cavendish wants to know the rate of change of solubility of his sweetenerwith temperature, from his data points.Can we find this for him?Since we already have the interpolating polynomial from the three data points,wecan differentiate the polynomial to get the derivative: Solubility y = 0.002x^2 + 0.1x + 6.3 where x is thetemperature. dy/dx = 0.004x +0.1This expression is ,of course,an approximate relation based on three values of yand is applicable only in the temperature range of 25 to 45 degrees Centigrade.-------------------------------------------------------------------------------------------

A few practice problems

1 Find the square root of 68 using Heron's iterative formula.2 Find the cube-root of 729 using the iterative formula given .3 Solve the equation: exp(0.1x)-x =0 using the method of successiveapproximation.Start with x =2.4 Prove for Fibo sequence: F(n+1).F(n-1)-F(n)F(n) = (-1)^n

5 Find the infinite series expansion for f(x) = 3/ (2x- 1) using the geometricseries. [around x=0]6 Find the approximate value of sin (18 degrees) using Taylor series.7 Find the approximate value of ln(1.3) using Taylor series to three decimalplaces.

8 Expand using the Taylor series expansion for exp(x) and thenintegrate term by term to find the error integral:

which is important for many applications in probability

9 Find the power series for using the Taylor series expansion given.10 Find the power series expansion for 11 Find the binomial series for (x+a)^8 using Pascal triangle.

12 The data on lead emissions in the USA for the years 1992,94 and 96 are asfollows:

year 1992 1994 1996 Lead emission[million tons] yo=3.80 y1=4.04 y2= 3.87

The year can be indexed as: x=-2 for 1992, x=0 for 1994 and x=2 for 1996.

Find the interpolating 'quadratic' polynomial for the data and find the interpolatedvalue of y for the year 1995.Find also the average value of lead emissions in theyears 1992-96. [Ans: y= 4.04 + 0.0175x - 0.051 x^2 For 1995, y= 4.007]13 Evaluate the stopping distance for a car when it moves at a speed x (miles perhour) and brake is applied: d (feet) = 0.14x.x -4.4x + 58.0 for x=60 mph usingHorner's rule. Ans:134.8 feet

------------------------------------------------------------------------------------- PART 7

Solving Algebraic equations

I shall give a method to solve an algebraic equation with an iterative procedure.

The Bisection Method .[due to Bolzano]

Suppose you want to solve y= f(x) =0. Assume that the function f(x) iscontinuous.Then choose two numbers x1 and x2 such that f(x1) is negative (<0)and f(x2) is positive (>0).Since the function is continuous,it passes through the xaxis [when y=f(x)=0] and the root x* must lie between x1 and x2.Now take theroot x* = (x1+x2)/2

Find f(x*); if f(x*) is negative,the root must be in the interval [x*,x2] If f(x*) ispositive,the root must be in the interval [x1,x*]Find the new root and proceed as before.

Let us take a simple example.

Example: Solve" x.exp(x) -2 =0Ist iteration:Let us try the first interval x:[0.5,1] f(0.5)= 0.5exp(.5)-2 = -1.176 f(1)= 2.72 - 2=0.72Since they are of opposite signs,we take x*= (0.5+1)/2= 0.75

2nd iteration: f(.75)=0.75exp(.75) - 2 = 2.117x0.75-2=-0.4122 <0The root must lie in the interval: x: [0.75,1] Take x* = (0.75+1)/2=0.875

3rd iteration: f(0.875)= 0.875.exp(.875)-2 =2.099-2=0.099 >0The root must lie in the interval:[0.75,0.875] Take x*=0.8125 f(0.8125)= 0.8125exp(0.8125)-2=-0.169

The root must lie in the interval: [0.8125,0.875] Take x* = 0.84375

Note that ,with each iteration,we are halving the interval.If the original intervalwas 1,then after 8 iterations,we will get an interval of (1/2)^8=0.125 or theroot obtained will be < |0.125| of the actual root.You may feel that this method is rather slow and involves lot of function-evaluation;this is a kind of trial and error method.But it is easy to do withcomputers and always works.Hence it is very popular with computer softwarepeople.

