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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.

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Presentation Transcript
slide1
Physics 218 Lecture 2

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

slide2
Overview of Calculus
  • Derivatives
  • Indefinite integrals
  • Definite integrals
slide3
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

slide4
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

slide8
Applications of derivatives
  • Maxima and minima
  • Differentials
  • area of a ring
  • volume of a spherical shell
  • Taylor’s series
slide9
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)

slide10
Indefinite integral

(anti-derivative)

slide13
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)

slide15
Example

Given:

Solve for x(t) using indefinite integral:

slide16
Given:

Solve for x(t) using definite integral

Using the fundamental theorem of calculus,

On the other hand, since

Therefore,

or

slide17
Integration techniques

Change of variable

Integration by parts

slide18
Gottfried Leibniz

1646-1716

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

slide20
“Moscow Papirus” (~ 1800 BC), 18 feet long

Problem 14: Volume of the truncated pyramid.

The first documented use of calculus?

slide21
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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