Chapter 4: Exponential and Logarithmic Functions
|
|
- Ronald Fox
- 5 years ago
- Views:
Transcription
1 Erciss. 7 EXERCISES. Chatr : Eotial a Logarithmic Fuctios. a. c.. a. c a. 0.5 /.69 c ( ). a ( ) 5 c /.96 5 ( ) ( ) a.. a. o [0, 5] by [0, 0] is highr tha. o [0, 0] by [0, 0] is highr tha 0.0. o [0, 6] by [0, 00] is highr tha for larg valus of. o [0, 000] by [0, 000] 0.0 is highr tha wh is ar Brooks/Col, Cgag Larig.
2 7 Chatr : Eotial a Logarithmic Fuctios c.. o [0, 0] by [0, 0,000] is highr tha for larg valus of. o [0, 5] by [0,,000,000] is highr tha 5 for larg valus of.. will c ay owr of for larg ough valus of.. a. For m (aual comouig), ( r ) t P + simlifis to P(+ r) t. Wh P 000, r 0., a t 8, th valu is ( + 0.) 000(.).59 Th valu is $.59. For quartrly comouig,, P 000, r 0., a t 8. Thus, ( + ) 000( ) 000(.05) 0.76 Th valu is $0.76. c. For cotiuous comouig, P 000, r 0.0, a t 8. Thus, rt P Th valu is $ a. P 00, r 0.0, a, comou wkly, which is 5 tims r yar. This givs a valu of ( + 0.0) 00(.0) 56 00(.0) $96 Th vig is qual to th amout ow aftr thr yars mius th amout loa. This is $96 $00.00 $096.. a. For m (aual comouig), ( r ) t P + simlifis to P(+ r) t. Wh P 000, r 0., a t 8, th valu is ( + 0.) 000(.) Th valu is $ For quartrly comouig,, P 000, r 0., a t 8. Thus, ( 0.0) 000(.0) Th valu is $ c. For cotiuous comouig, P 000, r 0., a t 8. Thus, rt P Th valu is $ For quartrly comouig, r 0.05 a (.05) 56 (.05) $, 6, 9,90 00 Brooks/Col, Cgag Larig.
3 Erciss Th stat rat of 9.5% (comou aily) is th omial rat of itrst. To trmi th ffctiv rat of itrst, us th comou itrst formula, P( + r), with r 9.5% a umbr of ays i a yar. Sic som baks us 65 ays a som us 60 i a yar, w will try both ways. If 65 ays, th r 9.5% Th P( + r) P(.0005).0969P. umbr of ays Subtractig givs , which rss as a rct givs th ffctiv rat of itrst as 9.69%. If 60 ays, th r Th P( + r) P( ).0969P a th ffctiv rat is also 9.69%. Thus, th rror i th avrtismt is 9.85%. Th aual yil shoul b 9.69% (bas o th omial rat of 9.5%). 9. If th amout of moy P ivst at 8% comou quartrly yils $,000,000 i 60 yars, th r a , 000, 000 P( + 0.0), 000, 000 P $869 0 ( + 0.0). For comouig aually, r 0.06 a 5. Prst valu P 000 $ ( + r) (+ 0.06). For 0% comou aually, r 0.0 a. Prst valu P 000 $75. ( + r) (+ 0.0) 8. To trmi th ffctiv rat of itrst, first w trmi r i th cass of 60 ays a 65 ays. For 60 ays, r 0.07 a P( + r) P P 60 Th ffctiv rat is 7.5%. For 65 ays, r 0.07 a P( + r) P P 65 Th ffctiv rat is 7.5%. Th rror i th aual yil shoul b 7.5%. 0. Th yil is $65,000, r , a , 000 P( ) 65, 000 P $, 5 7 ( ). For 6% comou aually, r 0.06 a. Prst valu P 000 $79.09 ( + r) ( ). For % comou aually, r 0. a 0. 0 P( + r).5( + 0.) $6.67 millio 00 Brooks/Col, Cgag Larig.
4 76 Chatr : Eotial a Logarithmic Fuctios 5. To comar two itrst rats that ar comou iffrtly, covrt thm both to aual yils. 0% comou quartrly: P( + r) P(.05) P(.08) Subtractig, Th ffctiv rat of itrst is 0.8%. 9.8% comou cotiuously: r P P P(.00) Subtractig : Th ffctiv rat of itrst is 0.0%. Thus, 0% comou quartrly is bttr tha 9.8% comou cotiuously. 7. Sic th rciatio is 5% r yar, r 0.5. a. P( + r) 5, 50( 0.5) $758 P( + r) 5, 50( 0.5) $, Sic 05 is 0 yars aftr 005, (0) billio % comou quartrly: P( + r) P P(.08) Th ffctiv rat of itrst is 8.%. 7.8% comou cotiuously: r P P P(.08) Th ffctiv rat of itrst is 8.%. Thus 8% comou quartrly (a ffctiv rat of 8.%) is bttr tha 7.8% comou cotiuously (a ffctiv rat of 8.%). 8. For.6% comou aually, r 0.6 a 0. 0 P( + r) 7( + 0.6) $ millio 0. Th oulatio of Chia is giv by.(.006) t billio, whr t is th umbr of yars aftr 005. I 05 (t 0), th oulatio of Chia will b 0.(.006).7 billio. Th oulatio of Iia is giv by.(.05) t billio, whr t is th umbr of yars aftr 005. I 05 (t 0), th oulatio of Iia will b 0.(.05).5 billio. Iia's oulatio will b largr i 05.. a. 00(5) (0.9997) If a mosquito brs 00 mosquitos o avrag, th th 00 chilr will br (00)(00) 00 90, 000 grachilr, a th 90,000 grachilr will br (90,000)(00) 00 7,000,000 grat grachilr. 00(0) (0.9997) Th roortio of light that trats to a th 0. of ft is giv by. a. If th th is ft, 0. 0.() or 6.7% If th th is 0 ft, 0. 0.(0)..% 0.0 or. a. 0.0() f() mg 0.0(8) f(8) 00 5 mg 5. a. f() 0.08(). mg 6. a. If th coloy oubls vry hour, th ish was 50% covr at A.M. 0.08(8) f(8) 0.8 mg If th ish was 50% covr at A.M., it was 5 (50%) covr at 0 A.M. 00 Brooks/Col, Cgag Larig.
5 Erciss (0) S(0) (0) (0) 0.90 or 90% 9. a. Aftr 5 miuts, t a 60.8(0.5) T(0.5) grs Aftr 0 miuts, t a 60.8(0.5) T(0.5) grs 0. a. (0) % () % + +. If S 00, 0, a r 0.0, th th R- (0.0)(0) Frost mol is I 00( ) 8. Th mol stimats thr will b about 8 wly ifct ol.. Sic 00 is 0 yars aftr 000, (0) 805 $50 thousa. W hav P $8000,, a th valu aftr yars is $0,9.7. 0, ( + r) 0, r ,9.7 r % Th air rssur is ( 0.00) a /00 rct of th origial rssur, whr a is th chag i altitu. 7. a ( 0.00) % So th air rssur cras by about %.. W hav P $9000, 8, a th valu aftr yars is $0, , r + 0,80.65 r % ( 0.05) t rct will still b rst aftr t yars. 50 a. ( 0.05) 0.8 8% 8. a. 00 ( 0.05) % o [0, 00] by [0, 00] o [ 00, 00] by [0, 700] Durig th yar 087 ( 8.08) 60 ft () $0,000 $59, It is ot ossibl to valuat f a with a gativ bas (a < 0) a 0.5, bcaus th fuctio is ot fi for all valus of. For aml, if a, a 0.5, 0.5 f (0.5) ( ), which is ot a ral umbr. 5. lim a lim 0. Th -ais (y 0) is a horizotal asymtot. 5. lim 0 a lim. Th -ais (y 0) is a horizotal asymtot. 00 Brooks/Col, Cgag Larig.
6 78 Chatr : Eotial a Logarithmic Fuctios 5. For vry larg valus of, will b largst bcaus it will hav th largst ot. 5. is a cotiually icrasig fuctio, so if y < y, th <. 55. That its growth is roortioal to its siz. 56. Smiaually, quartrly, aily, cotiuously, bcaus th bfit to th ositor icrass as th itrst is comou mor oft. 57. With 5% mothly, you must wait util th of th moth to rciv itrst, whil for 5% comou cotiuously you bgi rcivig itrst right away, so th itrst bgis arig itrst without ay lay. 59. Drciatio by a fi rctag givs th biggr cras i th first yar, a th straight li givs th biggr cras i th last yar. 58. Daily comouig givs lss rofit tha cotiuous comouig. Th bakr was icorrct. 60. Th cotiuously comou rat is lss tha 0%, bcaus 0% mor at th of th yar accouts for th itrst o th itrst as wll as th itrst o th ricil. EXERCISES.. a. log 5 log 5. a. log 7 log 5 5 log 8 log log 6 log c. log log c. / log log log log. log log 9. / log log. log log f. / log log f. / log log 9. a. 0 l( ) 0. a. 5 l( ) 5 / l l l c. / l l c. / l l. l / l l 5. l(l( )) l bcaus l( ). l l f. ( ) l l f. l(l ) l 0 5. f l(9) l 9 l9 + l l9 l 7. f l( ) l l l l 9. f l + l l l + l l 6. f l + l l l + l l 8. f l l l + l l l 0. f l( 5 ) l 5l l l 00 Brooks/Col, Cgag Larig.