Practice Problem Try this problem:

Solve and find the root as the golden ratio by the bisection method.Start withx:[1.5,1.7]

The Secant or Chord methodThis method is similar to the Bisection method but more efficient.To solve y=f(x)=0,to find the root x*, find two numbers x1 and x2,close to theapproximate root.Then join the two points f(x1) and f(x2) by a straight line andextend to the x axis.We are drawing a chord , a straight line with the slope: m= [f(x2)-f(x1)]/(x2-x1) y= mx+c f(x1)=mx1+c

Subtract: y-f(x1)=m(x-x1) [point-slope form]Now to find the root,set y=0 0= f(x1) + m(x-x1) mx= m(x1)-f(x1) x* = x1-f(x1)/m----------------------equation (1) {Note that in this method,f(x1) and f(x2) need not be of opposite signs ,like inBisection method.}

Secant method--see the chord line connecting xn and x(n-1)

Example: Solve xexp(x)-2=0As before ,take x1=0.5 x2=1 f(x1)= 0.5exp(.5)-2= -1.176 f(x2)=0.72Using equation (1): m= (0.72 +1.176)/0.5= 1.896/0.5=3.792 x* = 0.5 - (-1.176)/3.792=0.8101 f(x*)= 0.81exp(.81)-2=0.81(2.248)-2=-0.179

First iteration: x1=0.81 x2= 1 m= ( 0.72 +0.179)/0.19 = 1.995 x*= 0.81+0.179/1.995=0.8997 f(x*)=0.8997exp(.8997)-2=.2122

Second iteration: x1=0.81 x2=0.8997 m= (0.2122+0.179)/(0.0897)= 4.361 x*=0.81+0.179/4.361=0.85104 Compare this with bisection method....

One warning though: Bisection method will always work,but secant method failsin those cases when the slope m is close to zero,because m is in the denominatorand instead of getting close to root ,we may wander away.!

Once you know where the root is by the bisection method,then you can apply thesecant method.These methods are interesting because both use very simple mathematics. They are widely used in computer programming for solving equations.

Practice Problems1 Leonardo of Pisa [Fibonacci again] solved this equation in the year 1225 by amethod close to the secant method.You can now try with secant method: The root is near 1.35. Leonardo got the result: x= 1.368 808 107.

2 Math historians tell us that Isaac Newton solved the following equation usingCalculus [his Newton-Raphson method] in 1669. You can try to solve using thesecant method.Take the initial interval as x1=1.8 and x2=2.2. 3 Solve this problem using the secant method: with intial values:x=0.6 and x=0.8 {Ans [0.865] ----------------------------------------------------------------------------------------------------------------

PART 8Finding the area under a curve

You are asked to find the area under a parabola : y=-x.x +9 between x = -3and x =+3.This becomes easy once you learn the topic of "integration" in Calculus.But nowwe shall explore a simple numerical procedure,used later in Calculus of course.Iam discussing here the midpoint method: I cut up this area into small vertical strips, between close values of x.Then takethe middle value of the function y in each strip.I assume this area is close to thearea of a rectangular strip.Area of the strip a(1) = width. height = width .(y at the midpoint of x)For convenience ,I divide the x interval, [-3,3] into equal width of 1 units:[-3,-2], [-2,-1],[-1,0],[01],[1.2][2,3] along the x axis.I have six widths and so six rectangular strips.What are the y values?: Let us construct a small table for these calculations: X width midvalue of x y value area of strip [-3,-2] -2.5 y(-2.5)=- 6.25 +9 a0= 1(2.75) = 2.75 [-2,-1] -1.5 y(-1.5)= -2.25+9 a1 = 1(6.75) = 6.75 [-1,0] -0.5 y(-0.5) = -.5+9 a2= 1(4.5)= 8.5 [0,1] 0.5 y(0.5) = -0.5 +9 a3= 1(4.5) = 8.5 [1,2] 1.5 y(1.5) = -2.25+9 a4 = 1(6.75) =6.75 [2,3] 2.5 y(2.5) = -6.25 + 9 a5 = 1(2.75)=2.75Total area = a0+a1 +a2 +a3 + a4 +a5= 35.5 units.

Interestingly, Archimedes gave a formula for area under parabolas: Area=(2/3)width x apex height= (2/3)(6)(9)=36 units.The formula given by Archimedes gives the exact value! What we did by addingareas of strips is approximate.How do imporve our accuracy? Well,cut up into 12strips of 0.5 width or 24 strips of 0.25 width and add the areas.That is what wedo with numerical integration;the tedious caluculation is done swiftly with acomputer program! Can you write one program and run it? Try that!!

