https://interestingengineering.com/11-of-the-best-youtube-channels-for-understanding-mathematics

Mathematics demystified

August 13, 2019 – 7:23 am

August 10, 2019 – 10:14 pm

- MPEG 4, audio/video/speech recognition/speaker identification/face recognition/HDTV and mp3 —- all these use logarithms and trigonometry. The special jargon is — Fourier Series, and Fourier Transforms.
- In finance, compound interest is calculated by using a power function; the inverse problem of finding the duration of deposit is calculated using logarithms.
- All (digital) phones are touch phones and they use DTMF (Dual Tone Multi Frequency) standard — implemented using sines and cosines.
- Quadratic equations are used to design/model/develop certain kind of electronic amplifiers.
- Probability theory is used in computer networks, routing of telephone calls, and also in Wall Street — stock market !!
- There are ways to compute the numerical value of the irrational number up to a million digits and these ways are used to test the efficiency and efficacy of supercomputers.
- Quadratic equations are used to study projectile motion (or to put it playfully, suppose we throw a pebble at a certain angle from horizontal ground, (angle less than 90 degrees (which would mean vertically up)) — the projectile is subject only to the force of gravity of the earth — the path or curve or trajectory of the projectile is a parabole, which is characterized by a quadratic equation. This can be easily proved using laws of straight line motion in two dimensional using resolution of vectors.

More later,

Nalin Pithwa.

August 3, 2019 – 11:30 pm

Expertise in dealing with algebraic fractions is necessary especially for integral calculus, which is of course, a hardcore area of IITJEE mains or advanced maths.

Below is a problem set dealing with fractions; the motivation is to develop super-speed and super-fine accuracy:

A) Find the value of the following: (the answer should be in as simple terms as possible, which means, complete factorization will be required):

1)

2)

3)

4) 1

5)

6)

7)

8)

9)

More later,

Nalin Pithwa

June 26, 2019 – 12:02 am

You may have heard of a magic trick that goes like this. Take any number. Add 5. Double the result. Subtract 6. Divide by 2. Subtract 2. Now tell me your answer, and I will tell you what you have started with.

Pick a number and try it.

You can see what is going on if you let x be your original number, and follow the steps to make a formula for the number you end up with.

Have fun !!!

Shared by Nalin Pithwa,

Ref: Calculus and Analytic Geometry by G B Thomas, 9th edition.

June 25, 2019 – 10:59 pm

- Light travels along straight lines. In fact, the shortest distance between any two points is the path taken by a light wave to travel from the initial point to the final point. In other words, it is a straight line. (A slight detour: using this elementary fact, can you prove the triangle inequality?)
- Bodies falling from rest in a planet’s gravitational field do so in a straight line.
- Bodies coasting under their own momentum (like a hockey puck gliding across the ice) do so in a straight line. (Think of Newton’s First Law of Motion).

So we often use the equations of lines (called linear equations) to study such motions.

Many important quantities are related by linear equations. Once we know that a relationship between two variables is linear, we can find it from any two pairs of corresponding values just as we find the equation of a line from the coordinates of two points.

Slope is important because it gives us a way to say how steep something is (roadbeds, roofs, stairs, banking of railway tracks). The notion of a slope also enables us to describe how rapidly things are changing. (To philosophize, everything in the observable universe is changing). For this reason, slope plays an important role in calculus.

More later,

Nalin Pithwa.

PS: Ref: Calculus and Analytic Geometry by G B Thomas and Finney; or any other book on calculus.

