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http://apod.nasa.gov/apod/ap070819.html

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Overview of Calculus Derivatives Indefinite integrals Definite integrals

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Derivative is the rate at which something is changing Velocity: rate at which position changes with time Acceleration: rate at which velocity changes with time Force: rate at which potential energy changes with position

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Derivatives or Function x(t) is a machine: you plug in the value of argument t and it spits out the value of function x(t). Derivative d/dt is another machine: you plug in the function x(t) and it spits out another function V(t) = dx/dt

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Derivatives

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Differentiation techniques

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Derivatives

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Applications of derivatives Maxima and minima Differentials area of a ring volume of a spherical shell Taylor’s series

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Indefinite integral (anti-derivative) A function F is an “anti-derivative” or an indefinite integral of the function f if Also a machine: you plug in function f(x) and get function F(x)

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Indefinite integral (anti-derivative)

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Integrals of elementary functions

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Definite integral

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F is any indefinite integral of f(x) (antiderivative) Indefinite integral is a function, definite integral is a number (unless integration limits are variables) The fundamental theorem of calculus (Leibniz)

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“Proof” of the fundamental theorem of calculus

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Example Given: Solve for x(t) using indefinite integral:

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Given: Solve for x(t) using definite integral Using the fundamental theorem of calculus, On the other hand, since Therefore, or

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Integration techniques Change of variable Integration by parts

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Gottfried Leibniz These are Leibniz’ notations: Integral sign as an elongated S from “Summa” and d as a differential (infinitely small increment). 1646-1716

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Leibniz-Newton calculus priority dispute

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“Moscow Papirus” (~ 1800 BC), 18 feet long Problem 14: Volume of the truncated pyramid. The first documented use of calculus?

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Leonhard Euler 1707-1783 “Read Euler, read Euler, he is the master of us all” Pierre-Simon Laplace f(x), complex numbers, trigonometric and exponential functions, logarithms, power series, calculus of variations, origin of analytic number theory, origin of topology, graph theory, analytical mechanics, … 80 volumes of papers! Integrated Leibniz’ and Newton’s calculus Three of the top five “most beautiful formulas” are Euler’s “Most beautiful formula ever” “the beam equation”: a cornerstone of mechanical engineering

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