7 Erciss. 79. f l( 5 ) l 5 0. f l( ) + + l l f l f + l( ) 0 5. Th omai of l( ) is th valus of such that > 0 > > or < Th omai is { > or < }. Th rag is. 7. a. W us th formula P( + r) with mothly itrst rat % % 0.0 Sic oubl P ollars is P ollars, w solv P( + 0.0) P.0 l(.0 ) l l.0 l l l Sic is i moths, w ivi by to covrt to yars. A sum at % comou mothly oubls i about.9 yars. To fi how may yars it will tak for th ivstmt to icras by 50%: P( + r).5p ( + r).5 l( + r) l.5 l( + r) l.5 Now, r % % 0.0 thus, l.5 l l( + r) l Sic is i moths, w ivi by to covrt to yars yars A sum at % comou mothly icrass by 50% i about.7 yars. 6. Th omai of l( ) is th valus of such that > 0 > < < < Th omai is { } { y y 0}. < <. Th rag is 8. a. r Sic oubl P ollars is P ollars, w solv P( + 0.0) P.0 l(.0 ) l l. moths l.0 Th moy oubls i..95 yars To fi how may yars it taks to icras by 50%, which is.5p, w solv.5 P P( + 0.0).5.0 l.5 l(.0) l.5.7 moths l.0 Th moy icrass by 50% i.7. yars. 00 Brooks/Col, Cgag Larig.
8 80 Chatr : Eotial a Logarithmic Fuctios 9. a. W us P r with r Sic tril P ollars is P ollars, w solv 0.07 P P l l 5.7 yars 0.07 If P icrass by 5%, th total amout is.5p. W solv 0.07 P.5P l.5 l.5. yars If th rciatio is 0% 0. r yar, th r 0.. W us th itrst formula P( + r) a w solv P( 0.) 0.5P l 0.7 l 0.5 l yars l 0.7. W us th itrst formula P( + r) with r.% 0.0. Sic th icras is 50%, w must fi th umbr of yars to rach.5p. W solv P( + 0.0).5P.0.5 l.5 7. yars l.0 5. W wat to fi th valu of t that roucs (t) t t t l 0. t l ays a. W us P r with r Sic quarul P ollars is P ollars, w solv P P 0.06 l l. yars 0.06 A 75% icras givs.75p, a w solv P P. P( ) P.75 l yars l.09 l l.09 l l. yars l.09. W us P(+ r) with r 0.6, a w solv P( + 0.6) P t 0.0t.6 l.7 yars l t l ays (5) l or 58% 8. 0.t 900, 000, 000, 000( ) 0.t t 0. t l hours Brooks/Col, Cgag Larig.
9 Erciss t st 00( ) 0.t 00( ) 80 0.t t 0. 0.t 0. 0.t l 0. t l 0. wks t t l 0.t l ( ) t 6 hours 0.. To fi th umbr of yars aftr which.% 0.0 of th origial carbo is lft, w solv 0.000t t l 0.0 t l 0.0,00 yars To stimat th ag, w solv 0.000t 0.9 t l yars Th roortio of otassium 0 rmaiig aftr t t millio yars is. If th sklto cotai 99.9% of its origial otassium 0, th t t l l t l t l Thrfor, th stimat of th ag of th sklto of a arly huma acstor iscovr i Kya i 98 is aroimatly.7 millio yars. 5. To fi wh th raioactivity crass to 0.087t 0.00, w solv t l 0.00 t l ays l.7 yars l t t l millio yars Sic rai forsts ar isaarig at a aual rat of.8% 0.08, th r 0.08, a w solv ( 0.08) t t t l yars l 0.98 l(.6 ) l.5 l.6 l.5 l.5.7 yars l.6 00 Brooks/Col, Cgag Larig.
10 8 Chatr : Eotial a Logarithmic Fuctios o [0, 60] by [0, 5] a. 5 yars 55.5 yars. Lt t umbr of yars. Sic th rat is 6% comou quartrly, th r a th amout of moy is t t ( ).05.. o [0, ] by [0, 5] a.. yars.5 yars o [0, 0] by [0, ] a. 9.9 yars. yars. o [0, 6] by [0, ] a..6 yars 6.8 yars. 5. o [0, 0] by [0, ] a. About 9 ays About ays 6. o [0, 5] by [0, 6.] About vry ays o [0, 7000] by [0, ] About 500 yars o [0, 50] by [0.8, ] About 8 millio yars 7. If th amout of raioactiv wast is growig by.% aually sic 000, th th amout of wast yars aftr 000 is 5,000(.). W must fi th valu of such that (.).. l. l l. l l 6.5 l. Th amout will oubl i about 6.5 yars. 8. W must fi th valu of such that (.).. l. l l. l. yars l l. 0. of a yar is aroimatly moths, so th oublig tim for cll ho subscribrs is yars a moths. 00 Brooks/Col, Cgag Larig.
11 Erciss W must fi th valu of such that (.095)..095 l.095 l l.095 l l 7.8 yars l of a yar is aroimatly 0 moths, so th oublig tim for th fift-yar bo is 7 yars a 0 moths l W must fi th valu of such that (.0975) l.0975 l l.0975 l l 7.5 yars l of a yar is aroimatly 6 moths, so th oublig tim for th bo is 7 yars a 6 moths. 5. l( y) l + l y y 5. l ( ) 5. l l l y 55. log 0 y or log y 56. l y l 57. If l( ), th, but is always ositiv. So l( ) is ufi. 59. Cotiuous comouig woul giv th shortst oublig tim. Aual comouig woul giv th logst oublig tim. 58. If l 0, th 0, so woul b, which is ufi. 60. Th rug with a largr absortio costat will hav mor tim btw oss. 6. a. kt kt l kt l l t l k Th half-lif is l k. If k 0.08, half-lif l 9 hours Nt K for t 0, K> 0, b> 0. N(0) N0. Substitut 0 for t. N l N K K K bt a b(0) a 0 a a l K a l N0 l K + l l N l K a a a l K l N l 0 K ( N ) 0 00 Brooks/Col, Cgag Larig.
12 8 Chatr : Eotial a Logarithmic Fuctios 6. f H0 ( ) N Substitut 500 for N. f H 0 H0(0.999) (500) Rucig th frqucy by 6%, H 0.06 H H (0.999) 0.9 H H (0.999) (0.999) l 0.9 l(0.999) l 0.9 l(0.999) l gratios l y a for 0, a >, b> 0. Lt y. a a ( 0) a b b a b b b b l a l l a b l a b y, so y l a. b Th quilibrium oulatio is l a. b 65. a. For r 6%, th oublig tim is aroimatly 7 6 yars. P( ) P.06 l.06 l l.9 yars l If th itrst rat r is comou aually, th formula is P( + r) kp ( + r) k l( + r) l k l k l( + r) 66. a. For r %, th oublig tim is aroimatly 7 7 yars. P(.0) P l(.0) l l 69.7 yars l If th itrst rat r is comou cotiuously, th formula is r P kp r k r l l k l k r EXERCISES.. ( l ) l+ l+. l l l 6 ( ) l. l. l( + ) + 5. l l 6. l (l ) (l ) 00 Brooks/Col, Cgag Larig.
13 Erciss ( + ) l( + ) 6 ( + ) + 8. ( + )( ) l( + ) 8 ( + ) + 9. l( ) 0. l(5 ) 5 5. ( ). ( ) ( + ) (7) 5. ( ) 6. l( ) 7. ( ) ( l ) ()(l ) + l 9. l bcaus l 0. l bcaus l ( + ).. + l( + ) bcaus is a costat 6. 0 bcaus is a costat [l( + ) ] 8. [ l + ( + ) ] + + ( + ) ( + ) ( l + + 5) l + ( ) + ( ) + 0 l+ + (l + ) ( l + 7) ( ) l l. ( l ) l l + + l +. ( l) l ( l ) l l l. ( ) 00 Brooks/Col, Cgag Larig.
14 86 Chatr : Eotial a Logarithmic Fuctios. ( ) ( ) l l t 5. ( t ) ( t ) ( t ) t ( t ) 6. ( t / ) / ( t ) ( t) + + t t t + / / 7. ( t ) ( l t t l t ) ( t ) t+ / t + l t t t t 8. ( t+ lt) t+ lt + t 9. z z z t z z ( z) z z z z z z z 0. ( z) z z z z + z ( z) ( z) z ( z ) ( z ) ( z) z + z + z ( z ) z. ( z ) ( z ) ( z) z + z + ( z ) z. z l 5 5. a. f 6. a. f l 5 5 l f l+ f 5 l 5l l+ 5 0 ( ) 6 f () l 0 f () () l+ () () 7. a. f l( + 8) f + 8 () f () () a. f l f l+ l f () l 9. a. f l( ) f 50. a. 0 f (0) 0 (0) 0 f l( + ) + ( ) f f (0) Brooks/Col, Cgag Larig.
15 Erciss a. 5. a. / f 5l 5. a. f f 5l 5 + 5l+ 5 f ( ) f () 5l + 5 / f () f () 8.66 f ().778 f 5. a. f l( ) () f f f () f () () 9 f ().6 f ().05 / / /5 /5 /5 ( ) 5 5 /5 / /5 /5 5 5 /5 8 / ( ) /6 /6 5 /6 6 6 /6 /6 ( ) /6 ( )( ) /6 0 / ( k k k ) ( k ) k k k k k ( ) ( k ) k ( k) k ( ) ( k ) k ( k) k k k ( ) k k k k k 58. ( k ) k ( k ) k k ( ) k ( k) k ( ) k ( k) k k k ( ) ( ) k k k k k k k o [, ] by [, ] Thr is a rlativ maimum at (0, ); o rlativ miima. Thr ar iflctio oits at about (0.5, 0.6) a ( 0.5, 0.6). o [, ] by [, ] Thr is a rlativ miimum at (0, 0); o rlativ maima. Thr ar iflctio oits at about (, 0.9) a (, 0.9). 00 Brooks/Col, Cgag Larig.
