1 / 21

apod.nasa/apod/ap070819.html

Physics 218 Lecture 2. http://apod.nasa.gov/apod/ap070819.html. 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.

kendra
Download Presentation

apod.nasa/apod/ap070819.html

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Physics 218 Lecture 2 http://apod.nasa.gov/apod/ap070819.html

  2. Overview of Calculus • Derivatives • Indefinite integrals • Definite integrals

  3. 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

  4. 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

  5. Derivatives

  6. Differentiation techniques

  7. Derivatives

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

  9. 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)

  10. Indefinite integral (anti-derivative)

  11. Integrals of elementary functions

  12. Definite integral

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

  14. “Proof” of the fundamental theorem of calculus

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

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

  17. Integration techniques Change of variable Integration by parts

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

  19. Leibniz-Newton calculus priority dispute

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

  21. 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

More Related