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.
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
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
A function F is an “anti-derivative” or an indefinite integral of the function f
Also a machine: you plug in function f(x) and get function F(x)
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)
Solve for x(t) using indefinite integral:
Solve for x(t) using definite integral
Using the fundamental theorem of calculus,
On the other hand, since
Change of variable
Integration by parts
These are Leibniz’ notations: Integral sign as an elongated S from “Summa” and d as a differential (infinitely small increment).
Problem 14: Volume of the truncated pyramid.
The first documented use of calculus?
“Read Euler, read Euler, he is the master of us all”
“Most beautiful formula ever”
“the beam equation”: a cornerstone of mechanical engineering