Calculus

Discussion

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  ƒ(x) =  lim ƒ(x + Δx) − ƒ(x)
dx Δx → 0 Δx

keywords: derivative, differentiation, anything else?


[slide]

The slope of the line tangent to a curve y = ƒ(x) can be approximated by the slope of a line connecting ƒ(x) to ƒ(x + Δx). The smaller the distance between the points, the better the approximation. The limit of this procedure as Δx approaches zero is called the derivative of the function.

the integral

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

      n  

ƒ(xdx =  lim  ∑   ƒ(xi) Δx
Δx → 0
n → ∞
    i = 1  

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


[slide]

The area under a curve y = ƒ(x) can be approximated by adding rectangles of width Δx and height ƒ(x). The more rectangles (or equivalently, the narrower the rectangles) the better the approximation. The limit of this procedure as Δx 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.

ƒ(x) = 

d  ƒ(x)
dx =  d

ƒ(xdx
dx dx

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

etc.

Calculus was invented simultaneously and independently …

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

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

Symbology

Life, Liberty and the pursuit of Happineſs

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

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