The Physics
Opus in profectus


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the derivative

only straight lines have the characteristic known as slope

instantaneous rate of change, that is, the slope of a line tangent to the curve

d  f(x) = 
x → 0
  f(x + ∆x) − f(x)
dx x

keywords: derivative, differentiation, anything else?

The slope of the line tangent to a curvey = f(x) can be approximated by the slope of a line connectingf(x) tof(x + ∆x). The smaller the distance between the points, the better the approximation. The limit of this procedure asx approaches zero is called the derivative of the function.

the integral

area under the curve (area between curve and horizontal axis)

 f(xdx = 
x → 0
n → ∞
 f(xi) ∆x

keywords: integral, integration, indefinite integral, definite integral, limits of integration, more?

The area under a curvey = f(x) can be approximated by adding rectangles of width x and height f(x). The more rectangles (or equivalently, the narrower the rectangles) the better the approximation. The limit of this procedure asx approaches zero is called the integral of the function.

the fundamental theorem of calculus

Differentiation and integration are opposite procedures. The anti derivative is the integral. Proof of this is best left to the experts.

f(x) = 

d  f(x)

 dx =  d  


dx dx

The necessity of adding a constant when integrating (anti differentiating).


Calculus was invented simultaneously and independently…

newton leibniz
Isaac Newton
(1642–1727) England
(1646–1716) Germany
of Sorbian (Slavic) descent
fluxions, a term that is
not much used anymore
(elongated s) from Latinsumma, sum
d from Latindifferentia, difference
"Method of Fluxions"
published in 1969 (no joke)
Was it ever published by Leibniz?

The wordcalculus (Latin: pebble) becomes calculus (method of calculation) becomes "The Calculus" and then just calculus again.


Life, Liberty and the pursuit of Happineſs

f ƒ ſ s
short f long f integral
long s
(medial s)
short s
(final s)

Why these alternate versions of s and f are necessary is a matter of protracted discussion.

calculus of multiple variables

partial derivatives are good for…

fancy integrals

more fancy integrals