One mathematician Thomas Simpson [1710 - 1761] gave a better formula tofind area under the curves;It is called the Simpson RuleTake only three points: [-3,0] and [0,3]Find the y values at the three points: y(-3) =-( -3.-3) + 9 = 0 y(0) = 0+9 = 9 y(3) = 0Simpson gave the formula or Rule as follows:

Area = width {[ y(-3) + 4 y(0) + y(3) ]/6} = 6 [ 0 +4.9 +0]/6 = 36units..the same value given by Archimedes formula.Did Archimedes use the sameformula as Simpson...I do not know![Simpson method uses a parabolic or quadratic equation for the y =f(x) and findsthe integration.Since we have taken a parabola itself, Simpson's formula isexact.The term in the { } bracket is the weighted average of y values.But I amstill intrigued how Archimedes arrived at the 2/3(base)(height) formula byempirical methods.! You will learn more of such formulae in calculus /numericalmethods course.]

Practice Problem:1 Find the area under the curve: y=1/(1+x.x) for x=0 to x =1 by takingmidpoint values of width 0.25, using the midpoint method.[Hint: area = 0.25 [ y(.125)+ y(.375) + y(.625)+y(.875)] ]2 Do the same problem using the Simpson's rule with three points along the xaxis 1,0.2,05 and another 0.5,0.75,1.0.What do you find? The area under this curve should be equal to .How close isyour value? This provides yet another method for finding 'pi'.---------------------------------------------------------------------------------------------------------------- PART 9

Recurrence Relations and Difference Equations--Chaos and Fractals

Suppose you are given an Arithmetic series: 2,4,6,8,10......You could write the nth term : starting with That is, the nth term is previous term ,n-1 th term plus 2.This way of writing using the previous terms is called a recurrence relation.

Take the case of compound- interest formula.Suppose your bank is adding theinterest for the previous year to the previous year's principal amount and treatthat amount as the new principal for this year.We can write a recurrencerelation:

where the principal for (t+1) year is related to the principal of 't' th year and r isthe annual rate of interest. We have already seen one important recurrence relation: the Fibonaccisequence:

starting with We are adding the two previous terms , and this is called a second order relation.

In general, a difference equation [ a poor cousin of differential equations you willstudy in a Calculus course] can be wriiten as follows:Second order Difference Equation: where A and B are constants.How to solve this equation?Try the solution:

Substitute:

Divide by :

Now solve the quadratic equation: and find r1 and r2 the roots.[The roots may be real or complex.]The solution isthen : where c and d are constants.

You recall that we did just that for Fibo series: with A =1 andB=1to find x ,the Golden ratio.It all ties up neatly now!

Now let us see several applications of difference equations:

1 You can use them for interest related calculations and annuities as givenearlier;

2 Paul Samuelson, a great economist at MIT and Nobel Laureate for economics[year:1970] developed an equation relating consumer spending and investmentswith a relation:

where n refers to some reference year.Read about this in economics books ,including Samuelson's well-known text...Youmay attempt to get a Nobel prize with such mathematical models.!

3 Difference relations are widely used in population models.You know that Fiboseries is also a population model for rabbits.!When you have a biological system with hosts and parasites, you can writecoupled difference equations : Host population: Parasite population:

You can see how the parasite population increases as the host populationincreases,and the host population decreases as the parasites increase.! This iscalled Nicholson-Bailey model.Such 'prey-predator models' are widely used in biology and wild life conservationprojects.

Practice problem: 1 Develope the sequence : with x1= 46 x2= 1.8 x 46 + 10 = 92.8 x3= 1.8x92.8+10 ....... [You can use nested multiplication you learnt [Horner's rule] for suchproblems.]

2 When Sarah was born,her grandma, Florence, gave a gift of$5000.Amanda,Sarah's mother created a college fund which would yield 10%every year.Amanda decided to add $1000 every year from first birthday ofSarah.How much Sarah will have on her 19th birth day in the college fund?The difference equation is: starting with Solve for n+1 = 19.

Logistic map & Chaos theory

An interesting recurrence relation is this:

For a given value of r, one can find the x values starting with a given value forx.This leads to logistic growth map; "r" is the growth rate; the curve eventuallyleads to a stable value for r <1.

