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Chapter 3 7

# Chapter 3 7 - SECTION 3.1 DERIVATIVES OF POLYNOMIALS AND...

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Unformatted text preview: SECTION 3.1 DERIVATIVES OF POLYNOMIALS AND EXPONENTIAL FUNCTIONS 55. f(x) : 2 ~ in ifa: g 1 and ﬂea) = m2 — 2m + 2 ifa: > 1. Now we compute the right— and left-hand derivatives deﬁned in Exercise 2.9.46: f’_(1)=lim W: lim 2__(.1____ h—tO’ h h—+0- . f(1+h)—f(1) . (1+h)2—2(1+h)+2~1 I _ % I — Ml) * 333+ h h1-1—»0+ , so f is not differentiable at 1. But f/(az) = —1 form<1andf’(w):2mi2if:c>1. —172m ifzv<il 56.9(95): \$2 if—lgmgi :c if x > 1 lim g(—1-- h) _ g(—1) — lim [—1 _ 2(“14' h)] ‘ 1 — lim _—2h : 11m (—2) = ~2 and h_,o— h hqoe h h—.oe h hat)— —1 —— ~ ~1 — 2 — 1 —2 2 lim g( h) g( ) : lim “( 1+ h) 2 11m ——h+ h = hm (—2 —I— h): —2, h~»0+ h, h—>0+ h h—+O+ h h—>0+ so 9 is differentiable at —1 and g’(—1) : —2. _ 2 7 2 lim M : lim (1) 1 = lim 2h + h = lim (2 + h) = 2 and h_,0~ h 11—,0- h h—>O’ h h—aO‘ . 9(1)-9(1) . (1)-1 - h - - l H : 1 : — 2 2 I high It hi2): h [1133+ h ill—{151+ 1 1, so 9 (1) does not ex1st. Thus, 9 is differentiable except when m = 1, and —2 if33<—1 g'(a:)= 2w if —1§\$<1 1 ifx>1 57. (a)Notethat:v2—9<Oform2<9 (i) lml<3 <2> —3<a:<3.So 222—9 ifmS—3 f(a:): —1‘2+9if—3<ar<3 :> 952—9 ifm23 2x ifm<—3 f’(\$): —2\$ if~3<w<3 : {2m if|x|>3 2m ifcc>3 —2x if [ml < 3 f(3+h)—f(3) To show that f’(3) does not exist we investigate lim h by computing the left- and right—hand h—>O derivatives deﬁned in Exercise 2.9.46. \ 157 ...
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