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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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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

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)

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)

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

1646-1716

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

“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