PS: I strongly recommend the Thomas and Finney book : You can get it from Amazon India or Flipkart:

or Flipkart:

May 21, 2019 – 11:03 am

https://www.maa.org/press/maa-reviews/101-careers-in-mathematics

Shared by Nalin Pithwa — for spreading awareness in India also about career opportunities in maths/mathematics

March 1, 2019 – 9:35 pm

- Prove that the minimum value of for , is .
- A cylindrical vessel of volume cubic meters, open at the top, is to be manufactured from a sheet of metal. Find the dimensions of the vessel so that the amount of metal used is the least possible.
- Assuming that the petrol burnt in driving a motor boat varies as the cube of its velocity, show that the most economical speed when going against a current of c kmph is kmph.
- Determine the altitude of a cone with the greatest possible volume inscribed in a sphere of radius R.
- Find the sides of a rectangle with the greatest possible perimeter inscribed in a semicircle of radius R.
- Prove that in an ellipse, the distance between the centre and any normal does not exceed the difference between the semi-axes.
- Determine the altitude of a cylinder of the greatest possible volume which can be inscribed in a sphere of radius R.
- Break up the number 8 into two parts such that the sum of their cubes is the least possible. (Use calculus techniques, not intelligent guessing).
- Find the greatest and least possible values of the following functions on the given interval: (i) on . (ii) on (iii) on with and (iv) on (iv) on .
- Prove the following inequalities: (in these subset of questions, you can try to use pure algebraic methods, apart from calculus techniques to derive alternate solutions): (i) for (ii) (iii) for . (iv) for (iv) for .
- Find the interval of monotonicity of the following functions: (i) (ii) (iii) (iv)
- Prove that if , then
- On the graph of the function where , find the point such that the segment of the tangent to the graph of the function at the point intercepted between the point M and the y-axis, is the shortest.
- Prove that for and for any positive a and b, the inequality is valid.
- Given that for all real x, and , prove that for all , and that for all .
- If for all , prove that at most once in .
- Suppose that a function f has a continuous second derivative, , , for all x. Show that for all x.
- Show that has exactly one root in .
- Find a polynomial such that . Prove that there is only one solution.
- Find a function, if possible whose domain is , , for all , , if and , if .
- Suppose that f is a continuous function on its domain and . Prove that f has at least one critical point in .
- A right circular cone is inscribed in a sphere of radius R. Find the dimensions of the cone, if its volume is to be maximum.
- Estimate the change in volume of right circular cylinder of radius R and height h when the radius changes from to and the height does not change.
- For what values of a, m and b does the function: , when ; , when ; and , when satisfy the hypothesis of the Lagrange’s Mean Value Theorem.
- Let f be differentiable for all x, and suppose that , and that on and that on . Show that for all x.
- If b, c and d are constants, for what value of b will the curve have a point of inflection at ?
- Let and Find the critical points of g on
- Find a point P on the curve so that the area of the triangle formed by the tangent at P and the co-ordinate axes is minimum.
- Let Find all possible real values of b such that has the smallest value at .
- The circle cuts the x-axis at P and Q. Another circle with centre Q and variable radius intersects the first circle at R above the x-axis and line segment PQ at x. Find the maximum area of the triangle PQR.
- A straight line L with negative slope passes through the point and cuts the positive co-ordinate axes at points P and Q. Find the absolute minimum value of , as L varies, where O is the origin.
- Determine the points of maxima and minima of the function , with , where is a constant.
- Let be a fixed point, where . A straight line passing through this point cuts the positive direction of the co-ordinate axes at points P and Q. Find the minimum area of the triangle , O being the origin.
- Let . Show that the equations has a unique root in the interval and identify it.
- Show that the following functions have at least one zero in the given interval: (i) , with (ii) with (iii) , with
- Show that all points of the curve at which the tangent is parallel to axis of x lie on a parabola.
- Show that the function f defined by , with has a maximum value with .
- Show that the function f defined by with has a minimum value at which and a maximum at when .
- If for all , then show that for all .
- Prove that , if .

Happy problem solving ! Practice makes man perfect.

Cheers,

Nalin Pithwa.