[The logistic function or equation: y =f(x)= rx(1-x) reaches the maximium atx=0.5;it is a parabola.If y=4x(1-x) -->at the maximum,x=0.5,y=1]

For some values of r ,say r=1,the relation will lead to a 'fixed point' . After that,further numbers in the series will be the same:Let xo= 0.2, r=2 xo=0.2 x1==2*0.2*0.8=0.32 x2= 2*0.32*0.68=0.435 x3=2*0.435*0.565=0.4915

x4=2*0.49*0.51=0.5014x5= 2*0.5014*0.4986=0.4999This series,sometimes called time series,' reaches the fixed point of 0.50.It willnot change further.But if you choose r= 3.9,the story is different.

What happens when r is 3?The values of x will oscillate between some values..If rexceeds the magic number 3.57,chaos starts.The values of x keep changingalmost crazy---like a random stuff,but not exactly random. It is chaos!

Snow flakes and Koch Curve

Snowflakes,Koch curve and Fractals

It is easy to construct a snow flake on a paper.Draw an equilateral triangle ofside,say,6 cm length.Divide the sides into three parts.In the central 2cmlength,construct an equilateral triangle.Erase the original central part.Now thelength of each side is increased by a factor of 4/3.Continue doing this in all thethree sides..divide into three parts each side,construct an equilateral triangle inthe middle,erase the central part...Keep iterating this.By this the triangular areawill keep increasing.With each iteration, the number of sides is increasing by a factor (4).With eachiteration,the length of the side of a triangle decreases by 1/3.Therefore it is easyto work out the increase in the area of the snow-flake triangles by a recurrencerelation: Area after nth iteration:

With each iteration,the snow flake grows but at a lesser rate.As n appraochesinfinity,the area An tends to 8/5 of the original area of the triangle=8/5(initialarea)=1.6Ao ----- a neat result!You can create beautiful pictures of Koch curves , as these are called ,due to Kocharound 1904.

It is easy to see that for Koch curve, the area tends to a constant finite valuewith each iteration while the perimeter P increase to infinity: P = number ofsides,N x length of each line l.Now N increase in the ratio of 4/3 and so explodes as (4/3)^n where n is thenumber of iterations.

Study also constructing 'Sierpinski triangle' and carpet,suing a triangular grid.Youcan also construct pentagrams fractals.

Pentagram -Fractal

There is another curve called 'Anti-Koch curve' where the reverse is true.Theperimeter reaches a constant value while the area keeps increasing and tendingto infinity.How to draw this curve? Instead of drawing an equilateral triangle,drawa trough,triangle upside down at each side of the original triangle and iterate.

Fractals have two interesting properties: self-similarity and fractionaldimension.Benoit Mendelbrot, a mathematician[ hailing from Poland]] at IBM,studied self-similarity and coined the word fractal from latin: fractum or part.Hedeveloped the Fractal geometry, to study rough edges and jagged coastlines.Fractal dimension: For Koch curve, the fractal dimension is (log4/log3)=0.6020/0.4771=1.2617We have Fibonacci Fractals too.

Practice Problems1 Construct a Koch curve using an equilateral triangle of 6 cms,on paper withpencil.Try to approximate the area after 4 iterations by enclosing another biggertriangle.[You can use a triangular gird, download form www.math.rice.edu]2 Construct a square fractal using a procedure similar to Koch curve...3 Find an equation for the increase in perimeter of Koch curve with eachiteration.Does this series converge.? Ans:p(n)=3(4/3)^n.4 What would be the equivalent figure for 3-dimensional Koch 'curve' or prismusing a triangular prism.?5 Define fractal dimension.

Fractal Food: Study the shape of the vegetable 'Romansco Broccoli" and relate toproblem 4.This vegetable,a form of cauliflower, is called a Fractal food;the shapeof each cone follows the logarithmic spiral which we have learned earlier.[H NWhat I am writing here is recent history in math ,not 17th or 18th century math!Poincare didsome early work too--but this is 20th century math.Edward Lorenz ,a meterology professor at MIT, found that the logistic equation can lead tochaos.He found that small changes in the inital value led to tremendous randomness in thevalues of x found..He was trying to predict weather;but he founded the chaos theory.He spokeof "butterfly effect"---how a small change in initial value can result in chaos.Then comes thework of May and Feigenbaum and later fractals by Julia sets and Mendelbrot sets ,with beautifulcomputer graphic pictures you have seen,like the Maine coast in USA...The Koch curve givenhere is a simple way.Mandelbrot coined the word: "fractals".Most early workers toyed by putting r = 4 in Logistic map, including the great mathematicianand early computer wiz, John Von Neumann.Try reading about this relation when r =4 or

