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Dried Beans…We will now ace the most dreaded topic in the math world. CALCULUS. Woo…Well, its basically pretty conceptual & if you follow the ten step that follow, you’ll be in a better position to ace calculus!

Do remember that these are steps and not ways to ace calculus. So you’ll need to go step-by-step. You can move on to the next step only after the execution of the first(mathematicians are programmers).

1. Ace PreCalculus

Your precalculus must be very strong if you want to move on to Calculus.

To strengthen PreCalculus, you’ll need to master:

• Algebra
• Trigonometry
• Geometry

(including equations & inequalities, trigonometric identities, conics, the binomial theorem, sequences & series.)

If you’re planning to take up multivariable calculus as well, you also need to build up concepts of:

• Vectors
• Parametric coordinates & equations
• Matrices

as a part of precalculus itself.

Now,
once you are able to see functions all around you…you’re ready to go!

2. Functions are Calculus

The first course I took up for Calculus was a 6 month course. My instructor took 4 months just to teach functions & their graphs! That was scary, I had a competitive exam & I had to complete the rest of it in 2 months. Surprisingly, we could do it in half the time!

When the functions are with you, Calculus is with you. The rest is just application of functions. But, you need to be very good with them & especially with their graphs which help a lot in giving you an insight to the problem.

You can visualize the problem & what you’re being asked for & how to get it if you take a look at the graph of the functions involved in the problem.

To be a master of functions, know their nature. Their likes, dislikes & hobbies. Okay, technically you need to know their domain(they want these for the x), the range(they give these y’s) & their basic properties(the graph tells it all).

3. Understand the Basic Principles

Yes, its now time to know what you are trying to ace…Calculus. To understand the basic principles of Calculus, you jest need to ask yourself three questions:

What. Who. Why.

…And the answer is the same…

Mr. Newton & Mr. Leibniz gave us:

On your left is what we call the derivative(represented by d/dx or y’ or f’) which gives the slope of the tangent to the function at a point. Say you have the function f(x), then the slope of f(x) at x=a would be f’(a).

On the right is the integral of the function between two points(represented by the two lines without arrow heads) which gives the area under the curve of the function between these two points.

4. Learn to Apply The Basics

One you understand the above basic physical significance of calculus(integral & differential), you should learn ow to use it to solve problems.

In any calculus problem, You’ll be asked to find one of these… m(slope) or A(area) directly or inderectly. Once you know the significance of the integral & derivative, solution comes to you.

5. Understanding The First principles

Lets now take up the derivative. We talked about its physical significance, but how do we compute the derivative?

Well, we use something called the first principle. It directly gives us the derivative of a function & we can plug in different values of x into this derivative to get slopes of tangents at varius points.

And this is what we use to obtan the basic derivatives… You must derive all of the basic derivatives of common functions(the ones you find in any genuine basic derivatives table). And yes, you may also derive the basic rules of differentiation(sum/product/quotient) using the first principles.

You must practice a lot of problems on differnetiation to learn the derivatives of these functions & then extend the to any possible function using the rules of differentiation.

6. Apply those Derivatives

Once you are able to differentiate any function that you come accros, you would want to apply these derivatives in real time. So what can they be used for?

A)You can use these for finding slopes-something you’re already familiar with. The derivative itself is the slope of the tangent at a point. What next? You can then find the equations of tangents & normals using their slopes.

B)Derivatives can be used for finding maxima & minima of functions-derivatives can be used to find extremum by examining the slopes of tangents. Take a look at this curve below, you’ll find that the slope of the tangent vanishes(tangent becomes parallel to the x-axis) at points of minima & maxima.

All the x’s corresponding to these points(slope or derivative=0) are the points of extrema & the corresponding y’s, the maximum & minimum values of the function.

C) The third thing derivatives help us with is determining the nature of a function or curve. We can find the intervals in which the function increases or decreses.

Take a look at the image below. You’ll observe that the slope of the tangent is positive whenever the function increases while the slope is negative when the function decreases. Yeah, I know exactly how you feel!

Now the tangents 1 & 2 have a positive slope(observe that the angle they make with the x-axis < 90 degrees) & the function is increasing.
Also, the tangent 3 has a negative slope(angle < 90 degrees) & the function is decreasing.

Apart from the above, you can also use derivatives to sketch curves, in approximations & finding physical rates of change(which is what differential caluclus is all about).

7. Reverse the Process of Differentiation

Next, we need to reverse the differentiation process to obtain what is called the anti-derivative or indefinite integral. Indefinite, because of lack of the physical dimension to it.

If the derivaive of a function f(x) is F(x), then the integral of F(x) is f(x). Thats all it means. You’ll be able to get a deeper insight if you look at a genuine table of integrals.

Knowing some of the basic integrals, you can use various methods of indefinite integration to find integrals of almost all functions.

Yes…there is a reson why I used any function in case of differnetiation & alomost any function in case of anti-differentiation. The reson is simple. You can differentiate all functions(not considering specific values) but you cannot integrate all functions. There are a few non-integrable functions.

8. Add Sense to the Anti-Derivative

Its now time to go definite! This is easy, just take the anti-derivative of the function and plug in the values of the points which enclose the area.

