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alittlebitofcalculus-pdf-january2011-111112001007-phpapp01

# 3 limit step now we can take the limit as h 0 h l imit

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Unformatted text preview: + a n-- 1 ) x→a Substitute for x = a to get na n--1 66 estern wa Wester n w ay : f ’ (a) = ( a) L i m i t ( a + δ x ) n -- a n δ x → 0 ( a + δ x ) -- a f(x) − f(a) (a+ δ x) n -- a n = x−a ( a+ δ x)-- a In the numerator : (a+ δ x) n = an (1+ δ x )n a δ x ) n 1] n n : So: (a + δ x) -- a = a n [ (1 + a -- VERAGE step 1. AVERAGE step : ( from the right) In the denominator : (a+ δ x) -- a = δ x So: : (a+ δ x) n -- a n (a + δ x) -- a = n a [ ( 1 + δ x )n -- 1 ] a δx 2. TENDS TO step : expanding by the Binominal Theorem : a n [ ( 1 + n δ x + n( n -- 1) [ δ x ] 2 + . . . = a a 2! δx terms with higher powers of δ x ) -- 1] The +1 and --1 will cancel out. We are left with : a n [ n δ x + n( n -- 1) [ δ x ] 2 + . . . = a a 2! δx terms with higher powers of δ x ] / Since δ x → 0 but δx = 0 we may simplify the terms with δ x to get : n( n -- 1) δ x nn = a [ -- + 2! [ a2 ] + . . . terms with higher powers of δ x ] a L 3. LIMIT step : now take the LIMIT as δ x → 0 . δ ximit0 δ x = 0, → L imit δ x = 0 and L imit (terms with higher powers of δ x) = 0. δx → 0 δ x → 0 a2 n We are left with an -- = n a n--1 . a nn L imit (a+ δ x) -- a = n a n--1 δ x → 0 (a+ δ x)-- a (from the right) 67 In calculating f ’(a) when we wrote L imit we meant L imit+ , i.e. from the right. right x→a x→a We could have taken L imit , i.e. from the left . l eft x → a-n n (a -- δ x) -- a left ( from the lef t ) l ef ? ( a-- δ x ) -- a i.e. what is δL imit0 (a -- δ x)n -- an ? ( from the lef t ) left l ef x→ -- δ x IMPORTANT: notice the -- δ x in the denominator when approaching the instant a from the left : l eft x → a -- = a -- δ x as δ x → 0 a nd ( a -- δ x) -- a = -- δ x. What is δL imit0 x→ Now, r ather than c hoosing the par ticular instant a , we may w ant to i nstant find the INSTANTANEOUS RATE OF CHANGE of the function f(x) = x n a t any instant x . I n general, if we let x be any instant o ver which the i nstant instant function is WELL- BEHAVED (SINGLE...
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