9th - CHAPTER 7 INTEGRALS AND TRANSCENDENTAL FUNCTIONS 7.1...

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Unformatted text preview: CHAPTER 7 INTEGRALS AND TRANSCENDENTAL FUNCTIONS 7.1 THE I.( )GARITHI'N-‘l DEFINED AS AN INTEGRAL du. s 8. Let u 2 sec x— tanx 2:» du 2 [sec x tanx— sech] dx 2 (sec x)(tan x — sec x)dx 2:» sec xdx = I secxdx _ . du _ —'J,."2 _ 'J_ _ MB _ _ /—__ I m — (lnu) u du — 2(ln u) C — 2V 111(sec J: [an x) C Vilntsecx I [an x! _ 15. Let u = 1'N2 =5» du = 33F”? dr :5» 2 du = 1"”2 dr; dr 2 fer' - r“? dr: 2 fe” du = 23” — C 2 2c“ — C = 2M— C \- '1 _ ' "‘ . 26.1]H,dx_Jeinx, letu=e"‘—l 2:» dIIZ—C_xdx 2:» —du=e"‘dx; f E" dx:—f%du=—1n|u|—C=—ln[e"‘—l]—C e"}l 37 Jl—Dgf—xdx_f (12%] dx; [11: In): 2:» du =1—dx] ' ' L l —- Jam (1;) ax: film: (A) (931v) —c= aria—c 48. 5g: : e‘I see2 [we—l] :> y : fe“ see2 [fie—t] dt; letu=ae_I 2:» du: —?re_‘dt 2:» —%du=e"dt ::> y=—%fsec2udu= —%tanu—C g 2:» —];tan(?re_l“4)—C=% =:> —];tan(?r-]:]—C=% _ 1 — . — F [an ['rre 1] — C, y(ln 4) alt.» || ;thus, y = — 1; tan[7re_t] 'J _2 _ 51.%=l—%at(1,3)=> y=x—ln|x|—C;y=3atx=l 2:» C22 2:» yzx—lnlxl—Z 7.2 EXP( )NENTIAI. (}R( )W’TH ANI) DECAY dv dl —0.6y 2:» y = fine—M; y” = 100 2:» y = 100e‘0'fi‘t 2:» y = lOCIe‘E"S :5 54.88 grams when t = 1 hr 9-1 k‘ »aty Zandt 0.5wehave2 e°'5k »ln2 0.5k »k ln4. 7. y = ynekt and y” l » y :7 Therefore, ' = e““ ‘13" 2:» y = 624]“ = 42“ = 181474978 X 10” at the end of 24 hrs 13_ (a) Anemmys: Anew (b) 2A” 2 Aue‘fl-O“ :> 1112 :(0.04)t :> I I“? a: 17.33 years; 3A” 2 Anew-0“” 2:» 1113 =(0.04)t 0.04 2:» t = % m 27.47 years. 19. y = yne‘kt = yne“3k3"'3-"k: = yne‘3 = < $16 = (0.05)(yn) 2:» after three mean lifetimes less than 5% remains 25. From Example 5, the half—life ofcarbon—M is. 5700 yr 2:» .130“ = cue—“5m: 2:» k = % :3 0.CC01216 _ _ 1nt0.4451 N 2:» c = one 0-000121‘5‘ 2:» (0.445%: 2 Che D'Emmm 2:» t = —_D_ooflmfi m 6659 years 7.3 RELATIVE RATES 01-" GROWTH - ' - "l . '1 . . - (a) slower, 1an w = llm M 2 11m mi" 2 11m 242‘ = llm 24,0 0 x —. :30 e x —. :30 e x —. :30 e x —. :30 e x —. :30 E _ _ _ _ lnx—l t x l I I (b) slower, llm 2 11m “In—’2” = 11111 x x] = llm M‘— 2 11m 1”,” x—.:30 e x—.:30 E x—.:30 e x—.:30 x—.:30 E L) = lim 1 = lim i = x —. :30 e“ x —. :30 105‘ - : 1 I )0 I - 1 no a“ - 4x:1 a“ - 12x2 I - 24x I - . 7 7 7 7 7 (c) 510w“! x e' _ x 32" _ V x 2E2“ _ V3: 452‘ x 382‘ _ V x 15E?