Calculus
Where did the word “calculus” come from?
(a “calculus” is used for measuring distance)
What is calculus?
(3 minutes)
(3 minutes)
many parts of this article may be somewhat unintelligible at this point. But by the end of the semester it will make total sense to you.
The main thing to understand is that calculus developed over many centuries, but suddenly came together in the mid to late 1600’s when Isaac Newton developed "the science of fluxions" - which was actually the calculus we know today, but proven mostly with geometric shapes. He then published “Principia Mathenatica” which was an enormous breakthrough in understanding the world around us and is basically the “Classical Mechanics” we learn today as Physics. The methods of fluxions (today we would call it calculus) is used throughout Principia Mathematica.
About the same time in Germany, Gottfried Wilhelm Leibniz published his Nova Methodus pro Maximis et Minimis which described very similar concepts as Isaac Newton’s, but in a way that more closely resembles the symbols we use today. Newton started with what we now call differential calculus. Leibniz started with what we now call integral calculus, but they both ended up developing much of what is now called fundamental calculus.
Newton accused Liebniz of plagarizing his discoveries and it led to a long rift between England and continental Europe in mathematics, but most people agree now that each independently developed what we now call calculus.
“BeforeNewtonandLeibniz, the word ‘calculus’ was a general term used to refer to any body of mathematics, but in the following years, "calculus" became a popular term for a field of mathematics based upon their insights.” Ref:
In simple terms, what is calculus?My answer: For a long time you’ve known how to compute average speed; e.g., a car travels 90 miles in an hour and a half; what is its average speed?
But what if you have an equation for the distance something travels, like
d(t) = -16t2 + 100t + 60 (this could be for an object hurled almost straight up at 100 ft/sec from the top of a 60 ft cliff in the presence of gravity (that’s the
–16t2 term) and ignoring air friction). How would you determine how fast it is going after 1 second, 1.4367 seconds, 2 seconds, etc? This all requires being able to compute an instantaneous speed. And a question like this led to the development of differential calculus. /
The following answer was given to the question: how would you describe calculus in simple terms?
ref:
“There came a time in mathematics when people encountered situations where they had to deal with really, really, really small things. Not just small like 0.01; but small as ininfinitesimally small. Think of "the smallest positive number that is still greater than zero" and you'll realize what sort of problems mathematicians began encountering. Soon, this problem became more than just theoretical or abstract. It became very, very real.
For example - velocity. We know that average velocity is the change in position per change in time (i.e., 5 miles per hour). But what about velocityat a specific point in time? What does it mean to be going 5 mphat this moment?
One solution someone came up with was to say "it's the change in position divided by the change in time, where the change in time is an infinitesimally small amount of time". But how would you handle/calculate that?
Another problem came about trying to find the area under a curve. The current accepted solution was to divide the curve into rectangles, and add together the area of the rectangles. However, in order to find theexactarea under the curve, you'd need to divide it into rectangles that were infinitesimally tiny, and, therefore, add up an infinite amount of tiny rectangles -- to something that was finite (area).
Calculus came about as the system of math dedicated to studying these infinitesimally small changes. In fact, I do believe some people describe calculus as "the study of continuous changes".
Ref: answeredJul 20 '10 at 21:19 by Justin L.What if…?
I like the analogy about “what if an object was hurled in the air …”. A lot of human discovery has been based on questions that start with “what if …”.
What if we think all lengths are not the ratios of whole numbers (people once did think they were), and then look at a right triangle with sides of 1 and 1, like this
By the Pythagorean theorem the hypotenuse is the square root of 2. It turns out that Hippasus of Metapontum proved that the square root of 2 could not be the ratio of two integers, and was (supposedly) drowned by his fellow Pythagoreans who believed all numbers and lengths had to be the ratios of integers. But his discovery led to the irrational numbers, and that led to much, much more. /
What if we try to take the square root of a negative number? ( for example: -1 ?) – this question led to the entire theory of complex numbers (we couldn’t model electricity and magnetism very easily without them) and complex numbers also led to much, much more. /
Calculus is based on the questions, “What if we need to find instantaneous rates of change (e.g., speed) instead of average rates of change, and what if we need the exact area of something that isn’t a polygon or circle or other conic section – like the area under some curve like y = x3 - 4x2 + 3x + 2.5 for 0 < x < 3 ? Newton and Leibniz discovered differential and integral calculus, and it led to much, much more. /
In fact, Stephen King wrote a great book on creative writing titled, “On Writing – a Memoir of the Craft”. It meant a lot to me as I was (and still am) learning to write creatively. One of my favorite takeaways from his book is that almost all of his stories started with an idea: what if …? For example, Salem’s Lot started with the idea: “what if vampires moved to town and opened a store?”, or The Stand: “what if a biological weapon accident wiped out most of human kind?”