16 88 Chatr : Eotial a Logarithmic Fuctios o [ 5, 5] by [, ] Thr is a rlativ miimum at (0, 0); o rlativ maima. Thr ar iflctio oits at about (, 0.69) a (, 0.69). o [ 5, 5] by [, 9] Thr is a rlativ miimum at (0, ); o rlativ maima. Thr ar o iflctio oits o [, 8] by [, ] Thr is a rlativ maimum at about (, 0.5); rlativ miimum at (0, 0). Thr ar iflctio oits at about (0.59, 0.9) a (., 0.8). o [, 5] by [, ] Thr is a rlativ maimum at about (, 0.7); o rlativ miimum. Thr is a iflctio oit at about (, 0.7) o [, ] by [, ] Thr is a rlativ maimum at about ( 0.7, 0.7); rlativ miimum at about (0.7, 0.7). Thr ar o iflctio oits. (Th fuctio is ot fi at 0.) o [, ] by [, ] Thr ar rlativ miima at about ( 0.6, 0.8) a (0.6, 0.8); thr ar o rlativ maima (Th fuctio is ot fi at 0.) Thr ar iflctio oits at about ( 0., 0.07) a (0., 0.07) 67. y y ( y y ) () yy ' ( y ' + y ) 0 y'( y ) y y y ' y 68. y l y 0 l (0) ( y y) y ' y yy ' (l y + ) 0 y'( y ) l y y l y yl y y ' y y y 69. f 0, f ' 0, 000(0.95) 0.95 f ' () f '() 7800 f '() Aual salary icrass by about $980 r tra yar of calculus t f() t 5 0.t 0.t f '( t) 5(0.) 0.t f '( t). 0.(0) f '(0). f '(0).7866 Th umbr of thousa mgawatts icrass by about. r tra yar. 00 Brooks/Col, Cgag Larig.
17 Erciss a. To fi th rat of growth aftr 0 yars, valuat V (0). 0.05t Vt t 0.05t V ( t) 000 (0.05) (0) 0 V (0) Th rat of growth aftr 0 yars is $50 r yar. To fi th rat of growth aftr 0 yars, valuat V (0). 0.05t V () t (0) 0.5 V (0) Th rat of growth aftr 0 yars is $8. r yar. 7. a. To fi th rat of chag at t 0 valuat V (0). 0.5t Vt () 0, t V () t 0,000 ( 0.5) 0.5t (0) V (0) Th valu is crasig by $500 r yar. To fi th rat of chag aftr yars, valuat V (). 0.5() V () Th valu is crasig by $78.05 r yar t Pt t 0.075t P ( t) 6.5 (0.075) t Pt () () 75 t P t ( 0.) 5 I 05, t (0) a. P (0) (0) P (0) It is crasig by 5% r tim uit. I 05, th worl oulatio is growig by millio ol r yar. P () 5 0.() 8. It is crasig by 8.% r tim uit t At (). A ( t). ( 0.05) 0.06 a. 0.05t 0.05t A Th amout rmaiig aftr 0 hours is crasig by 0.06 mg r hour. 0.05() 0. A () Th amout rmaiig aftr hours is crasig by 0.05 mg r hour. 0.05(0) 0 (0) Tt T () t 5.5t.5t a. T (0) 5 Th tmratur is crasig by 5 grs r hour. T.5() () 5 7 Th tmratur is crasig by 7 grs r hour S wkly sals (i thousas) 0. S 900 ( 0.) rat of chag of sals r wk a. 0. S () 90 rat of chag of sals r wk aftr wk 90(0.908) 8. thousa sals r wk S (0) 90 90(0.679) thousa sals r wk aftr 0 wks 78. Nt N ( t) 50, 000 ( 0.) 0, t 50, 000( ) 0.t 0.t a. 0.(0) N (0) 0, 000 0, 000 ol r hour 0.(8) N (8) 0, ol r hour 00 Brooks/Col, Cgag Larig.
18 90 Chatr : Eotial a Logarithmic Fuctios 79. To fi th maimum cosumr itur, solv E 0, whr E D. 0.0 E D E ( 0.0) Sttig E 0, w gt (00 ) 0 $00 W us th sco rivativ tst to show that 00 is a maimum E 5000 ( 0.0) ( 0.0) E (00) < 0 so E is maimiz. 80. To fi th maimum cosumr itur, solv E 0, whr E D() E D E ( 0.05) (0 ) $0 Us th sco rivativ tst to show that 0 is a maimum E 8000 ( 0.05) ( 0.05) E (0) < 0 so E is maimiz ric fuctio (i ollars) a. R rvu fuctio To maimiz R(), iffrtiat R ( 0.0) 00 ( 0. ) R 0 wh 5, which is th oly critical valu. 0. W calculat R () for th sco rivativ tst R 00 ( 0.0)( 0. ) + 00 ( 0.0) ( 0.0)( 0.+ ) ( 0. ) ( 0. ) At th critical valu of 5, R (5) 80 < 0, so R() is maimiz at At 5, 00 (5) Th rvu is maimiz at quatity 5000 a ric $ Brooks/Col, Cgag Larig.
19 Erciss a. R l To maimiz, iffrtiat a solv R () 0. R l l 0 l l Now us th sco rivativ tst. R < 0 for > 0, so R is maimiz. At, l Th rvu is maimiz at 0,086 uits a ric $. 8. a..5t T (0.5) T(0.5) t.5t T 0 (.5) 05.5(0.5) T (0.5) 05.8 Aftr 5 miuts, th tmratur of th br is 57.5 grs a is icrasig at th rat of.8 grs r hour..5t T () T() t.5t T 0 (.5) 05.5() T () 05. Aftr hour, th tmratur of th br is 69. grs a is icrasig at th rat of. grs r hour. 8. a. ( 0.5t N 00,000 ) ( 0.5(0.5) N ) ( t) (0.5) 00,000, t N 00, 000 ( 0.5) 00, (0.5) N (0.5) 00, ,880 Aftr 0 miuts, th umbr of ol who hav har th bullti is about,0 a is icrasig at th rat of 77,880 r hour. ( 0.5t N 00,000 ) ( 0.5() N ) ( t) () 00, , 7 N 00, 000 ( 0.5) 00, 000 N () 00, 000, Aftr hours, th umbr of ol who hav har th bullti is about 55,7 a is icrasig at th rat of, r hour t 0.5() 85. A s of 0 mtrs r sco aftr. scos. 86. o [0.9.9] by [0,0] Aftr 0. scos, Lwis s acclratio was mtrs r sco. Aftr 9. scos, his acclratio was 0.00 mtrs r sco. 87 a. 88. Th fuctio caot b iffrtiat by th owr rul bcaus h bas, is a costat a th variabl, is i th ot. o [5, 80] by [0, 0] f(5).; f (5) 0.8 At ag 5, th fastst ma s tim is hours. miuts a icrasig at about 0.8 miuts r yar. c. f(80).9; f (80) 7.76 At ag 80, th fastst ma s tim is hours 5.9 miuts a icrasig at about 7.76 miuts r yar. 00 Brooks/Col, Cgag Larig.
20 9 Chatr : Eotial a Logarithmic Fuctios a. f '( ) f f 9. c a. 9. l 9. l f f ' f c. l5 5 l5 0 5 l ' Fals: f l f. It os ot ivolv a f atural log. 97. l a 98. l a 99. bt a Nt K for t 0, k> 0, b> 0. bt N a K t bt bt bn( a ) bt ( a t ) bt ( ) a K a b bt a bn(l ) l K bn a bt K bn l K ( N ) If N > K, l K 0 N crasig. If N < K, K oulatio will b icrasig. < a th oulatio will b l > 0 a th N 00. y a for 0, a >, b> 0. b b y a( + ( b)) a ( b) y 0 wh b 0. y 0 at. b Sco rivativ tst b b y a( b) ( b) + a ( b) b b b ab ( + b ) ab ( b ) b b b b ab b y ab ( b ) ab ( ) < 0 ( a >, b> 0) This shows that is i a maimum. b Th oulatio is maimiz wh th artal stock is b. 00 Brooks/Col, Cgag Larig.
21 Erciss To maimiz R(r), solv R (r) 0. R () r a b 0 r a br 0 a b r Usig th sco rivativ tst, R () r a < 0 r so R(r) is maimiz. 0 a. y log a for a > 0 a > 0 y a y l l a l yl a l y l a l loga l a 0. To fi th maimum coctratio, solv A (t) 0, whr 0. 0.t 0.6t At () ( ) t 0.6t 0.t 0.6t ( 0.) ( 0.6) 0.6t 0.t A () t t 0.6t t 0.6t l(0. ) l(0. ) l(0.) + ( 0. t) l 0. + ( 0.6 t) 0.t l 0. l 0. t.0 l0. l0. 0. Now us th sco rivativ tst. 0.6t 0.t A ( t) 0. ( 0.6) 0. ( 0.) 0.t 0.6t (.0) 0.6(.0) A (.0) so A(t) is maimiz. Th maimum coctratio occurs at about.0 hours. y l l a yl a l y l a l y a loga y log l a l a 0. ( l a) a (l a) sic l 05.. l. (l.) l0.6 (l0.6) a. 0 (l 0)0 (l ) (l ) c. (l ) () (l ). 5 (l 5)5 (6 ) 6(l 5) 5. (l ) ( ) ( l ) Brooks/Col, Cgag Larig.