higher..In 1970,May and Feigenbaum discovered that this logistic map leads to Chaos when r isclose to 4.You can demonstrate that by starting with x= 0.3,r=4 and repeat the recurrencerelation given earlier.Write a computer program! The relation is well-behaved if r is less than 1;that is what the population experts likeVerhulst used.

Read the following two books for starter:James Glieck---Chaos-making a new science---Penguin books{more historical,vey little math}Edward Lorenz---Essence of chaos theory--Washington Univ press.{right balance of math andstory}]

Refer to this website: www.math.rice.edu ----Cynthia Lanius web pages Click here-->Website on Koch curves,fractals ]---------------------------------------------------------------------------------------------------------------- PART 10

Mathematical models -- A few samples

We can study many dynamic systems in real life using linear relations[straightlines] and quadratic equations.These equations resemble recurrence relations,buta bit more complicated.Consider this scenario:Farmers have large yields of corn this year.After six month,when they areharvested and brought to the market,the supply is huge and the price falls.Therefore, the equation of price versus quantity is a straight line with negativeslope: price = P = a - b QBut this involves the time factor.Let t be counted in six-month period. The price depends on quantity grown in the previous time period.This we call a'delay or lag equation".This equation is the demand equation of economists.Now, traders usually increase the price when the demand quantity goes up.This isa sraight line with positive slope: P = c + d QAgain there is a time shift : This is called the supply equation.The two lines will intersect at the equilibrium value: [Q*,P*]

But things are not that simple.We do not always have the equilibrium valueoperating at a given time..You start with some value of Q,Q1 away from Q*.Then you get a value of P1 .Then for that P1,certain quantity Q2 is madeavailable in the market.But for this quantity Q2,there is a price p2 from thedemand equation...These changes keep happening over time intervals,till youreach the equilibrium value...This leads to a powerful model ,called Cobwebmodel.Though mathematically simple,the real-world implications are huge.

Note the the equilibium may be reached after some time period.This point is thefixed point discussed earlier.Next year's yield may be high or low.Another cyclestarts.

Population Models are the most widely used models, in several subjects, not justdemography.Most of them are modified forms of exponential growth-- modified insome way.

Recall the exponential function: -----equation(1)where P(0) is the population at some time t=0 as reference year.'k' is the growthrate.The doubling time, the time for the population to double itself, is given by:

Taking log both sides:

For instance, if k=2%= 0.02, then doubling time

If the growth rate is 2%, the population doubles itself in about 35 years or justone generation!.This did happen for human population in the early 20th centuryand may happen in third world countries and African states.!A thumb rule to remember is this: if the growth rate is k %,the doubling time isnearly 70/k.

Biologists study 'drososphila' or fruit fly for genetic studies---which doubles itspopulation in a few days. Bacteria double their population in 20 mins to onehour.!

Equation (1)implies that the population keeps growing at the same rate forever.It would lead to infinite population.But in real world,the population--humanor fish or tiger or cell phones,gets limited.

Thomas Malthus [1776-1834],an english economist and demographer,told thathuman population would get limited due to famine,pestilence [diseases likeplague which used to wipe out nearly 50% of population of a city in medievaltimes] and wars.He was one of the early demographers.His statement ran asfollows: population increases in geometric ratio or series,while food productiongrows in arithmetic ratios.He inspired Charles Darwin.Mankind ,due to science andtechnology, has almost eliminated large scale famine and terrible diseases.{Agrave thought-->What about nuclear holocaust in future?}

The resource limitation, for instance, food for a fish population in a lake,couldrestrict the population growth.The population grows exponentially but tends toreacha maximum value.

We use the "modified exponential form":

----------equation (2)When t =0, P(t) =0.The population grows and, as t tends to infinity, exp(-kt)goes to zero, P(t)= N,the maximum value or stable value for thepopulation.When P(t) is close to N, we note that the birth rate and death rate ofthe species are the same! N is the population that can be 'sustained' by the lakefor fish population.! This model may not hold for all population scenarios.

