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Newton's Contributions to Calculus

distance over time, finding tangents, and areas of/ under a curve
by

Bielka Liriano

on 13 December 2012

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Transcript of Newton's Contributions to Calculus

Foundation Development of Calculus Infinitesimals His Major contributions Core Newton thought of the variables of a function (x and y) as a relationship of change over time.
Infinitesimals
Power Series
Binomial Theorem Short Biography: Newton's findings were possible not only because he had an astonishing capability to concentrate on a given topic until he thoroughly understood it, but also because of the work around finding tangents, area's under curves, and finding the maximum and minimum values of functions; which were solved through geometric constructions by mathematicians before him. Isaac Newton fluxions = derivative f ' (x)

fluent= formula f(x)

development of binomial theorem:

(0 n) n * (n-1) * (n-2)* (n-3) ...(n-k+1)
k! and his contributions to Calculus Born December 25th, 1642 in Woolsthorp, England.
Attended a local grammar school in Grantham when he was 13, where he not only learned basic arithmetic, but also plane trigonometry and geometric constructions.
By the time he was 21, Newton was studying mathematics independently, and attended Trinity College. Mastered Euclid's work in order to understand Trigonometry,
The Key to Mathematics of William Oughtred (covered the essentials of arithmetic and algebra),
and Wallis' Arithmetica Infornitorum as well as Viete's work (he was known for introducing the use of systematic algebraic notation). Binomial Theorem Newton viewed the variables of a function (x and y) as change over time. Meanwhile Leibniz thought of the variables x and y as ranging over sequences of infinitely close values. He introduced dx, dy, (limit symbol) and the summation symbol . Leibniz thought of a curve as a polygon with infinitely many sides, where at each intersecting point the the value of y could be drawn to the axis. Is a very useful tool when expanding a function such as (a+b)^n How long do you think it would take to expand then combine like terms of (a+b)^5? Pascal's Triangle Despite their different approaches both men developed ways of determining the tangents, area, maximum and minimum values of a curve in an innovative way (more generalized) using algebraic notation. However, Leibniz's computations seem to be a bit more generalized than Newton's. Newton relied on the power series in determining the linear equation of a tangent or finding the area under a curve, he did not provide proofs, and repeatedly used the same functions to demonstrate his work; whereas Leibniz generated a general formula that can be used to find the area and the tangent of a curve for an infinite amount of functions.
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