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alittlebitofcalculus-pdf-january2011-111112001007-phpapp01

# X and a nd instant instant particular we may now take

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Unformatted text preview: - ≡ x = a − δx B → A ≡ B = A + δx To express the concept x coincides with a , we may say : Limit from the right : x = → { a + δx } right δx 0 Limit from the left : x = → { a − δx } l eft δx 0 Likewise, to express B coincides with A we may say : B = left Limit { A + δx } δx → 0 right x1 − δ x x1 x1 + δ x x axis Geometry: point x1 Analysis: instant x1 Algebra: real number x1 left t1− δ t right t1 t1 + δ t Geometry: point t1 Analysis: instant t1 Algebra: real number t1 18 t axis 6. δ x and a nd INSTANT INSTANT particular We may now take an Analysis view of the par ticular point x1 on the x-axis and define par it as : particular The par ticular point x 1 on the x-axis is where par L imit ( x − δ x) = x = Limit ( x + δ x) 1 δx → 0 1 δx → 0 1 And the general point x on the x-axis is defined by dropping the subscript. general A general point x on the x-axis is where general point L imit ( x− δ x) = x = Limit ( x + δ x) → δx → 0 δx 0 Likewise, we may also define the par ticular instant t1 on the time axis as : particular par The par ticular instant t1 on the time axis is where particular par L imit ( t − δ t) = t = Limit ( t + δ t) 1 δt → 0 1 δt → 0 1 And the general instant t on the time axis is defined by dropping the subscript. general A general instant t on the time axis is where general instant L imit ( t− δ t) = t = Limit (t + δ t) δt → 0 δt → 0 Thus we see the TIME AXIS is identical to the x-axis except for a change in name. GEOMETRY x-axis ANALYSIS ≡ time axis x t Each point x on the x-axis corresponds to an instant t on the TIME AXIS. point instant 19 When we relate this to the set of real numbers R in Algebra we have : ALGEBRA GEOMETRY and ANALYSIS x1 < x2 for x1 , x2 ∈ R ≡ t1 < t2 for t1 , t2 ∈ R ≡ x1 x2 t1 t2 x-axis time axis Now when we say t2 TENDS TO t1 on the TIME AXIS, denoted by t2 → t1 , we let instant t2 go instant , instant , instant , . . . all the way to instant t1 in a instant instant instant instant continuous continu...
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