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Calculus is the language of Physics which is the foundation of Engineering. Without Calculus there would be no electronics, no skyscrapers, no engines, no electricity, no nothing.
Une très vieille (et très belle) calculatrice au musée.
A very old (and very beautiful) calculator at the museum.
Students in Abby Ross' Calculus Class review their work towards the end of spring semester, 2019. Photography by Glenn Minshall.
Working on my calculus. I work in pen because it doesn't smear, like pencil does.
Who makes mistakes, anyway? No need to erase mistakes, if I don't make them.
The equation W=Fxcos(theta) is ok if you have a constant force. But work is really one of those things that requires calculus to understand. If a force varies based on location, that affects how much work was done. So Newton figured out that by finding the area under the curve on a (x,F) graph, you can deal with such non-constant forces. This is called taking the integral of F with respect to x, and it is written mathematically as shown on the bottom left.
Getting areas under curves is hard. That's why Newton, Leibniz, (and apparently Archimedes in ancient Greece 1800 years ago, but we lost his book until recently) invented integral calculus to do it. In AP 1, you won't be able to get the area under the parabola shown at bottom center. But you will be able to get the area under the trapezoid shown at bottom right, simply by using geometry (I'd break it into 3 pieces to do it.) If you're given a non constant force, you'll know because there'll be a graph (or they ask you to plot one.) Get the area under the curve and you've got work. If they tell you its a constant force, feel free to use W=Fxcos(theta). Note that this COMES from the area under a horizontal curve.
Obviously in a calculus based physics class we could get the area under both curves.
Gross photo of a bivalved nephrectomy specimen showing: A large staghorn calculus in renal pelvis with extension into renal calyces. The renal cortex is largely atrophic. Extensive fibrosis involves the perirenal adipose tissue. Jian-Hua Qiao, MD, FCAP, Los Angeles, CA, USA.
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Students in Abby Ross' Calculus Class review their work towards the end of spring semester, 2019. Photography by Glenn Minshall.
I taped all of those pieces of paper to my wall to help me study (as you can see behind me) but maybe I just should have been studying instead of taking pictures...
First published published by the English Language Book Society in association with the English Universities Press Ltd in 1962
This sixth EBLS edition published in 1970
Contas com pedras, antes dos números existiam as pedras.
"Nas diferenças que a natureza estabelece por si mesma, a pedra não é dura nem mole, e junto do magma dos vulcões foi líquida e deixou de o ser. A pedra só é dura e durável em confronto com os seres humanos, que deverão ter visto nela o contrário da fragilidade da carne, que lhes terá dado a ver outras formas de durar que não as orgânicas, mostrando que há outras temporalidades dentro do tempo. (...)"
Extrato de Pedra de José Bragança de Miranda
www.arte-coa.pt/index.php?Language=pt&Page=Saberes&am...
Aqueduto Mosteiro de Pombeiro | Felgueiras
o input foi: sin(10^-3)/10^-3
que é o limite de sen(h)/h com h tendendo a 0.
o wolfram|alpha derivou a parada e fez mais algumas
vale notar que o wolfram|alpha também aceitaria como limites... tente:
lim (sin x)/x as x->0
Who invented Calculus as infinitesimal calculus, is a mathematical discipline focused on limits, functions, derivatives, integrals, and infinite series. Ideas leading up to the notions of function, derivative, and integral were developed throughout the 17th century, but the decisive step was made by Isaac Newton and Gottfried Leibniz. Publication of Newton's main treatises took many years, whereas Leibniz published first (Nova methodus, 1684) and the whole subject was subsequently marred by a priority dispute between the two inventors of calculus.