alittlebitofcalculus-pdf-january2011-111112001007-phpapp01

# Imagine a and b located one meter apart on the x axis

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: ference ∆ = C a−x C as x → a = 0.5, be it from the left or from the right. left right ∆ = C a−x C = 0.01, 0.001, 0.0001, 0.00001, . . . After a while it becomes infinitely small and we denote it by δ. We may say : infinitel infinitely 1 δ = /10n as n → ∞ 1 We can see that LIMIT { /10n } = 0 . n→∞ Let us look at another example. Imagine A and B located one meter apart on the x-axis . At each second let B jump 1/2 distance towards A. We may say that B → A. / / A ← 18 14 a 1 /2 B x-axis a + 1 meter The difference ∆ = C A − BC = 1 2 , 1 4 , 1 8 , . . . /// After a while it becomes infinitely small and we denote it by δ. We may say : infinitel infinitely 1 δ = /2n as n → ∞ 1 We can see that LIMIT { /2n } = 0 . n→∞ By definition: An infinitely small q uantity whose LIMIT is zero is infinitely called an INFINITESIMAL. Note that the LIMIT is zero, i.e. the quantity δ TENDS TO zero but T ENDS does not become z er o. δ does not v anish. zer ero vanish. d oes 16 We may now use the infinitesimal δ and say: x → a+ ≡ x = a + δ x → a-- ≡ x = a − δ B→A ≡ B=a+δ To express the concept x coincides with a, we may say : from the right : x = Limit { a + δ } right δ→0 from the left : x = Limit { a − δ } l eft δ→0 Likewise, to express B coincides with A we may say : B = Limit {a+δ} δ→0 As B jumps from 1 to 1 2 to 1 4 to 1 8 . . . to A it skips a lot points in between. / / / points Likewise, as x → a it skips a lot of points in between. This is because δ is a points discrete infinitesimal . We can think of the number line or x-axis as smooth and CONTINUOUS.There are no gaps or breaks. In Calculus we want to study the behavior of a function at each and every point on the x-axis or instant on the TIME AXIS. So we want x to TEND TO a point in a smooth and CONTINUOUS manner. For the infinitesimal that TENDS TO zero in a smooth and CONTINUOUS manner we have the special notation δ x. Geometrically, we can think of δ x as a small CONTINUOUS line segment representing an infinitely small CHANGE in x. The length of this small line segment δ x is always > 0 no matter where we are on the x-axis. δ x gets smaller and smaller. The length of δ x TENDS TO zero but does not become zero. δ x does not vanish. However, the LIMIT of δx is zero. 17 We may let : 1 δx = /10x as x → ∞ 1 or δx = /2x as x → ∞ Now we may express x → a in a continuous manner by : continuous x → a+ ≡ x = a + δx x → a-...
View Full Document

## This note was uploaded on 11/29/2012 for the course PHYSICS 105 taught by Professor Tamerdoğan during the Fall '09 term at Middle East Technical University.

Ask a homework question - tutors are online