22 9 Chatr : Eotial a Logarithmic Fuctios 08. a. 5 (l 5)5 (l ) (l ) c. (l ) () ( l ) (l 9)9 (0 ) 0(l 9) 9. 0 (l0)0 ( ) ( l0)0 09. a. log (l ) log 0( ) (l 0)( ) c. log ( ) (l )( ) 0. a. log (l ) log ( + ) (l )( + ) c. 0 log ( ) (l 0)( ) EXERCISES.. a. l f ( t) l t l t l f( t) l t t t t For t, th rlativ rat of chag is. For t 0, th rlativ rat of chag is a. l f ( t) l t l t l f( t) l t t t t For t, th rlativ rat of chag is. For t 0, th rlativ rat of chag is a. 0.t l f( t) l00 l00 + l l t l f( t) 0. t 0.t. a. 0.5t l f( t) l00 l00 + l l00 0.5t l f( t) 0.5 For t 5, th rlativ rat of chag is 0.. For t, th rlativ rat of chag is 0.5. t 0.5t 5. a. t t l f ( t) l t l f( t) t For t 0, th rlativ rat of chag is (0) 0. t 7. a. l f ( t) l t l f( t) t t For t 0, th rlativ rat of chag is (0) a. 8. a. t t l f ( t) l t l f( t) t For t 5, th rlativ rat of chag is (5) 75. t t l f ( t) l t l f( t) t For t 5, th rlativ rat of chag is (5) Brooks/Col, Cgag Larig.
23 Erciss a. l f( t) l 5 t / l 5( t ) l 5 + l( t ) l f( t) 0 + t ( t ) For t 6, th rlativ rat of chag is (6 ) a. l f( t) l00 t+ l00 + l( t + ) l f( t) 0 + t ( t+ ) For t 8, th rlativ rat of chag is. (0) 0. A l A l(. ) l. + l l l A A. (0.5) l A 0.5 A 0.5. For t 0, th rlativ rat of chag is l A (0) 0.5 or 50%. Th stock is arciatig by 50% r yar.. B l B l( ) l + l l + 0. l B B (0.) l B 0. B 0. For t, th rlativ rat of chag is l B () 0. or %. Th stock is arciatig by % r yar.. 0.0t l Nt l( ) 0.0t. (0.0) l Nt t 0.0t For t 0, th rlativ rat of chag is 0.0(0). (0.0) 0.0(0) or 0.7%.. 0.0t l Nt l(0. +. ) 0.0t. (0.0) l Nt t 0.0t 0.+. For t 0, th rlativ rat of chag is 0.. (0.0) or 0.77%. 5. a. 0.0t l Pt l( +. ) 0.0t. (0.0) 0.0t l Pt t t 0.0t For t 8, th rlativ rat of chag is 0.0(8) or.%. 0.0(8) +. If th rlativ rat of chag is.5% 0.005, th 0.0t t t 0.0t t t t l 0.06 ( 0.05 ) t l ( 0.05) 5. Th rlativ rat of chag will rach.5% i about 5. yars. 6. a. 0.05t l Pt l(6 +.7 ) 0.05t.7 (0.05) 0.05t l Pt t 0.05t 0.05t For t 8, th rlativ rat of chag is (8) 0.05 or.5% (8) Th rlativ rat of chag will rach.5% i about 8. yars. 00 Brooks/Col, Cgag Larig.
24 96 Chatr : Eotial a Logarithmic Fuctios 7. D 00 5, 0 8. a. a. Elasticity of ma is D ( 5) 5 E D Evaluatig at 0, E (0) Th lasticity is lss tha, a so th ma is ilastic at 0. ( 8) E (5) 0 5 (5) 5 E(5) Sic E(5) >, th ma is lastic. 9. a. ( ) E 0. a. (0) (0) 00 E(0) Sic E (0), th ma is uitary lastic. ( ) E (5) (5) 75 E(5) Sic E (5) <, th ma is ilastic.. a. ( 00) E. a. ( 500) E Sic E (), th ma is uitary lastic. Sic E (), th ma is uitary lastic.. a. E / [ (75 ) ]( ) / (75 ) (75 ). a. / [ (00 ) ]( ) E / (00 ) 00 (50) 50 [75 (50)] 50 E(50) Sic E (50) >, th ma is lastic E(0) Sic E (0) <, th ma is ilastic. 5. a E 6. a E Sic E (0) >, th ma is lastic. Sic E (5) >, th ma is lastic. 0.0 (000 )( 0.0) 0.05 (6000 )( 0.05) 7. a. E a. E E (00) 0.0(00) E (00) 5 Sic E (00) >, th ma is lastic. Sic E (00) >, th ma is lastic. 00 Brooks/Col, Cgag Larig.
25 Erciss Th ma fuctio is D ( ) To trmi whthr th alr s to rais or lowr th ric to icras rvu, w to trmi th lasticity of ma. D E D ( )( 0.00) ( ) Wh th cars sll at a ric of $,000, E 0.00(, 000) (,000) (, 000) 5 Sic E 8>, to icras rvus th alr shoul lowr rics.. To trmi th lasticity of ma, w cosir / { [50,000(.75 ) ( )] } E / 50, 000(.75 ) (.75 ) Wh th far is 75 cts, E (.5).5 (.75.5).5.5 Sic ma is lastic, raisig th far will ot succ.. D (0 + ) 0 0+ D E D 0+ E (6) Sic E <, icrasig rics shoul icras 8 rvus. Ys, th commissio shoul grat th rqust. 0. To trmi th lasticity of ma, w cosir ( ) E Wh th ric is $5, E (5) 5 5 Sic ma is lastic, th iscouts will succ.. To trmi th lasticity of ma, w cosir / [0,000(75 ) ( )] E / 80,000(75 ) (75 ) Wh th ric is 50 cts, E(50) 50 (75 50) Sic E(50), raisig rics will ot succ.. To trmi th lasticity of ma, w cosir (9.5 )( 0.0) E Wh th ric is $0 r barrl, E (0) 0.0(0).8 Sic ma is lastic, it shoul lowr rics. 5. To trmi th lasticity of ma, w cosir 0.06 (.5 )( 0.06) E Wh th ric is $0 r barrl, E (0) 0.06(0) 7. Sic ma is lastic, it shoul lowr rics. 6. E.859 (.509)( 0.859) Brooks/Col, Cgag Larig.
26 98 Chatr : Eotial a Logarithmic Fuctios 7.. (7.88)( 0. ) E a. E 0.75 Sic ma is ilastic, raisig rics will rais rvu. c. $, ( ) 9. a. E a. E Sic ma is ilastic, raisig rics will E(0) rais rvu. c. $0,0 c. E(89.999) , E( ) , 999, As aroachs 90, E aroachs ifiity.. Th mioit is 5. E(5) For D a b, ( b) b a b a b E b(0) (b) For 0, E (0) a b(0) 0. (c) Wh th valu aroachs b, E aroachs ifiity. a a b, b b a a a a ( b ) a b a a a a a ( ) a a b b () For th mioit, E.. 5. rct chag i ma E rct chag i ric % % or 5% f. l f l... f l f l. f l f l f. l f l l 6. ( ) E ( )( ) E E 00 Brooks/Col, Cgag Larig.
27 Rviw Erciss for Chatr ( ) E ( )( ) E E 8. Dma will icras wkly. 9. Dma will icras strogly. 50. Elastic for hatig oil a ilastic for oliv oil. (It is commoly us.) 5. Ilastic for cigartts (thy ar habit-formig) a lastic for jwlry. c c 5. a. + E c c If E( ), th w kow from art (a) that D c c. 5. c a ( c) E c 5. c If S a, th c a c S ( a )( c) Es c S c a 55. If S a, th S ( a ) Es S a REVIEW EXERCISES FOR CHAPTER. a.. a. For quartrly comouig, r a 8. P( + r) 0, 000( + 0.0) 0, 000(.0) $8,85. For comouig cotiuously, r 0.08 a 8. r P (0.08)(8) 0, 000 $8, 96.8 If th rciatio is 0% r yar, r 0% 0. a t. Th formula for th valu is t t V 800,000( 0.) 800,000(0.8) Aftr yars, its valu is V 800, 000(0.8) $7, 680. For 6% comou quartrly, r a. P( + r) ( ).06 For 5.98% comou cotiuously, r a. r P % comou cotiuously has a highr yil. 0.t. For Drug A, Ct (). Aftr hours, 0.8 C() t For Drug B, Ct (). Aftr hours, C().0 Drug B has a highr coctratio. 00 Brooks/Col, Cgag Larig.
28 00 Chatr : Eotial a Logarithmic Fuctios Th valu t corrsos to / 8 0 a C () 65, 56 mgabits. São Paulo will ovrtak Tokyo i about 60 yars, or urig th yar r 0% 5% 0.05 a. P( ) P.05 l.05 l l.05 l l l Sic is i half yars, w ivi by to covrt to yars. About 7. yars. P(.05).5P.05.5 l.05 l.5 l.05 l.5 l.5 l half yars 8.. yars 8. a. For comouig cotiuously, r 7% P P 0.07 l l 9.9 yars 0.07 To icras by 50%, P P.5 l yars Th roortio of otassium 0 rmaiig aftr t t millio yars is. Sic 97.% 0.97 rmais, t t l 0.97 t l millio yars ol t. W wat to solv Nt,000,000( ) with N(t) 500, t, 000, 000( ) 500, t t t t l 0.5 t l hours 0. Sic 99.9% rmais, t t l millio yars ol Sic th rat of icras is % 0.0 r yar, r 0.0. To fi wh ma will icras by 50% 0.5,.5 P P( + 0.0).5 (.0) l.5.7 yars l.0 00 Brooks/Col, Cgag Larig.
29 Rviw Erciss for Chatr 0. a. If th itrst rat is 6.5% comou quartrly, r ( ) (000) +.5 l.5 l ( ) 5. quartrs 6. yars If th itrst rat is 6.5% comou cotiuously, r (000) l.5 6. yars l l 6.. a. To rach 0% 0. of th ol, (t) t t t l 0.7 t l0.7 ays 0.0 To rach 0% 0. of th ol, t () t t 0.6 t l ays 0.0 ( ) l( ) ( ) 7. l ( ) / + ( ) 8. l + / ( + ) + / / l l l /. l ( l ). ( l ) l + l. ( ). 5. l ( ) + 6. l / l / l l l / 7. (5 + l + ) 0+ l + 0+ l + 8. ( + l ) 6 + l l 00 Brooks/Col, Cgag Larig.