“ 2 V/fi = 0 (d) slower, x E1190 e" _ -e . - e" _ - L _ (e) slowel, x E, — x E1190 9,, — 0 (f) faster, lim E: = lim 3: = :30 x —. r30 6 x —. :30 can _ _ -l l (g) slower, smce for all reals we have —1 < cos x < l 2:» e l < em“ < e] 2:» 2—,, C e K. and also an 2—; so by the Sandwich Theorem] we conclude that x limr3O 3:? = 0 1 7 - 1_ 7 1 (ll) eljx—x—ll — x lingo E — e . x3} 5 _ . _ 1— _ (a) same, x 4—K, _ X (1 x32] — l (b) same, lim = lim 10 = 10 x —. :30 X' x —. :30 (c) slower, lim “Ix = lim 1—: = x —. (x) X‘ X —. (x) e lnxz I 2 _ ' lUEJoX _ - [lulu] _ 1 - Zlnx _ 2 - X] _ l - i _ (d) slower, x 11hr x! — x llrrr x3 — mm K111-11; x2 — mm x 11hr 2x — mm x but X; (e) faster, x l_1rngo .2 : x (x — 1) : go . - - 1 _ (f) slowel, x LungO —.g— — x 11m:3O my — 0 X _ . 1.1]‘ . rm 1.1mm“ - unrnmlr faster, llm —3—c . 2 11m ‘ . ‘ = 1m . = (g) x —- :30 X' x —- :30 3" x —- :30 3 (11) same, lim fl: lim (1 —fl] :1 x —. :30 x -00 lnx (a) same, lim 13m = lim 1'”) = lim 1— : i x—.go lnx 31—00 lnx x—.go In? In? - 111 EX - (b) same, x 1“ — l_1ht . 1 F . (0) same, 11111 3—K 2 11m 1m = x—.go lnx x—.go lnx x—_go.’ 3 . /; . l '2 . . /; (d faster, llm 3‘— : 11m ‘— 2 11m 2 11m x = 1m 3‘.— = :30 ) Iii—.001“ M x—.:30 i] x—.:302V}x x—.:30 3 (e) faster, lim — 2 11m L = lim 3: = :30 x —. :30 1M x —. :30 a x —. :30 - 5 [mt - ( same, 11m = 11111 5 = 5 f) x —. :30 l“ x —. :30 (-‘J I - x _ - ] _ (g) Slows]! x lnx _ xlnx _ (11) faster, lit] 13‘ = lim e: 2 11m xe" = :30 x —. :30 M x —. :30 [:l x —. :30 . r-v . rr-n K . 1 1‘ rr-n? . hm \lnf, : hm ‘ln ,lnl2)...(ln 21 : lml Elntln 2'3)! (In 2. = \ln ‘1']:2... In.“ (In 2? = 0 x —- :30 x x —- :30 -" - - —- :30 o x 2. ' - 2x _ - 2 _ 2 mt. 2:» (1n _) grows slower than 3: , K l_u-n€30 2, — x —Unm, — K l_ltn€30 —(ln 2333, — 0 2:» x grows slower than _ , . ' K . x . \ llm 3— : llm 2 = 0 2:» 2" ows slowerthan e". Therefore, the slowest tothe fastest Is: (1112 ‘, x3, 2" E" 3 gr 3: —. :30 and e" so the order is c, b, a, cl X—-€3O | 10. (a) true; = xfg < 1 if): b l (or sufficiently large) '— 4 (b) true; *3”) :1— (c) false; lim : Hm ( _ 1): 1 (d) true; 2 — cos x é 3 =5» 2 l 3’” é 3 ifx is sufficiently large . e‘ H _ x x _ L g. . . (e) true, E, — e; and E; . Oas x . :30 2:» l E, < _ le ls suffielently large I . - xlnx _ - ln_ _ - _ (fl mle’xl—uan_xl—mioo X_xl—lm0o 1—0 (g) true; Infill“ < 13—: = 1 if x is sufficiently large 1 . - lnx _ - (El _ - - i 'J _] (h) falsc‘xl—ungo mil—U—xl—lmgo _,_,2" ] _xl—lmc3o —"'__xllm0o (5—?!) _§ x-—l ...
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9th - CHAPTER 7 INTEGRALS AND TRANSCENDENTAL FUNCTIONS 7.1...

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