Another well-known model is that the growth rate increases first and then startsdecreasing: For this we use the Logistic equation we have seen before in Chaos theory.This iscalled Verhulst model. y= r x (1-x) = rx - rx.xHere the maximum growth rate is reached when x= 1/2, that is half of the maxpopulation predicted.. 'r' is called the growth rate; y varies from 0 to 1.You can also write logistic equation as a quadratic equation: Growth rate = ----------equation (3)

The first term gives the exponential growth, but the second term ,which isnegative, reduces the growth rate and is proportional to the square of thepopulation.By adjusting the value of b,we can get realisitic growth rate for anypopulation .This is the process of modeling.We pick some equations and then adjust theparameters and constants to fit the real-world data or experimental data.

Consider the human population;The growth rate [birth rate - death rate], 'a, inthe above equation was nearly 2% per year, with a doubling time of about 35years.

In most countries, the growth rate has reduced to 1.1% or less.For USA, it isaround 1%.Some countries may have a negative growth too...the populationkeeps decreasing.!Demographers take the average growth rate for World human population as1.4%.[base on a UN report.]Then the doubling time =0.69/0.014= 70/1.4= 50, nearly 50 years.If the trends don't change, we will have a human population of 12.4 billion [fromthe present 6.2 billion in 50 years],that is by 2059.A pictorial representation of growth rates for various countries based on a UNreport is given here..

Possibly, the growth rate will decrease in all countries to 1.1% or less and thehuman population may stabilise around 8 billion.I do not expect that large scalefamine or disesases or wars will wipe out large population,the ""Malthusian devils"will not operate in the next 50 years.!Nuclear holocaust is another problem.

Many of these models are often used by ecologist/environmentalists/wild-lifeconservationists to study the population of animals/insects and so on.Note that adecrease in bee population can fail certain crops totally and reduce cropproduction.Again locusts in large numbers can destroy crops too.There is adelicate balance between species which can be easily altered by pesticides andother chemicals.Mathematical modelers can help a great deal in these studies.

Stopping Distance for cars and trucks.When you apply brakes to a moving vehicle,the vehicle stops after some distance,called stopping distance.This distance has two components: distance movedduring your reaction to apply the brake ,called reaction distance R and the actualbraking distance d which depends on the friction between the tires and roadsurface and any gradient or incline the road may have.

A simple model would be a linear one--straight line relationship between stoppingdistance and speed. Stopping distance S= R + d = vt + kvwhere v is the speed in miles per hour and t is the reaction time ,usually given inseconds and k a constant.

Experiments show that the linear equation is not that good and one uses aquadratic equation relating braking distance with a v.v term.

This stopping distance is of great use for three sets of people: vehicle designers, the traffic cops who investigate the vehicle speed and road conditions duringaccidents ,and highway engineers.[Traffic cops measure the length of scratchmarks on the road to estimate the stopping distance and the speed of thevehicle.]

In one student exercise,taking average values of reaction time t and brakingdistance at several speeds,the following equation was arrived at, with the data:

speed: 32 64 96 (kmph) reaction time: 6 12 18 (secs) braking distance: 6 18 55 (metres)The equation is: S = 0.006 v.v +0.125 v + 2.38 metres

[You can get the coefficients by using the interpolation formula we got earlier:See Part 7.]

What if we neglect the first term: S= 0.125 v +2.38 ? What is the % error for aspeed of 100 kmph.? Find this out!

Vehicle designers use more complex models. One such equation is: S = 1.47 vt - v.v/ ( 30 (f+-G))where v is the speed in mph, time in seconds, f is the coefficient of frictionbetween tires and the road and G is the gradient.Such models are developedbased on extensive testing by auto companies.

In some cases, airplane pilots overshoot the runway and stop the aircraft in aground beyond th e runway.This may happen during poor visibility conditons andalso when the landing speed is high or the runway is covered with snow or wetwith rain.The aircraft engineers again model the stopping distance by a quadraticequation..The pilots add 15% more distance for a wet runway.You can read up onthat.

Epidemic Model -- S-I -R model

We can develop a simplified model for spread and control of epidemics, like therecent SARS epidemic in 2003. This model is due to Kermack and Mac Kendrick(1927).

