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Francois Vieta invented symbolic geometry by combining Greek mathematics and Islamic Algebra.
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Descartes and Fermat invented analytic geometry. Analytic geometry: curves could be represented by equations, or every equation creates a curve.
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Fermat's method: there is only one solution for the maximum instead of two. Claimed this method could be used to find tangent lines, as well. -
Johann Hudde took Fermat's method and created his own to find the maximum and minimum. -
Fermat, Descartes, John Wallis, Isaac Barrow, and many others decided that: a maximum was found by competing the slope of the tangent and asking when it was zero. Idea of derivative came from extrema, tangent, area, limit, continuity, and function by interacting with theses concepts a certain way. -
Sir Isaac Newton invented the idea of Calculus because the math he knew was not sufficient enough to solve the problems he was interested in. (x')
Goltfried Leibnez had the same ideas around the same time. (dy/dx, integralydx)
Took the ideas of extrema, tangent, and areas and narrowed them down to two categories: derivatives and integrals. -
Newton presents "the calculus". Derivatives are found in areas and tangents, therefore they are inverses of each other.
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"indefinitely small quantity o" What is "o"? -
Taylor series were invented to solve differential equations using f(x+h) in terms of f(x) -
Euler could use differential equation to describe the vibrating of a spring. People were surprised that derivatives could have sines and cosines. -
Didn't believe Newton's ideas were good enough anymore, claims that it is all just algebra, including Euler's methods. Thought he proved that all functions had a power series expansion. Believed that all functions were the sum of a Taylor Series and have infinitely many derivatives. (nth order) -
Took Lagrange's method and put it more into a definition of a derivative. (Improved Lagrange's method)