30 0 Chatr : Eotial a Logarithmic Fuctios 9. ( ) 6 () ( ) ().. Rlativ miimum at (0,.) Rlativ maimum at (0, 6).. a. 0. S S 500 ( 0.) 50 0.() S () 50 6 Sals ar icrasig by 6,000 aftr wk. 0.(0) S (0) Sals ar icrasig by 55,000 aftr 0 wks.. a. 0.08t At () t 0.08t A ( t).5 ( 0.08) (0) A (0) Th amout of th rug i th bloostram immiatly aftr th ijctio is crasig by 0. mg r hour. 0.08(5) A (5) Th amout of th rug i th bloostram aftr 5 hours is crasig by 0.08 mg r hour. 5. Pt () 00 00l( t+ ) P ( t) 00 rat of chag t + P (5) rat of chag aftr 5 scos Th rat of chag aftr 5 scos is crasig by % r sco. 6. a. 0.t Tt () t 0.t T 5 ( 0.).5 0 T (0).5.5 grs r hour 7. a. 0.t Nt 0, 000( ) 0.t 0.t N ( t) 0,000 ( 0.) () N () Aftr hour, th rat of chag i th umbr of iform ol is icrasig by 6667 r hour. 0.5 T (5).5. grs r hour 0.(8) N (8) Aftr 8 hours, th rat of chag i th umbr of iform ol is icrasig by 86 r hour. 00 Brooks/Col, Cgag Larig.
31 Rviw Erciss for Chatr t V() t 50t 0.08t 0.08t V t t t 0.08t 0.08t ( 0.08) 00t t W st V (t) 0 to maimiz V(t). 0.08t 0.08t 0 00t t 0.08t 0 t (5 t) A critical valu is t 5. Now w us th sco rivativ tst. 0.08t 0.08t 0.08t 0.08t V ( t) t ( 0.08) 8t t ( 0.08) 0.08t 0.08t 0.08t 0.08t 00 8t 8t + 0.t 0.08t 0.08t 0.08t 00 6t + 0.t V (5) < 0 so V is maimiz. Th rst valu is maimiz i 5 yars a. 0.5 R 00 R ( 0.5) ( 0.5 ) R 0 wh, which is th oly critical valu R 00 ( 0.5)( 0.5 ) + 00 ( 0.5) ( 0.5)( 0.5+ ) ( 0.5 ) R () 50 < 0, so R is maimiz at. Thus, quatity 000 a ric () 00 $7.58 maimiz rvu. 0. a. R (5 l ) 5 l R 5 l l R 0 wh, which is th oly critical valu. R < 0 wh so R is maimiz. 5 l( ) Thus, th quatity a th ric $ maimiz rvu. 00 Brooks/Col, Cgag Larig.
32 0 Chatr : Eotial a Logarithmic Fuctios. To fi th maimum cosumr itur, solv E 0, whr E D. 0.0 E (5,000 ) E 5, , 000 ( 0.0) , E 0 wh 50, which is th oly critical valu. W us th sco rivativ tst to show that 50 is a maimum E 5, 000 ( 0.0) ( 0.0) (50) 0.0(50) E (50) (50) < 0 so E is maimiz. Th ric $50 maimizs cosumr itur... o [ 5, 5] by [ 5, 5] Th fuctio has a rlativ maimum at about (,.69) a a rlativ miimum at (0, 0). Thr ar iflctio oits at about (,.7) a (6,.). o [ 5, 5] by [ 5, 5] Th fuctio has a rlativ maimum at about ( 0.7, 0.) a a rlativ miimum at about (0.7, 0.). Thr ar iflctio oits at about ( 0., 0.07) a (0., 0.07).. To fi th maimum cosumr itur, solv E 0, whr E D E (00 ) E ( 0.00) E 0 wh th ric is about 769, which is th oly critical valu E ( 0.00) (769) 0.00(769) E (769) (769) 0.00(769) 0.6 < 0 so E is maimiz. Th ric $769 maimizs cosumr itur R (0 ) R ( ) R () 0 wh th quatity is about 0. R 0 ( ) R (0) (0) (0) 0.5 < 0 so R is maimiz. Th quatity 0 maimizs rvu (0) (0) (0) 00 Brooks/Col, Cgag Larig.
33 Rviw Erciss for Chatr t Gt () t 0.0t (0.0) l Gt 0.0 t t 0.0t If t 0, th th rlativ rat of chag is % G () t 0.0(0) 0. For t 0, % Gt () 0.0(0) D ( ) E D 6 6 E() Sic E() <, ma is ilastic. Raisig rics will icras rvu. / 00(600 ) D ( ) 00 E D 00(600 ) / 00(600 ) (600 ) E (50) (600 50) 500 Sic E <, ma is ilastic. Raisig rics will icras rvu.. D ( 0.58 ) E D 0... Dma is ilastic. 5. Rlativ rat of chag of P l P. l P (9) At 9, % (9) (9) 5. a. Sic ma is lastic, th alr shoul lowr th ric. c. E(8.7) ; thus at $8700 lasticity quals. o [0, 7] by [0, 5] D E D ( ) Th ma is lastic at $0,000 bcaus E(0) Brooks/Col, Cgag Larig.
1985 AP Calculus BC: Section I
985 AP Calculus BC: Sctio I 9 Miuts No Calculator Nots: () I this amiatio, l dots th atural logarithm of (that is, logarithm to th bas ). () Ulss othrwis spcifid, th domai of a fuctio f is assumd to b
More informationChapter 2 Infinite Series Page 1 of 11. Chapter 2 : Infinite Series
Chatr Ifiit Sris Pag of Sctio F Itgral Tst Chatr : Ifiit Sris By th d of this sctio you will b abl to valuat imror itgrals tst a sris for covrgc by alyig th itgral tst aly th itgral tst to rov th -sris
More informationReview Exercises. 1. Evaluate using the definition of the definite integral as a Riemann Sum. Does the answer represent an area? 2
MATHEMATIS --RE Itgral alculus Marti Huard Witr 9 Rviw Erciss. Evaluat usig th dfiitio of th dfiit itgral as a Rima Sum. Dos th aswr rprst a ara? a ( d b ( d c ( ( d d ( d. Fid f ( usig th Fudamtal Thorm
More informationOption 3. b) xe dx = and therefore the series is convergent. 12 a) Divergent b) Convergent Proof 15 For. p = 1 1so the series diverges.
Optio Chaptr Ercis. Covrgs to Covrgs to Covrgs to Divrgs Covrgs to Covrgs to Divrgs 8 Divrgs Covrgs to Covrgs to Divrgs Covrgs to Covrgs to Covrgs to Covrgs to 8 Proof Covrgs to π l 8 l a b Divrgt π Divrgt
More informationProbability & Statistics,
Probability & Statistics, BITS Pilai K K Birla Goa Campus Dr. Jajati Kshari Sahoo Dpartmt of Mathmatics BITS Pilai, K K Birla Goa Campus Poisso Distributio Poisso Distributio: A radom variabl X is said
More informationWorksheet: Taylor Series, Lagrange Error Bound ilearnmath.net
Taylor s Thorm & Lagrag Error Bouds Actual Error This is th ral amout o rror, ot th rror boud (worst cas scario). It is th dirc btw th actual () ad th polyomial. Stps:. Plug -valu ito () to gt a valu.
More informationH2 Mathematics Arithmetic & Geometric Series ( )
H Mathmatics Arithmtic & Gomtric Sris (08 09) Basic Mastry Qustios Arithmtic Progrssio ad Sris. Th rth trm of a squc is 4r 7. (i) Stat th first four trms ad th 0th trm. (ii) Show that th squc is a arithmtic
More informationAPPENDIX: STATISTICAL TOOLS
I. Nots o radom samplig Why do you d to sampl radomly? APPENDI: STATISTICAL TOOLS I ordr to masur som valu o a populatio of orgaisms, you usually caot masur all orgaisms, so you sampl a subst of th populatio.
More informationWashington State University
he 3 Ktics ad Ractor Dsig Sprg, 00 Washgto Stat Uivrsity Dpartmt of hmical Egrg Richard L. Zollars Exam # You will hav o hour (60 muts) to complt this xam which cosists of four (4) problms. You may us
More informationMONTGOMERY COLLEGE Department of Mathematics Rockville Campus. 6x dx a. b. cos 2x dx ( ) 7. arctan x dx e. cos 2x dx. 2 cos3x dx
MONTGOMERY COLLEGE Dpartmt of Mathmatics Rockvill Campus MATH 8 - REVIEW PROBLEMS. Stat whthr ach of th followig ca b itgratd by partial fractios (PF), itgratio by parts (PI), u-substitutio (U), or o of
More informationHow many neutrino species?
ow may utrio scis? Two mthods for dtrmii it lium abudac i uivrs At a collidr umbr of utrio scis Exasio of th uivrs is ovrd by th Fridma quatio R R 8G tot Kc R Whr: :ubblcostat G :Gravitatioal costat 6.
