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Overview of Calculus Derivatives Indefinite integrals Definite integrals
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
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
Applications of derivatives Maxima and minima Differentials area of a ring volume of a spherical shell Taylor’s series
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)
Indefinite integral (anti-derivative)
Integrals of elementary functions
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)
“Proof” of the fundamental theorem of calculus
Example Given: Solve for x(t) using indefinite integral:
Given: Solve for x(t) using definite integral Using the fundamental theorem of calculus, On the other hand, since Therefore, or
Integration techniques Change of variable Integration by parts
Gottfried Leibniz These are Leibniz’ notations: Integral sign as an elongated S from “Summa” and d as a differential (infinitely small increment). 1646-1716
Leibniz-Newton calculus priority dispute
“Moscow Papirus” (~ 1800 BC), 18 feet long Problem 14: Volume of the truncated pyramid. The first documented use of calculus?
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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