As you see here, you may find area under a curve or the area between two curves(subtract the two shaded areas in the image).

9. Put Everything Together

Put the two of them together. What do you get? Rates of Change & Areas under curves going hand in hand?

Weird? But thats what calculus is all about. Thats the whole beauty of calculus. We come across a lot of weird & beautiful relation in math & Calculus gives another one.

You might now wish to do some basic differential calculus to see how exactly the two process of differentiation & integration are related to each other. When you mix em’ up together, you’ll get what is called a differential equation.

10. Practice & Keep Smiling

It all just converges to two things…The two things math always wants from you. Smile & Practice. Give it these two, and it will optimize the output for you.

Hope you enjoyed this post! Good Luck. Keep Smiling! Keep Practicing!

Posted via Math model may decrease phantom traffic jams – Science- msnbc.com

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Stuck in a traffic jam? Math comes to the rescue. Everything has an equation for it. Including traffic jams!

A Jamiton, lets call it J, until they find a symbol for it. It would represent something strikingly similar to a detonation wave produced by explosion(read news article for more info).

Lets take the Jam coefficient to be tau. Rest just represent the usual quantities.(v is for velocity & rho for density) Using the variables, we could form a general equation for such a jam!

Lets say it would look something like this:

(does that make sense? caution : it does not represent the actual equation-which is still to come )

Remember, its a phantom jam(has no particular reason for its being!).

Ok. So what do you do with this equation? Simple, optimize the variables to minimize the probability of a phantom jam!

Then, its all for the engineers, however they might wish to do it.

The Atharva Veda, the ‘fourth Veda’ is a sacred Hindu text written by The Aryans. It is usually referred to as ‘The Book Of Spell‘. It deals with Medicine, Warfare & all other Sciences & has a magical essence to it. Its magical character can be seen in its hymns, rituals & spells which can create, heal, preserve or destroy. The Atharva Veda, written in 2nd Millennium BC has a more efficient substitute for everything that can be done today and the things we dream of achieving tomorrow. It can cure diseases in a single prayer, it has charms for securing prosperity, to evert the evil, for obtaining a wife, a husband, & even a charm to give birth to a son, a daughter with specific features which corresponds to ‘In vitro fertilisation’ minus its complications. It also has a charm to shrink or extend time, a concept corresponding to ‘Time Travel’ on which we have seen a number of movies & our physicits are still confused if it is practically posiible.

Apart from these, I would like to introduce something with i can explain with proofs which are the only basis for the formation of mathematical concepts. It is the concept of ‘Vedic Mathematics‘, which again is a part of The Atharva Veda. It is simply a system of mathematics by the Atharvans, which can change our thoughts, beliefs and the overall scope of mathematics. These days, man has become dependant on all kinds of machines for even his basic routine works. We use the calculator for the very basic calculations we need at home, at school, in the market, at our workplace, and we kill the mathematician who lives inside each one of us. Vedic Mathematics consists of methods of mental calculations apart from its other mathematical discipines.

Here is the trick for cubing numbers using vedic mathematics.

You will need to know about bases to learn this method.

Here is the introductory video on bases.

To explain the method, i will take two cases…

Case 1 : The number is > the base

Lets start with the number 104 which has the base 100.
The answer will be divided into three parts(three since we’re cubing).
The answer would come out to be…

So, the answer would be 1124864.

In the first part, we took 4 & multiplied it be 3.
In the second part, we took the square of 4 & multiplied it by 3.
& in the third part, we simply took the cube of 4.

Now, 4 here is the quantity by which the original number(104) exceeds the base(100) i.e. 104-100

And the three in the first & second part of the answer just shows that we are cubing the number. So those 3’s would stay there for every number.

Lets take another example,

I made it 08 in the third part as the base is hundred which means that there must be 2 digits in the second & third part of the answer.

So, the answer would simply be 1061208.

Again, the three’s in the method appear just because we are cubing.
& 2 is just 102(number)-100(base).

Lets try a different base now, lets take 1007 which has the base 1000.
Then, applying the same method, we have;

The second case is when the number we want to cube is less than the base. Try to figure it out yourself. I will be posting it as well…

A lot of people are confused about the divisibility rules.

So i have put them here together…

By 1-I’m confused as well.

By 2-the last digit divisible by two

By 3- the sum of the digits of the number divisible by three

By 4-the last two digits are divisible by four

By 5-the last digit is 5 or 0

By 6-if the number is divisible by 2 and by 3, then its divisible by 6

By 7-i would suggest you to simply perform the division, however, if you have huge numbers-double the last digit of the number and subtract it from the number without its last digit.

Example-for 55212674, subtract 8 from 5521267 . Repeat this procedure until you get a number that  is/is not divisible by seven(& you’re sure of it). Then the divisibility of this number is the same as that of this number.

Lets say you have a number 11324.

1132-8=1124

112-8=104

10-8=2

& since 2 is not divisible by7, 11324 is also not divisible by seven.

By 8-the last three digits are divisible by 8. Don’t ask me what to do for a 2-3 digit number.

By 9-the sum of the digits of the number divisible by nine

By 10-You know it!