More informationWindowing in FIR Filter Design. Design Summary and Examples
Lctur 3 Outi: iowig i FIR Fitr Dsig. Dsig Summary a Exams Aoucmts: Mitrm May i cass. i covr through FIR Fitr Dsig. 4 ost, 5% ogr tha usua, 4 xtra ays to comt (u May 8) Mor tais o say Thr wi b o aitioa
More informationTriple Play: From De Morgan to Stirling To Euler to Maclaurin to Stirling
Tripl Play: From D Morga to Stirlig To Eulr to Maclauri to Stirlig Augustus D Morga (186-1871) was a sigificat Victoria Mathmaticia who mad cotributios to Mathmatics History, Mathmatical Rcratios, Mathmatical
More informationSession : Plasmas in Equilibrium
Sssio : Plasmas i Equilibrium Ioizatio ad Coductio i a High-prssur Plasma A ormal gas at T < 3000 K is a good lctrical isulator, bcaus thr ar almost o fr lctros i it. For prssurs > 0.1 atm, collisio amog
More information1973 AP Calculus BC: Section I
97 AP Calculus BC: Scio I 9 Mius No Calculaor No: I his amiaio, l dos h aural logarihm of (ha is, logarihm o h bas ).. If f ( ) =, h f ( ) = ( ). ( ) + d = 7 6. If f( ) = +, h h s of valus for which f
More informationln x = n e = 20 (nearest integer)
H JC Prlim Solutios 6 a + b y a + b / / dy a b 3/ d dy a b at, d Giv quatio of ormal at is y dy ad y wh. d a b () (,) is o th curv a+ b () y.9958 Qustio Solvig () ad (), w hav a, b. Qustio d.77 d d d.77
More informationChapter Five. More Dimensions. is simply the set of all ordered n-tuples of real numbers x = ( x 1
Chatr Fiv Mor Dimsios 51 Th Sac R W ar ow rard to mov o to sacs of dimsio gratr tha thr Ths sacs ar a straightforward gralizatio of our Euclida sac of thr dimsios Lt b a ositiv itgr Th -dimsioal Euclida
More information07 - SEQUENCES AND SERIES Page 1 ( Answers at he end of all questions ) b, z = n
07 - SEQUENCES AND SERIES Pag ( Aswrs at h d of all qustios ) ( ) If = a, y = b, z = c, whr a, b, c ar i A.P. ad = 0 = 0 = 0 l a l
More informationPURE MATHEMATICS A-LEVEL PAPER 1
-AL P MATH PAPER HONG KONG EXAMINATIONS AUTHORITY HONG KONG ADVANCED LEVEL EXAMINATION PURE MATHEMATICS A-LEVEL PAPER 8 am am ( hours) This papr must b aswrd i Eglish This papr cosists of Sctio A ad Sctio
More informationNarayana IIT Academy
INDIA Sc: LT-IIT-SPARK Dat: 9--8 6_P Max.Mars: 86 KEY SHEET PHYSIS A 5 D 6 7 A,B 8 B,D 9 A,B A,,D A,B, A,B B, A,B 5 A 6 D 7 8 A HEMISTRY 9 A B D B B 5 A,B,,D 6 A,,D 7 B,,D 8 A,B,,D 9 A,B, A,B, A,B,,D A,B,
More informationThey must have different numbers of electrons orbiting their nuclei. They must have the same number of neutrons in their nuclei.
37 1 How may utros ar i a uclus of th uclid l? 20 37 54 2 crtai lmt has svral isotops. Which statmt about ths isotops is corrct? Thy must hav diffrt umbrs of lctros orbitig thir ucli. Thy must hav th sam
More informationUnit 6: Solving Exponential Equations and More
Habrman MTH 111 Sction II: Eonntial and Logarithmic Functions Unit 6: Solving Eonntial Equations and Mor EXAMPLE: Solv th quation 10 100 for. Obtain an act solution. This quation is so asy to solv that
More informationChapter (8) Estimation and Confedence Intervals Examples
Chaptr (8) Estimatio ad Cofdc Itrvals Exampls Typs of stimatio: i. Poit stimatio: Exampl (1): Cosidr th sampl obsrvatios, 17,3,5,1,18,6,16,10 8 X i i1 17 3 5 118 6 16 10 116 X 14.5 8 8 8 14.5 is a poit
More informationFermi Gas. separation
ri Gas Distiguishabl Idistiguishabl Classical dgrat dd o dsity. If th wavlgth siilar to th saratio tha dgrat ri gas articl h saratio largr traturs hav sallr wavlgth d tightr ackig for dgracy
More informationPrecalculus MATH Sections 3.1, 3.2, 3.3. Exponential, Logistic and Logarithmic Functions
Precalculus MATH 2412 Sectios 3.1, 3.2, 3.3 Epoetial, Logistic ad Logarithmic Fuctios Epoetial fuctios are used i umerous applicatios coverig may fields of study. They are probably the most importat group
More information+ x. x 2x. 12. dx. 24. dx + 1)
INTEGRATION of FUNCTION of ONE VARIABLE INDEFINITE INTEGRAL Fidig th idfiit itgrals Rductio to basic itgrals, usig th rul f ( ) f ( ) d =... ( ). ( )d. d. d ( ). d. d. d 7. d 8. d 9. d. d. d. d 9. d 9.
More informationDiscrete Fourier Transform (DFT)
Discrt Fourir Trasorm DFT Major: All Egirig Majors Authors: Duc guy http://umricalmthods.g.us.du umrical Mthods or STEM udrgraduats 8/3/29 http://umricalmthods.g.us.du Discrt Fourir Trasorm Rcalld th xpotial
More informationA Simple Proof that e is Irrational
Two of th most bautiful ad sigificat umbrs i mathmatics ar π ad. π (approximatly qual to 3.459) rprsts th ratio of th circumfrc of a circl to its diamtr. (approximatly qual to.788) is th bas of th atural
More informationWhere k is either given or determined from the data and c is an arbitrary constant.
Exponntial growth and dcay applications W wish to solv an quation that has a drivativ. dy ky k > dx This quation says that th rat of chang of th function is proportional to th function. Th solution is
More informationy = 2xe x + x 2 e x at (0, 3). solution: Since y is implicitly related to x we have to use implicit differentiation: 3 6y = 0 y = 1 2 x ln(b) ln(b)
4. y = y = + 5. Find th quation of th tangnt lin for th function y = ( + ) 3 whn = 0. solution: First not that whn = 0, y = (1 + 1) 3 = 8, so th lin gos through (0, 8) and thrfor its y-intrcpt is 8. y
More informationSolid State Device Fundamentals
8 Biasd - Juctio Solid Stat Dvic Fudamtals 8. Biasd - Juctio ENS 345 Lctur Cours by Aladr M. Zaitsv aladr.zaitsv@csi.cuy.du Tl: 718 98 81 4N101b Dartmt of Egirig Scic ad Physics Biasig uiolar smicoductor
More informationTime : 1 hr. Test Paper 08 Date 04/01/15 Batch - R Marks : 120
Tim : hr. Tst Papr 8 D 4//5 Bch - R Marks : SINGLE CORRECT CHOICE TYPE [4, ]. If th compl umbr z sisfis th coditio z 3, th th last valu of z is qual to : z (A) 5/3 (B) 8/3 (C) /3 (D) o of ths 5 4. Th itgral,
More information3 a b c km m m 8 a 3.4 m b 2.4 m
Chaptr Exris A a 9. m. m. m 9. km. mm. m Purpl lag hapr y 8p 8m. km. m Th triangl on th right 8. m 9 a. m. m. m Exris B a m. m mm. km. mm m a. 9 8...8 m. m 8. 9 m Ativity p. 9 Pupil s own answrs Ara =
More informationMultiple Short Term Infusion Homework # 5 PHA 5127
Multipl Short rm Infusion Homwork # 5 PHA 527 A rug is aministr as a short trm infusion. h avrag pharmacokintic paramtrs for this rug ar: k 0.40 hr - V 28 L his rug follows a on-compartmnt boy mol. A 300
More informationChapter 3 Exponential and Logarithmic Functions. Section a. In the exponential decay model A. Check Point Exercises
Chaptr Eponntial and Logarithmic Functions Sction. Chck Point Erciss. a. A 87. Sinc is yars aftr, whn t, A. b. A A 87 k() k 87 k 87 k 87 87 k.4 Thus, th growth function is A 87 87.4t.4t.4t A 87..4t 87.4t
More information(Reference: sections in Silberberg 5 th ed.)
ALE. Atomic Structur Nam HEM K. Marr Tam No. Sctio What is a atom? What is th structur of a atom? Th Modl th structur of a atom (Rfrc: sctios.4 -. i Silbrbrg 5 th d.) Th subatomic articls that chmists
More information2. SIMPLE SOIL PROPETIES
2. SIMPLE SOIL PROPETIES 2.1 EIGHT-OLUME RELATIONSHIPS It i oft rquir of th gotchical gir to collct, claify a ivtigat oil ampl. B it for ig of fouatio or i calculatio of arthork volum, trmiatio of oil
More informationHow many neutrons does this aluminium atom contain? A 13 B 14 C 27 D 40
alumiium atom has a uclo umbr of 7 ad a roto umbr of 3. How may utros dos this alumiium atom cotai? 3 4 7 40 atom of lmt Q cotais 9 lctros, 9 rotos ad 0 utros. What is Q? calcium otassium strotium yttrium
More informationPeriodic Structures. Filter Design by the Image Parameter Method
Prioic Structurs a Filtr sig y th mag Paramtr Mtho ECE53: Microwav Circuit sig Pozar Chaptr 8, Sctios 8. & 8. Josh Ottos /4/ Microwav Filtrs (Chaptr Eight) microwav filtr is a two-port twork us to cotrol
More informationa 1and x is any real number.
