alittlebitofcalculus-pdf-january2011-111112001007-phpapp01

So we may straight away say that fx is not

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Unformatted text preview: of 0 : left Limit x → 0− = L imit δx → 0 x x = 0 − δx f(x) − f(0) L imit f(0 − δx) − f(0) = δx → 0 x−0 (0 − δx) − 0 −(0 − δx) − 0 L imit + δx L imit = δx → 0 = δx → 0 (−1) = −1 − δx − δx Approaching 0 from the right : f(x) = x right Near 0 and to the right of 0 : x = 0 + δx right Limit x → 0+ f(x) − f(0) L imit f (0 + δx) − f(0) = δx → 0 (0 + δx) − 0 x−0 (0 +δx) − 0 L L imit + δx L imit = δximit0 = δx → 0 = δx → 0 (+1) = +1 → + δx +δx left derivative Limit − f(x) − f(0) ≠ right derivative Limit + f(x) − f(0) right x→0 x→0 x−0 x−0 we say : f(x) is not differentiable at x = 0. differentiable 88 [meter s / sec] Example 3: v(t) is the speed of a car weighing one ton that star ts from rest (0 kmph) and accelerates steadily due East (along the x-axis) for 10 secs and then levels off at 90 kmph (25 m/sec). Is v(t) differentiable at t = 10 secs? differentiable v(t) 30 25 m/sec 25 20 15 S P 10 E E5 D v(t) = a . t for t ≤ 10 secs where a is the constant acceleraton. 25 m/sec for t ≥ 10 secs. 0,0) (0,0 ) 5 10 15 TIME [sec] 20 25 Approaching t = 10 secs from the left : v(t) = a.t = 2.5 t meter/sec2 left Near t = 10 secs and to the left of t = 10 secs : left Limit t → 10− = L imit δt → 0 t = 10 − δt v(t) − v(10) L imit {2.5 (10 − δt)} − 25 m/sec = δt → 0 t − 10 (10 − δt) − 10 (25 − 2.5δt) − 25 L imit − 2.5 δt L imit = δt → 0 = δt → 0 (2.5) − δt − δt = 2.5 m/sec2 So the acceleration from the left or left derivative = 2.5 m/sec2 . acceleration left 89 t Approaching t = 10 secs from the right: v(t) = 25 meters/sec, constant. right Near t = 10 secs and to the right of t = 10 secs : right Limit t → 10+ t = 10 + δt v(t) − v(10) L imit 25 − 25 m/sec L imit 0 = δt → 0 = δt → 0 = 0 m/sec2 t − 10 (10 + δt) − 10 δt So the acceleration from the right or right derivative = 0 m/sec2, which is what acceleration right we expect since the speed or v(t) =...
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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.

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