By 11-start by making two sums-The first sum is the sum of the first, third, fifth..(odd) digits and the other sum is the sum of the second, fourth..(even) digits. then subtract the sums-if the difference is divisible by 11, then the number also is divisible by 11.

By 12 -check divisibility by 4 & 3.

By 13-very similar to the divisibility by 7 method.This time just quadruple the last digit of the number and add it to the number without its last digit. Repeat this procedure until you get a number that you know for sure is or is not divisible by 13. ..divisibility is same for this number and the original number.

By 14- check divisibility by 7 and by 2.

By 15- check divisibility by 5 and by 3

By 16-last four digits are divisible by 16

By 17-similar to 7 & 13, this time the number-Penta it! follow the same procedure after that.

By 18- check divisibility by 9 and by 2.

By 19-similar to 7,13,17, guess what now(?)

By 25-last two digits divisible by 25

for others you need to check divisibility using their factors…

Music, Meditation, Martial Arts…All converge into Mathematics. The 4 M’s, All correspond to the purest form of art & thought. Mathematics is Music, which leads us to a state of self-realization, the meditative state.

The Martial Arts, more than defense techniques, are very much related to the culture of the East. In the Vedic Age in India, What some people would achieve by long & deep meditation, others would attain through Indian classical music.

As a matter of fact, all mathematicians understand the language of music. When we read about the lives of some of the greatest mathematicians, we find that most of then played a musical instrument!

Every musical note corresponds to a particular physical part of the human body & vibrates each atom of the body at frequencies required to attain the meditative state…

The statement single handedly highlights the beautiful relation between the various arts.

Every student interested in mathematics should take up some form of music-any instrument & some martial art. This would not only develop the brain, but would also improve your math skills. Try it out! Start learning something new today!

As seen at :: Detrix

Vedic Mathematics provides us with excellent methods of mental mathematics & fast calculations. Its methods allow us to perform our daily calculations faster than a calculator. Its methods also help in increasing concentration & the overall intelligence of the individual. The methods are found in the ancient Indian texts called the ‘Vedas’. Vedic Mathematics changes the complete scope of mathematics adding a new dimension to it. Moreover, it simply makes Mathematics interesting & fun.

The methods of Vedic Mathematics are described as ’sutras’ (meaning basic formulas) and it consists of 16 sutras in all. These sutras give rise to the fastest possible mathematics, the Vedic Mathematics.

An online session/class on Vedic Mathematics is being held by Dev Von De at WizIQ on Oct 31 2009.

In this online session, the presenter will demonstrate & explain the methods for:

-Basic calculation techniques
-All about Bases
-Multiplication Using Bases
-Squaring numbers Using Bases
-General Squaring(Square ANY number)
-General Multiplication(Multiply ANY two numbers)

The session will enable you to multiply any two numbers in a single step, no matter how long they may be! It will allow you to square any number. You will be able to solve equations that would otherwise take 10-20 steps in a SINGLE step.
Lets see what else you’ll be able to do:

1. The products 875 X 991, 203 X 566, 9991 X 9989 in a single step.
2. The squares (999998)², (693)², (588)², or any number for that matter in a single step.

And More…

You can view the recording on the introductory using the link below: An Introduction To Vedic Mathematics

For some tutorials on basic mental Mathematics, visit Dev Von De’s blog at PentaMath.com

You may also find the following article by Dev on Vedic Mathematics useful:
The Vedic System Of Mathematics

To join the session, visit the following link::

http://www.wiziq.com/online-class/188122-The-Secret-Techniques-Of-Mental-mathematics

The Gamma function is one of the “special” functions in math… special due to its significance in analysis.

Lets start by defining the gamma function. Now, if you know the factorial, you know the gamma function. The gamma function is simply a generalization of our good old factorial.

Firstly, this is how we can define it mathematically…

(n being positive)

Now, how is it related to the factorial function…Lets have a look…

We integrate the above definition by parts & it gives us…

When we plug in the upper limit(infinity), the resulting form calls for the L Hopital’s rule & the value of the limit comes out to be 0.

Which gives us…

Now, take up the RHS…& apply the above definition furthur…
(assuming that n is a positive integer)

So, we have…

Hence, we see here that the factorial is nothing but a particular case(when n is a positive integer) of the generalized gamma function which can also take up other values for n.

Ever tried solving cubic equations mentally?
Hmmm…Here is a Vedic Sutra(method of vedic math) I’ve been using to solve cubic equations in one single step.

Well, not all cubics can be solved using this method. But there are special type of cubics which we will call “semming cubics” which can be solved using this method.

It gets that name simply because these cubics always turn out to be first degree equations on some solving.

So… How would you solve the following equation?

Well, it does look like a cubic. However, someone who knows this method would just say… “The answer is 6″.

And yeah, it very much is. The trick here is to observe that 6(on the R.H.S) is the mean of 3,9(on the L.H.S)

So, in any such case, you just take the mean number of the numbers on the L.H.S.

Or simply, If you have…

Such that c is the mean of a,b. Then, the answer would just be c!

You can almost solve all equations using vedic mathematics…this is just one of them.