The population is divided into three groups:S,the susceptible,who may get the disease by contact with infectives and whohave no immunity;I,the infectives who have contracted the disease and may transmit tosusceptibles;R,the recovered or removed, those who have succumbed and died, or who havebeen isolated [quarantined] or those who have developed immunity.Thetransition is shown as follows: S ---> I ----> R

Let us assume that the transition from Susceptible to Infective ,S--> I dependson the contact of S with I and is proportional to the product SI.[This term is oftencalled 'interaction term' in these models.]Take a time interval of one day. "a" and "b" are constants. Changes from oneday to the next is written down:Change in S, the number of susceptible = ----------equation 1Change in I, the number of infectives = equation2 Change in R, the recovered or removed = equation3

We increment the previous values of S,I and R for the next day.

equation 4

Note that the number R,depends on the number infected and b is a constant.Thenumber of infectives I gets increased by those getting infected from S, andreduced by bI,the number removed.The total change is zero:add the three equations,1,2 &3.

How to solve these equations?...[These are difference equations, but you can usedifferential equations approach also, taught in Calculus courses.]

Start with some initial values of S,I and R;S,the overall population without immunity for the disease,say 1 millionpeople.S=1 .Let I ,the infective, be 100 or 10^-4 million.R=0;Take a =0.001 and b= 0.1.[b=0.1 means that 1/10 of infectives are removed--isolated or dead].You have to get the solution with a set of values [S,I,R] for each day:The calculations are easy to perform in a tableau...See the table below.

This is a simple numerical method of solving such equations.It is the same asEuler method for solving differentialequations.{dx/dt = f(x) is approximated to: and here ...

}

We can understand the trend of the epidemic--how it progresses.

Find the ratio of I/S: We are interested in the ratio of I to S,which should notincrease! This ratio will increase with the onset of the epidemic,reach a maximumand then start decreasing! Dividing the first two equations, [1/2] we get an equation that is independent of time.This shows that the change in infectivesis proportional to the ratio of b/a, the two constants.... One powerful method is to decrease b or isolate or quarantine infectives.Then theproduct IS will decrease.[If R=as/b, called basic number,then if R < 1,epidemic does not occur..it is calledendemic--only a few are infected and their number remains constant...If R > 1, we get full-blown epidemic like SARS.]

Now let us return to solve the system of equations using a tabular calculation:Take s= 1 million, a=0.001, b=0.1,I=100 , R=0.Run the cycle of calculations:

Solving the Difference equations -SIR model

Day S I R S'=S+ I'=I+

1 1 10^-4 0 1-0.1 10^-4 + .1-0.0001

2 0.9 0.1 10^-5 0.9-0.0009 0.1+ 0.0009-0.01

3 0.8991 0.09 0.01 0.8991-0.00008 0.09+0.00008-0.009

4

5

0.8983

0.89823

0.081

0.0729

0.009

0.0081

0.8983-0.00007

----------------

0.081+0.00007-0.0081

-----------------

You can continue with the table.Note that the increase in I ,the infectives is quite

high in the first two days and then starts decreasing slowly from day 3 ,for thisset of parameters.This is typical---the epidemic rages fast in the first few days orweeks and then slowly tapers off.You can see the 'alarming figures' which setsthe panic during the onset of any epidemic....The value of I decreasesapporximately by 0.09 million, every day after peak. If you continue, you mayreach a figure of I=10-4 ,after 15 days or so....then the epidemic has died down.

Practice problems:1 Plot I versus # of days using the above table.2 Change the parameters to this set: a=0.0001.Observe the slow onset ofepidemic.3 Change the parameters to : a=0.0001, b=0.2 Observe the variation in I.4 Since the progress of epidemic is sensitive to 'a' and 'b', pick a few pastepidemics and collect the data for these parameters and analyse the trends asobserved.

During SARS epidemic in 2003, isolation was quickly and effectively done and theepidemic died down.During the flu epidemic in 1917,this could not be done and many millions died.Note that change in I infectives,or S = b/a . That is the point when I reaches a maximum ,before I starts todecrease.Let S = 30000,when the I was maximum. Then if b/a =30000.We can nowproceed to solve the equations with a =1,b=30000= 3x10-2 million,S=1

This method of solving the equations can be easily programmed for acomputer...Write a program for this!