Calcls Nots Eponnts an Logarithms Eponntial Fnction: Has th form y a, whr a 0, a an is any ral nmbr. Graph y, Graph y ln y y Th Natral Bas (Elr s nmbr): An irrational nmbr, symboliz by th lttr, appars
More informationCalculus & analytic geometry
Calculus & aalytic gomtry B Sc MATHEMATICS Admissio owards IV SEMESTER CORE COURSE UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION CALICUT UNIVERSITYPO, MALAPPURAM, KERALA, INDIA 67 65 5 School of Distac
More informationChapter Taylor Theorem Revisited
Captr 0.07 Taylor Torm Rvisitd Atr radig tis captr, you sould b abl to. udrstad t basics o Taylor s torm,. writ trascdtal ad trigoomtric uctios as Taylor s polyomial,. us Taylor s torm to id t valus o
More information3.1 Atomic Structure and The Periodic Table
Sav My Exams! Th Hom of Rvisio For mor awsom GSE ad lvl rsourcs, visit us at www.savmyxams.co.uk/ 3. tomic Structur ad Th Priodic Tabl Qustio Par Lvl IGSE Subjct hmistry (060) Exam oard ambridg Itratioal
More informationLECTURE 13 Filling the bands. Occupancy of Available Energy Levels
LUR 3 illig th bads Occupacy o Availabl rgy Lvls W hav dtrmid ad a dsity o stats. W also d a way o dtrmiig i a stat is illd or ot at a giv tmpratur. h distributio o th rgis o a larg umbr o particls ad
More information2008 AP Calculus BC Multiple Choice Exam
008 AP Multipl Choic Eam Nam 008 AP Calculus BC Multipl Choic Eam Sction No Calculator Activ AP Calculus 008 BC Multipl Choic. At tim t 0, a particl moving in th -plan is th acclration vctor of th particl
More information3) Use the average steady-state equation to determine the dose. Note that only 100 mg tablets of aminophylline are available here.
PHA 5127 Dsigning A Dosing Rgimn Answrs provi by Jry Stark Mr. JM is to b start on aminophyllin or th tratmnt o asthma. H is a non-smokr an wighs 60 kg. Dsign an oral osing rgimn or this patint such that
More informationMSLC Math 151 WI09 Exam 2 Review Solutions
Eam Rviw Solutions. Comput th following rivativs using th iffrntiation ruls: a.) cot cot cot csc cot cos 5 cos 5 cos 5 cos 5 sin 5 5 b.) c.) sin( ) sin( ) y sin( ) ln( y) ln( ) ln( y) sin( ) ln( ) y y
More informationMath 34A. Final Review
Math A Final Rviw 1) Us th graph of y10 to find approimat valus: a) 50 0. b) y (0.65) solution for part a) first writ an quation: 50 0. now tak th logarithm of both sids: log() log(50 0. ) pand th right
More informationare given in the table below. t (hours)
CALCULUS WORKSHEET ON INTEGRATION WITH DATA Work th following on notbook papr. Giv dcimal answrs corrct to thr dcimal placs.. A tank contains gallons of oil at tim t = hours. Oil is bing pumpd into th
More informationFirst order differential equation Linear equation; Method of integrating factors
First orr iffrntial quation Linar quation; Mtho of intgrating factors Exampl 1: Rwrit th lft han si as th rivativ of th prouct of y an som function by prouct rul irctly. Solving th first orr iffrntial
More informationStatistics 3858 : Likelihood Ratio for Exponential Distribution
Statistics 3858 : Liklihood Ratio for Expotial Distributio I ths two xampl th rjctio rjctio rgio is of th form {x : 2 log (Λ(x)) > c} for a appropriat costat c. For a siz α tst, usig Thorm 9.5A w obtai
More informationEMPLOYMENT AND THE DISTRIBUTION OF INCOME. Andrés Velasco
EMPLOYMENT AND THE DISTRIBUTION OF INCOME Adrés Vlasco Ju 2011 Motivatio My xpric as Fiac Miistr Chil: hatd discussios o iquality. But... Focus oly o th wag distributio Discussios o th shap of th wag distributio
More informationPhysics 302 Exam Find the curve that passes through endpoints (0,0) and (1,1) and minimizes 1
Physis Exam 6. Fid th urv that passs through dpoits (, ad (, ad miimizs J [ y' y ]dx Solutio: Si th itgrad f dos ot dpd upo th variabl of itgratio x, w will us th sod form of Eulr s quatio: f f y' y' y
More informationCompound Interest. S.Y.Tan. Compound Interest
Compoud Iterest S.Y.Ta Compoud Iterest The yield of simple iterest is costat all throughout the ivestmet or loa term. =2000 r = 0% = 0. t = year =? I =? = 2000 (+ (0.)()) = 3200 I = - = 3200-2000 = 200
More information22/ Breakdown of the Born-Oppenheimer approximation. Selection rules for rotational-vibrational transitions. P, R branches.
Subjct Chmistry Papr No and Titl Modul No and Titl Modul Tag 8/ Physical Spctroscopy / Brakdown of th Born-Oppnhimr approximation. Slction ruls for rotational-vibrational transitions. P, R branchs. CHE_P8_M
More informationCHAPTER 4 Integration
CHAPTER Itegratio Sectio. Atierivatives a Iefiite Itegratio......... Sectio. Area............................. Sectio. Riema Sums a Defiite Itegrals........... Sectio. The Fuametal Theorem of Calculus..........
More informationENGG 1203 Tutorial. Difference Equations. Find the Pole(s) Finding Equations and Poles
ENGG 03 Tutoial Systms ad Cotol 9 Apil Laig Obctivs Z tasfom Complx pols Fdbac cotol systms Ac: MIT OCW 60, 6003 Diffc Equatios Cosid th systm pstd by th followig diffc quatio y[ ] x[ ] (5y[ ] 3y[ ]) wh
More informationCalculus II (MAC )
Calculus II (MAC232-2) Tst 2 (25/6/25) Nam (PRINT): Plas show your work. An answr with no work rcivs no crdit. You may us th back of a pag if you nd mor spac for a problm. You may not us any calculators.
More informationCDS 101: Lecture 5.1 Reachability and State Space Feedback
CDS, Lctur 5. CDS : Lctur 5. Rachability ad Stat Spac Fdback Richard M. Murray ad Hido Mabuchi 5 Octobr 4 Goals: Di rachability o a cotrol systm Giv tsts or rachability o liar systms ad apply to ampls
More information1997 AP Calculus AB: Section I, Part A
997 AP Calculus AB: Sction I, Part A 50 Minuts No Calculator Not: Unlss othrwis spcifid, th domain of a function f is assumd to b th st of all ral numbrs for which f () is a ral numbr.. (4 6 ) d= 4 6 6
More informationRational Functions. Rational Function. Example. The degree of q x. If n d then f x is an improper rational function. x x. We have three forms
Ratioal Fuctios We have three forms R Ratioal Fuctio a a1 a p p 0 b b1 b0 q q p a a1 a0 q b b1 b0 The egree of p The egree of q is is If the f is a improper ratioal fuctio Compare forms Epae a a a b b1
More information5.1 The Nuclear Atom
Sav My Exams! Th Hom of Rvisio For mor awsom GSE ad lvl rsourcs, visit us at www.savmyxams.co.uk/ 5.1 Th Nuclar tom Qustio Papr Lvl IGSE Subjct Physics (0625) Exam oard Topic Sub Topic ooklt ambridg Itratioal
More informationSolution to 1223 The Evil Warden.
Solutio to 1 Th Evil Ward. This is o of thos vry rar PoWs (I caot thik of aothr cas) that o o solvd. About 10 of you submittd th basic approach, which givs a probability of 47%. I was shockd wh I foud
More informationCase Study Vancomycin Answers Provided by Jeffrey Stark, Graduate Student
Cas Stuy Vancomycin Answrs Provi by Jffry Stark, Grauat Stunt h antibiotic Vancomycin is liminat almost ntirly by glomrular filtration. For a patint with normal rnal function, th half-lif is about 6 hours.
More informationProd.C [A] t. rate = = =
Concntration Concntration Practic Problms: Kintics KEY CHEM 1B 1. Basd on th data and graph blow: Ract. A Prod. B Prod.C..25.. 5..149.11.5 1..16.144.72 15..83.167.84 2..68.182.91 25..57.193.96 3..5.2.1
More informationPartial Derivatives: Suppose that z = f(x, y) is a function of two variables.
Chaptr Functions o Two Variabls Applid Calculus 61 Sction : Calculus o Functions o Two Variabls Now that ou hav som amiliarit with unctions o two variabls it s tim to start appling calculus to hlp us solv
More informationPipe flow friction, small vs. big pipes
Friction actor (t/0 t o pip) Friction small vs larg pips J. Chaurtt May 016 It is an intrsting act that riction is highr in small pips than largr pips or th sam vlocity o low and th sam lngth. Friction
More informationz 1+ 3 z = Π n =1 z f() z = n e - z = ( 1-z) e z e n z z 1- n = ( 1-z/2) 1+ 2n z e 2n e n -1 ( 1-z )/2 e 2n-1 1-2n -1 1 () z
Sris Expasio of Rciprocal of Gamma Fuctio. Fuctios with Itgrs as Roots Fuctio f with gativ itgrs as roots ca b dscribd as follows. f() Howvr, this ifiit product divrgs. That is, such a fuctio caot xist
More information4. Money cannot be neutral in the short-run the neutrality of money is exclusively a medium run phenomenon.
PART I TRUE/FALSE/UNCERTAIN (5 points ach) 1. Lik xpansionary montary policy, xpansionary fiscal policy rturns output in th mdium run to its natural lvl, and incrass prics. Thrfor, fiscal policy is also
More informationCorrelation in tree The (ferromagnetic) Ising model
5/3/00 :\ jh\slf\nots.oc\7 Chaptr 7 Blf propagato corrlato tr Corrlato tr Th (frromagtc) Isg mol Th Isg mol s a graphcal mol or par ws raom Markov fl cosstg of a urct graph wth varabls assocat wth th vrtcs.
More information6.1 Integration by Parts and Present Value. Copyright Cengage Learning. All rights reserved.
6.1 Intgration by Parts and Prsnt Valu Copyright Cngag Larning. All rights rsrvd. Warm-Up: Find f () 1. F() = ln(+1). F() = 3 3. F() =. F() = ln ( 1) 5. F() = 6. F() = - Objctivs, Day #1 Studnts will b
More informationHow much air is required by the people in this lecture theatre during this lecture?
