Tag Archives: Calculus

Calculus

by DAVID BROWN | CLEARNFO.com | Jan 03, 2016

Math

Math

Now I have taken many math classes at the University: from linear & abstract algebra to Calculus 1-3, Topology and more but one day while in Advanced Calculus we had been studying a cut theory for weeks, trying to figure out the difference between a 1 and a 2, and it occurred to me how utterly simple and basic all this math stuff was conceptually; and that even a first grader could understand it; and in fact it is my opinion that we should teach Calculus to all our children in elementary school. To put this into context, I’m not saying we should teach them how to solve complex equations, derivatives or integrals. It is a fact that I was a lousy math student in public school since I didn’t have good number sense or a calculator. I made straight ‘Fs’ in math until I got to the University, where I made straight ‘As’. The difference? The higher you get into math, the less it has to do with numbers and the more it has to do with concepts. What I am saying, is that the concepts are so simple and profound that they apply to everything and should be taught to our children at a very early age. In fact I have taught all my kids calculus –without them knowing it—by the first or second grade. Simply put Calculus is magical, philosophical and depends on faith. It solves an ancient unsolvable problem of determining the exact area under a curve. It just says as you make each square smaller and smaller and as the size goes to zero the limit of the equation that describes the curve is exactly equal to this. Kids can get this stuff and see the beauty and usefulness of math.

Albalone shellserveimage
When you walk along the sea shore, you can pick up an Abalone Shell and tell your kids that if you had the equation for this shell, you could tell EXACTLY how many drops of sea water it would hold.

 

"a boy playing on the seashore"

“a boy playing on the seashore”

“I do not know what I may appear to the world; but to myself I seem to have been only like a boy playing on the seashore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me.”

–Isaac Newton (developed Calculus in the 17th century)


The unsung Magic of Calculus

Misc Musings on the surety of science or why I am skeptical of ‘Well-Established’ facts.

by DAVID BROWN | CLEARNFO.com | December 12, 2014

Math-for-Piano-PerformanceNewton’s 3 Laws of Motion were proven science for 100s of years and helped put a man on the moon and are still used today … only one problem: they are incorrect. Newton’s laws of motion, together with his law of gravitation, provide a useful basis for explaining the motion of everyday objects under everyday conditions. However, when applied to extremely high speeds or extremely small objects, Newton’s laws break down.

How about the Heisenberg Uncertainty Principle? It put a big crack down the middle of the well-behaved universe.

How about Zeno’s paradox Achilles and the tortoise? How about the a priori? The fallacy of logic: If A=B and B=C, then A=C, right? But the entire logic is based on an unexamined assumption of the IF which is the a priori which can never be proven.

How about the Precambrian explosion of life forms in the geologic record?

How about the three sets of laws that don’t agree? The big laws (Relativity), the medium laws (Newtonian) and the really small laws (Quantum). They are all correct except when they are not.

In ancient times mathematicians ran into a logical brick wall which no one could overcome. This had to do with the interplay of the linear, logical, discontinuous mind-set of the day –that many of us today still possess– and the reality of infinity. Corollaries to this conundrum would be the asymptote (y – 1/x), Zeno’s paradox or just trying to find the area under a curve for y = x^2 or y equals x squared.

The area of a square or rectangle is easy enough to determine. If the box is 4 ft. by 5 ft. you just multiply 4 X 5 and the area is 20 sq. ft. If however you want to determine the area under a curve defined for example by the equation of y = x^2 (Note: y is also called the function at x or f(x)), you can subdivide the area under the curve into smaller and smaller boxes but you can never determine the exact area under the curve because you can never make the boxes small enough since there are an infinite number of box sizes before you can get them small enough to determine the exact area under the curve.

The Calculus solved this problem by saying that the limit of f(x) is exactly = to the area as the change in x goes to zero. So, we never really get there, we just say that the limit or the point beyond which it can go is limited by the curve.

Mathematics…is man’s way of cutting up and compartmentalizing the world he is able to experience into digestible units and then describing relationships that seem to have some meaningful use. The result of this activity can produce useful understanding and some ability to control / manipulate our environment.

Science…I believe in the empirical method. I also believe that many in our scientific community have let their emotions and political point of view turn their science into a religion. You either believe certain “well-established facts” or you do not get tenure or you are ridiculed or…. These “well-established facts” represent a dogma or a credo similar to any religion. The result is that many place their minds in a box and their ability to take in new information and process this information critically and honestly is therefore necessarily aberrated.

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Additional reading …

“However, accelerating expansion requires a mysterious source of energy in space acting against gravity, dubbed “dark energy.” Calculations show that the energy required is equivalent to 73% of the total mass-energy of the universe! Historians will look back at science today with disbelief and amusement at the ‘science’ of today. Following equally mysterious ‘black holes’ and ‘dark matter,’ if we continue to discover darkness at the present rate we shall soon know nothing!”  A Nobel Prize for the Dark Side

Defying textbook science, study finds new role for proteins