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chapter 2 59

chapter 2 59 - SECTION 2.8 DERIVATIVES 123(a h)2 1 a2 1 16...

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Unformatted text preview: SECTION 2.8 DERIVATIVES 123 (a+h)2+1 a2+1 16. Jum)_’1Li£\$f(a-|—h]i—f(a)_}1Lig%J (a+h)—: (1—2 : lim (a2+2ah+h2+1)(a—2)— (a2+1)(a,+h—2) h—->0 h(a + h — 2)(a — 2) _ (a3—2a2+2a2h~4ah+ah2—2h2+a—2)~(a3+a2h—2a2+a+h—2) ZIPS?) h(a+h—2)(a—2) th—4ah+ah2—2h2—h 1_ h(a2—4a+ah—2h—1) — . _ 1111—“ 111% h(a+h—2)(a—2) h—>0 h,(a+h—2)(a—2) . a2—4a+ah—2h—1 a2~4ail :llm ——— h—>O (a+h—2)(ai2) — ((1—2)2 17. Use Deﬁnition 2 with f(x — 1/x/a: +2 1 1 f’(a)~1im f(a—»h)—f(a)klim (a+h)+2 \/a+2_ h—>0 h h—’0 h Lem :hm mm _.1m \/£l_+2—\/a+h+2 x/a+2+\/Zl+h+2] ’HO hino hf+h+2ﬁ+2 x/EL+2+\/a+h+2 :hm (a+2)—(a a+h+2) ’HOhx/a+h+2\/a+2(\/a+2+\/a+h+2) =lim #h h—+0hx/a+h+2\/a+2(\/LT+2+\/a—+h+2) —1 gﬂ¢a+h+2ﬁ+2(\/a+2+x/a_+h+2) ~1 1 («11+ 2)2 (2M) : 2(a+2)3/2 3(a+h)+1~ 3a+1 13. f’(a):liin1]f(a;hf)li(a) :53) h 11- (x/ga+3h——1~\/3a~—1)(\/3a+3h+1+\/?Ta+1) 1“ mmwm) (3a+3h —— 1) ~ (3a _ 1) —1i1n Iim haoh(\/3a+3h+1+\/3a+1) h—»0h(\/3a+3h+h1 11+¢zm+ 1) . 3 3 —11m h~o¢3a+3h+1+¢3a+1 2\/§a+1 Note that the answers to Exercises 19 —24 are not unique. 10 _ 19. By Deﬁnition 2. lim W : f’(1),where f(:c) : \$10 11—»0 h and a : 1. . ' ~ 1 10 _ 0r: By Deﬁnltlon 2‘ Ami) (+11% : f’(0). where f(g;) : (1 + 55)“) and a : 0‘ ...
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