The SIR epidemic model has other uses too. Modeling forest fires,marketingstrategies and pest control models are modified form of SIR models.

H NThe population studies, demography, and the epidemic studies have occupiedhundreds of mathematicians and stat experts in many countires.Earlier,Malthusand Vilfredo Pareto did pioneering work---they were part demographers and parteconomists.Verhulst,Gomberg and others did extensive modelling in thesefields..These models have been widely employed in business schools too...Leslie came up with a 'model or theory,using 'Leslie matrices'.He included thereproductive age group in the population,say % of women in the age group 15 to45 years,with three 10 year intervals, and their fertility rates.Note that thefertility rate may change a lot due to social mores,economic pressures andgovernment intervention.In a country with aging population as in Japan,thepopulation growth will automatically decrease.Some governments regulate theimmigration policies based on the growth rate: Canada ,Australia,New Zealand forinstance.

Practice problems

1 A small lake can support a particular fish population of nearly 5000 fishes.If thepopulation initially was 200 at time t=30 days, find the growth equation.Whenthe population would reach a figure of 4500.Hint: 200 = 5000(1-exp(-30k)) Solve for k.Use the equation to find t for 4500.

2 Comparing China and India, find the doubling time if the pop growth rate is 0.6 for china and 1.4 for India.

3 For the population of an island, the growth rate R = aP - bp.p is quite valid. a= 0.005 and b= 0.0001.Find the growth rate for p= 1million and 5 million population.

4 The stopping distance [D] for a truck is given by the following data: Speed: 30 40 50 [mph] D: 20 32 48 [feet.]Fit a quadratic equation using the intrepolation method given earlier.

5 Develop a population model taking into account the age groups in the population,with % ofpopualtion in the age group [15 to 30] and another age group [30 to 45] with different fertilityrates,say 3% and 2% in the equation. [Hint: replace a by weighted avarage of fertility ratetaking % population in the age groups.You can get plenty of data from open sources,especiallyUN /UNICEF reports.]6 Develop an epidemic model with a =0.001, The decrease in S is mainly due deaths in earlyepidemics like the Plague during the medieval renaissance time in Italy.Taking s=0.75 of intialpopulation of say 1 million,find the possible value of b and I =100, S= 10^6

At the height of epidemic, Therefore, b=1000.Now we can solve the equation step by step.,till we reach S = 0.75x10^6.

Suggested Reading and References

1 William Dunham---- Journey through Genius --Penguin Books

2 Donald E Knuth ----- Fundamental Algorithms--Pearson Education

3 Roland Larson et al-- "Calculus" and also "Precalculus" -----Houghton Mifflin

4 Steven Chapra---Applied numerical methods with MATLAB---

5 Victor Katz ---The history of Mathematics--Addison-Wesley

6 'e'-the story of a number--Eli Maor

7 Jane Muir -- of Men and numbers --Dell Publishing8 Edward Lorenz---Essence of Chaos..Washington Univ press

------------------------------------------------------------------------------------------------------------------

Historical Note Madhava of Sangamagrama [a village in Kerala, near Kochi,India] lived in the years 1350 to1425.His school of mathematics discovered many results in mathematical analysis andcalculus.Madhava found the infinite series expansion we have given here ,for sine x and arctanx.He had developed the Taylor series expansion and had discussed the correction term too.Hecalculated the value of 'pi' upto 11 decimal places using his infinte series.He introduced theconcept of limits which is central to Newton-Leibniz calculus,almost 200 years beforeNewton.He is also credited with concepts of convergence and ratio test,usually ascribed toCauchy.He/his students had also used methods similar to secant method,approximationssimilar to Taylor series expansion and Rolle's theorem.He had also worked on continued fractionwhich was known to Hindu mathematicians since Aryabhatta in 5th century.There are somesurmises that the Jesuit missionaries who were located in those parts of Kerala might havetaken these concepts to Europe.There is gradual acceptance of Madhava's works that Gregory series is now called Madhava-Gregory series, and so on.Some have stated that Madhava is the original founder ofMathematical analysis,though Newton and Leibniz developed calculus futher with their notations.The Madhava school flourished for nearly 200 years [14th to 16 th centuries] and many moreoutstanding mathematicians contributed from that school.[See the corresponding web-sites ] --------------------------------------xxxxxxxxxxxxxxxxxxxxxx-------------------------------------------------