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7.3_1 - SECTION 13 THE NATURAL EXPONENTIAL FUNCTION 267 1...

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Unformatted text preview: SECTION 13* THE NATURAL EXPONENTIAL FUNCTION 267 1 83. Iff(:c) = 111(1—0—93), then f'(a:) = 1—1—03’ so f’(0) = 1. Thus, lim 111(1_+$) = lim M = lim m = f’(0) = 1. $60 $ IE—>0 :2: $90 :1: — 0 7.3* The Natural Exponential Function R 1. (a) e is the number such that 1n 6 = 1. (c) (b) e z 2.71828 The function value at w = 0 is 1 and the slope at a: = 0 is 1. 3. (a) lne‘fizx/i (101131”: (61“2)3=23=8 5-(a)21nx—1 lnm—% —> .1:_e1/2_\/E (b)e_z 5 > m—ln5 —> :c——1n5 1 7.1n(1na:)=1 (i) 61’1“”)261 4:) lnx=e =6 4:; elm=ee <:> {17:68 9.2lnm=ln2+ln(3m—4) => lnm2=1n[2(3m—4)] => Inm2=ln(6w—8) => m2=6m—8 : m2—6m+8=0 => (x—2)($—4)=0 => m=2or$=4,b0tharevalidsolutions. 11. emzCebw 4:1 1ne”=ln[C(ebm)] (i) aw=lnC+bac+lnebm (i) am=lnC+bm (i) 1110 a—b ax—b:c=lnC (i) (a—b)$=lnC <=> :17: 13. 112+?“ = 100 :> 111 (62W) : 111100 => 2 +511 2 111100 => 5$ = 111100 — 2 => 0: = gun 100 — 2) 2 0.5210 15.ln(e””—2)—3 —> e"8 2—113 e1_e3+2 => w=ln(es+2)%3.0949 17. (a)e$<10 : lnem<ln10 => m<ln10 2 m€(—oo;ln10) 1 (b)lna:>—1 => elm>e—1 => m>e_ => w€(1/e,oo) 19. y y ...
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