3 NTEGRATON tgrtio is us to swr qustios rltig to Ar Volum Totl qutity such s: Wht is th wig r of Boig 747? How much will this yr projct cost? How much wtr os this rsrvoir hol? How much ir is rquir y th
More informationReliability of time dependent stress-strength system for various distributions
IOS Joural of Mathmatcs (IOS-JM ISSN: 78-578. Volum 3, Issu 6 (Sp-Oct., PP -7 www.osrjourals.org lablty of tm dpdt strss-strgth systm for varous dstrbutos N.Swath, T.S.Uma Mahswar,, Dpartmt of Mathmatcs,
More informationGRAPHS IN SCIENCE. drawn correctly, the. other is not. Which. Best Fit Line # one is which?
5 9 Bt Ft L # 8 7 6 5 GRAPH IN CIENCE O of th thg ot oft a rto of a xrt a grah of o k. A grah a vual rrtato of ural ata ollt fro a xrt. o of th ty of grah you ll f ar bar a grah. Th o u ot oft a l grah,
More informationEuropean Business Confidence Survey December 2012 Positive expectations for 2013
Dcmbr 2012 Erpa Bsiss Cfic rv Dcmbr 2012 Psitiv xpctatis fr 2013 Lasrp a Ivigrs EMEA hav rctl cmplt thir latst Erpa Bsiss Cfic rv. Th fiigs sggst a psitiv start t 2013 a a mr ptimistic tlk cmpar t that
More informationDifferentiation of Exponential Functions
Calculus Modul C Diffrntiation of Eponntial Functions Copyright This publication Th Northrn Albrta Institut of Tchnology 007. All Rights Rsrvd. LAST REVISED March, 009 Introduction to Diffrntiation of
More informationOn the approximation of the constant of Napier
Stud. Uiv. Babş-Bolyai Math. 560, No., 609 64 O th approximatio of th costat of Napir Adri Vrscu Abstract. Startig from som oldr idas of [] ad [6], w show w facts cocrig th approximatio of th costat of
More informationDFT: Discrete Fourier Transform
: Discrt Fourir Trasform Cogruc (Itgr modulo m) I this sctio, all lttrs stad for itgrs. gcd m, = th gratst commo divisor of ad m Lt d = gcd(,m) All th liar combiatios r s m of ad m ar multils of d. a b
More informationMor Tutorial at www.dumblittldoctor.com Work th problms without a calculator, but us a calculator to chck rsults. And try diffrntiating your answrs in part III as a usful chck. I. Applications of Intgration
More informationAdditional Math (4047) Paper 2 (100 marks) y x. 2 d. d d
Aitional Math (07) Prpar b Mr Ang, Nov 07 Fin th valu of th constant k for which is a solution of th quation k. [7] Givn that, Givn that k, Thrfor, k Topic : Papr (00 marks) Tim : hours 0 mins Nam : Aitional
More informationExercises for lectures 23 Discrete systems
Exrciss for lcturs 3 Discrt systms Michal Šbk Automatické říí 06 30-4-7 Stat-Spac a Iput-Output scriptios Automatické říí - Kybrtika a robotika Mols a trasfrs i CSTbx >> F=[ ; 3 4]; G=[ ;]; H=[ ]; J=0;
More informationUNIT #5. Lesson #2 Arithmetic and Geometric Sequences. Lesson #3 Summation Notation. Lesson #4 Arithmetic Series. Lesson #5 Geometric Series
UNIT #5 SEQUENCES AND SERIES Lesso # Sequeces Lesso # Arithmetic ad Geometric Sequeces Lesso #3 Summatio Notatio Lesso #4 Arithmetic Series Lesso #5 Geometric Series Lesso #6 Mortgage Paymets COMMON CORE
More informationElectrochemistry L E O
Rmmbr from CHM151 A rdox raction in on in which lctrons ar transfrrd lctrochmistry L O Rduction os lctrons xidation G R ain lctrons duction W can dtrmin which lmnt is oxidizd or rducd by assigning oxidation
More informationChemical Physics II. More Stat. Thermo Kinetics Protein Folding...
Chmical Physics II Mor Stat. Thrmo Kintics Protin Folding... http://www.nmc.ctc.com/imags/projct/proj15thumb.jpg http://nuclarwaponarchiv.org/usa/tsts/ukgrabl2.jpg http://www.photolib.noaa.gov/corps/imags/big/corp1417.jpg
More informationSCHUR S THEOREM REU SUMMER 2005
SCHUR S THEOREM REU SUMMER 2005 1. Combinatorial aroach Prhas th first rsult in th subjct blongs to I. Schur and dats back to 1916. On of his motivation was to study th local vrsion of th famous quation
More informationLectures 2 & 3 - Population ecology mathematics refresher
Lcturs & - Poultio cology mthmtics rrshr To s th mov ito vloig oultio mols, th olloig mthmtics crisht is suli I i out r mthmtics ttook! Eots logrithms i i q q q q q q ( tims) / c c c c ) ( ) ( Clculus
More information1973 AP Calculus AB: Section I
97 AP Calculus AB: Sction I 9 Minuts No Calculator Not: In this amination, ln dnots th natural logarithm of (that is, logarithm to th bas ).. ( ) d= + C 6 + C + C + C + C. If f ( ) = + + + and ( ), g=
More information(c) Write, but do not evaluate, an integral expression for the volume of the solid generated when R is
Calculus BC Fial Review Name: Revised 7 EXAM Date: Tuesday, May 9 Remiders:. Put ew batteries i your calculator. Make sure your calculator is i RADIAN mode.. Get a good ight s sleep. Eat breakfast. Brig:
More informationSOLVED EXAMPLES. Ex.1 If f(x) = , then. is equal to- Ex.5. f(x) equals - (A) 2 (B) 1/2 (C) 0 (D) 1 (A) 1 (B) 2. (D) Does not exist = [2(1 h)+1]= 3
SOLVED EXAMPLES E. If f() E.,,, th f() f() h h LHL RHL, so / / Lim f() quls - (D) Dos ot ist [( h)+] [(+h) + ] f(). LHL E. RHL h h h / h / h / h / h / h / h As.[C] (D) Dos ot ist LHL RHL, so giv it dos
More informationPower Spectrum Estimation of Stochastic Stationary Signals
ag of 6 or Spctru stato of Stochastc Statoary Sgas Lt s cosr a obsrvato of a stochastc procss (). Ay obsrvato s a ft rcor of th ra procss. Thrfor, ca say:
More informationThomas J. Osler. 1. INTRODUCTION. This paper gives another proof for the remarkable simple
5/24/5 A PROOF OF THE CONTINUED FRACTION EXPANSION OF / Thomas J Oslr INTRODUCTION This ar givs aothr roof for th rmarkabl siml cotiud fractio = 3 5 / Hr is ay ositiv umbr W us th otatio x= [ a; a, a2,
More informationCDS 101: Lecture 5.1 Reachability and State Space Feedback
CDS, Lctur 5. CDS : Lctur 5. Rachability ad Stat Spac Fdback Richard M. Murray 7 Octobr 3 Goals: Di rachability o a cotrol systm Giv tsts or rachability o liar systms ad apply to ampls Dscrib th dsig o
More informationFrequency Measurement in Noise
Frqucy Masurmt i ois Porat Sctio 6.5 /4 Frqucy Mas. i ois Problm Wat to o look at th ct o ois o usig th DFT to masur th rqucy o a siusoid. Cosidr sigl complx siusoid cas: j y +, ssum Complx Whit ois Gaussia,
More informationMathematics 1110H Calculus I: Limits, derivatives, and Integrals Trent University, Summer 2018 Solutions to the Actual Final Examination
Mathmatics H Calculus I: Limits, rivativs, an Intgrals Trnt Univrsity, Summr 8 Solutions to th Actual Final Eamination Tim-spac: 9:-: in FPHL 7. Brought to you by Stfan B lan k. Instructions: Do parts
More informationCIVE322 BASIC HYDROLOGY Hydrologic Science and Engineering Civil and Environmental Engineering Department Fort Collins, CO (970)
CVE322 BASC HYDROLOGY Hydrologic Scic ad Egirig Civil ad Evirotal Egirig Dpartt Fort Collis, CO 80523-1372 (970 491-7621 MDERM EXAM 1 NO. 1 Moday, Octobr 3, 2016 8:00-8:50 AM Haod Auditoriu You ay ot cosult
More informationCh. 24 Molecular Reaction Dynamics 1. Collision Theory
Ch. 4 Molcular Raction Dynamics 1. Collision Thory Lctur 16. Diffusion-Controlld Raction 3. Th Matrial Balanc Equation 4. Transition Stat Thory: Th Eyring Equation 5. Transition Stat Thory: Thrmodynamic
More informationSection 7 Fundamentals of Sequences and Series
ectio Fudametals of equeces ad eries. Defiitio ad examples of sequeces A sequece ca be thought of as a ifiite list of umbers. 0, -, -0, -, -0...,,,,,,. (iii),,,,... Defiitio: A sequece is a fuctio which
More informationY 0. Standing Wave Interference between the incident & reflected waves Standing wave. A string with one end fixed on a wall
Staning Wav Intrfrnc btwn th incint & rflct wavs Staning wav A string with on n fix on a wall Incint: y, t) Y cos( t ) 1( Y 1 ( ) Y (St th incint wav s phas to b, i.., Y + ral & positiv.) Rflct: y, t)
More informationFirst derivative analysis
Robrto s Nots on Dirntial Calculus Chaptr 8: Graphical analysis Sction First drivativ analysis What you nd to know alrady: How to us drivativs to idntiy th critical valus o a unction and its trm points
More informationy cos x = cos xdx = sin x + c y = tan x + c sec x But, y = 1 when x = 0 giving c = 1. y = tan x + sec x (A1) (C4) OR y cos x = sin x + 1 [8]
DIFF EQ - OPTION. Sol th iffrntial quation tan +, 0
More information