February 22, 2019 – 1:40 pm

January 15, 2019 – 2:19 am

The only way I can pay an honour to an outstanding generous mathematician, Prof Ali Nesin, is to share information about him and his Mathematics Village is to share information collected from the internet:

http://www.nesinkoyleri.org/eng/

https://www.ams.org/notices/201506/rnoti-p652.pdf

https://en.wikipedia.org/wiki/Nesin_Mathematics_Village

https://interestingengineering.com/nesin-mathematics-village-learn-enjoy-math

http://mathematics-in-europe.eu/?p=1568

http://www.hurriyetdailynews.com/a-brief-introduction-to-turkeys-mathematics-village-70193

I can’t help myself noticing similarities between Prof Anand Kumar of India’s Super 30 http://www.super30.org and Prof Ali Nesin 🙂

Regards,

Nalin Pithwa

January 14, 2019 – 1:33 pm

*Just paying yet another tribute to Sir Michael Atiyah (re-sharing one of the articles I have collected about him):*

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Maths and the bomb Sir Michael Atiyah at 80

The Times

April 21 2009

When, five years ago, he shared the £480,000 Abel Prize, the equivalent of a Nobel prize in the world of mathematics, Sir Michael Atiyah might have listened to his wife’s urgings to put his feet up and settle into a comfortable life. But that would not have been his style. “Some mathematicians retire,” he concedes with a smile. “I don’t think I have.”

This week, Sir Michael’s 80th birthday and a life dedicated to science and political activism is celebrated in a series of events. A three-day conference celebrating his contribution to geometry and physics, at the University of Edinburgh Informatics Forum, ends today, his birthday. Tomorrow and on Friday his Sir Michael’s role in promoting disarmament is recognised with readings and lectures dedicated to exposing the folly of nuclear weapons.

Much has been achieved at an age when contemporaries might have settled for a quiet life. In 1995, as president of the Royal Society and aged 67, Sir Michael made a stinging attack on Britain’s nuclear weapons policy.

Subsequently he accepted the presidency of the influential Pugwash disarmament conferences, which unite scientists in opposition to the arms race.

**He still believes passionately in the cause, which, he says, is more important to the world than maths, “because if we blow ourselves up, there will be no mathematics anyway”. **

Sir Michael discovered his aptitude for mathematics during his boyhood in the Sudan. His Lebanese father was an Oxford graduate and a civil servant, his mother was Scottish and he grew up regarding himself as British, studying at Manchester Grammar School and Cambridge University.

The key professional encounters in his life came in the United States in the 1950s, when he joined the Institute for Advanced Study, at Princeton University, a gathering place for the world’s most brilliant mathematical minds. Here he forged relationships which have endured, and much of his greatest work has come from what he calls the “dialogues of ideas” established there.

His greatest achievement has been the Atiyah-Singer theorem, which secured his fame and prize money, shared with his collaborator, Isadore Singer, of the US. At the time, he said he couldn’t think what to do with his share; the sporty red Lexus parked outside the Informatics building suggests he has since given it more thought.

In simple terms, the theorem provided a kind of analytical bridge which could be shifted between disciplines. “The theorem technique enables you to get to an answer by-passing all the intervening calculations,” he says. The idea “was something where you could calculate numbers of solutions by very indirect methods which applied in a very wide range of situations: geometry, algebra, physics…”

**Maths, he says, is something he plays out in his mind as he walks around his flat and his garden, and he jots things down – “the dull stuff” – only when he has to check something. **

**“Walking helps the physiological process. You have to maintain a very high pitch of concentration when you do mathematics. It’s illumination – shining the mind’s eye on a problem and really seeing through it. **

**“The old clichés about the beauty of maths are true. It has beauty within it, but not all parts are equally beautiful. Beauty in mathematics is the thing that helps you in the search for truth.” **

Some people, he believes, are born with mathematical brains, although they might choose other careers. One former student won the Nobel Prize for Economics, another is the best-paid hedge fund manager in the US. So was Sir Michael never tempted to use his mathematical skill in a wider world? Could he have solved the global financial crisis?

“Economics is a combination of gambling, psychology and who knows what,” he says. “The current crisis? I think people made a bloody mess. You can foretell that the bubble will burst – the question is when. If you gambled on it you might win or lose a lot of money. I just didn’t gamble.”

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Regards,

Nalin Pithwa.