Chapter 10 An Introduction to Calculus: Limits, Derivatives, and Integrals

Transcription

1 86 Capter 0 A Itroductio to Calculus: Limits, Derivatives, ad Itegrals Capter 0 A Itroductio to Calculus: Limits, Derivatives, ad Itegrals Sectio 0. Limits ad Motio: Te Taget Problem Eploratio.. v ave s t ft sec ft>sec. Te are te same.. As te slope of te lie joiig (a, s(a)) ad (b, s). - + or + 6. m # + - # # + - # Sectio 0. Eercises.. m - - Quick Review m m v ave s t v ave s t , miles mi per our.7 ours 0 km 0 km per our. ours + 7 s + - s. s'() s + - s. s'() # + - # s + - s. s'() a a + a + a (a+a)a s + - s 6. s'() # # f - f0 7. Tr - 0 f - f 8. Tr - 9. No taget 0. No taget - - -

2 Sectio 0. Limits ad Motio: Te Taget Problem 87. (c) 9 [ 7, 9] b [, 9] m. [ 0, ] b [ 7, ] m. f + - f 8. (a) m Sice (, f())(, 0) te equatio of te taget lie is (-). (c) m. [ 0, ] b [, ] [, ] b [, ] m f0 + - f0. (a) f'(0) Te iitial velocit of te rock is f'(0)8 ft/sec. f0 + - f0 6. (a) f'(0) Te iitial velocit of te rock is f'(0)70 ft/sec. f f - 7. (a) m Sice (, f( ))(, ) te equatio of te taget lie is - (+). f + - f 9. (a) m Sice (, f())(, ) te equatio of te taget lie is +(-),or-. (c)

3 88 Capter 0 A Itroductio to Calculus: Limits, Derivatives, ad Itegrals f + - f 0. (a) m # +. Sice (, f()) a, te equatio of te taget b (c) lie is At : m, at : m, at 0, m does ot eist.. [, ] b [, ] At : m0., at : m0., at 0, m0.. f + - f f + - f a + b [, ] b [, ] - + # f f f + - f f f - ƒ - + ƒ - # We >0, wile we <0,. Te it does ot eist. Te derivative does - ot eist f f # # f'() f'() f'() f'() # #

4 Sectio 0. Limits ad Motio: Te Taget Problem (a) Betwee 0. ad 0.6 secods: ft/sec Betwee 0.8 ad 0.9 secods: ft/sec f() , time i secods [ 0., ] b [ 0., 8] (c) f().9 ft (a) -7. ft/sec s(t) 6.0t +.t+0. [0,.] b [0, ] st + - st (c) s'(t) -6.0t + +.t t +.t t (.0t ).0t+.; s ().0() At t, te velocit is about 0.60 ft/sec.. (a) 9 6. (a) 0 Sice te grap of te fuctio does ot ave a defiable slope at, te derivative of f does ot eist at. Te fuctio is ot cotiuous at. (c) Derivatives do ot eist at poits were fuctios ave discotiuities. From te grap of te fuctio, it appears tat te derivative ma eist at. Usig te first defiitio of te derivative ad takig secat lies o te left of (so tat f()+(-) ), we ave f - f S - S S - S

5 90 Capter 0 A Itroductio to Calculus: Limits, Derivatives, ad Itegrals Now, takig secat lies o te rigt of (so tat f()-(-) ), we ave f - f S - S ( - )0. S - S Sice te its are te same, f () eists at ad f ()0. 7. (a) 9. Aswers will var. Oe possibilit: Aswers will var. Oe possibilit: Sice te grap of te fuctio does ot ave a defiable slope at, te derivative of f does ot eist at. Te fuctio is ot cotiuous at. (c) Derivatives do ot eist at poits were fuctios ave discotiuities. 8. (a). Aswers will var. Oe possibilit: π From te grap of te fuctio, it appears tat te derivative ma eist at 0. si - f0 + - f0 f (0) si - Tis it caot be foud usig algebraic teciques. Te table of values below suggests tat tis it equals 0. si Te grap supports tis sice it appears tat tere is a orizotal taget lie at 0. Tus, f'(0)0.. Aswers will var. Oe possibilit:. Sice f()a+b is a liear fuctio, te rate of cage for a is eactl te slope of te lie. No calculatios are ecessar sice it is kow tat te slope af'(). f 0@. f'(0) Lookig at secat lies, we see tat tis it S0 does ot eist. If te secat lie is to te left of 0,it will ave slope m, wile if it is to te rigt of 0, it will ave slope m. At 0, te grap of te fuctio does ot ave a defiable slope.

6 Sectio 0. Limits ad Motio: Te Area Problem 9. False. Te istataeous velocit is a it of average velocities. It is ozero we te ball is movig. 6. True. Bot te derivative ad te slope equal f - fa. Sa - a 7. For Y +-, at 0 te calculator sows d/d. Te aswer is D. 8. For Y -, at te calculator sows d>d -7. Te aswer is A. 9. For Y, at te calculator sows d/d. Te aswer is C. 0. For Y, at te calculator sows - d/d 0.. Te aswer is A.. (a). (a) Te average velocit is ^s 8 ft/sec. ^t Te istataeous velocit is (96+)96 ft/sec 6. (a) g m/sec t Average speed: ^ ^t m/sec (c) Sice t, te istataeous speed at t is m/sec 7. [.7,.7] b [.,.] No, tere is o derivative because te grap as a corer at 0. No. (a) [.7,.7] b [.,.] No, tere is o derivative because te grap as a cusp ( spike ) at 0. Yes, te taget lie is 0.. (a) Sectio 0. Limits ad Motio: Te Area Problem [.7,.7] b [.,.] No, tere is o derivative because te grap as a vertical taget (o slope) at 0. Yes, te taget lie is 0.. (a) [.7,.7] b [.,.] Yes, tere is a derivative because te grap as a overtical taget lie at 0. Yes, te taget lie is. Eploratio. Te total amout of water remais gallo. Eac of te gal 0 teacups olds 0. gallo of water. 0. Te total amout of water remais gallo. Eac of te gal 00 teacups olds gallo of water Te total amout of water remais gallo. Eac of te gal,000,000,000 teacups olds ,000,000,000 gallo of water.. Te total amout of water remais gallo. Eac of te teacups olds a amout of water tat is less ta wat was i eac of te billio teacups i step. Tus eac teacup olds about 0 gallos of water.

7 9 Capter 0 A Itroductio to Calculus: Limits, Derivatives, ad Itegrals Quick Review ,,,,,,, 8,, ,,,,,,,,, [ ] (+) ± (+)+() +(-) (+)+(+)+(+)+...+(+)+(+) Tus a k + (+), ad k a k + (+). k. 0 [ ] 6. [ (-) + ] c d 6 7. (7 mp)( ours)8 miles 8. a gal (0 mi)600 gallos mi b ft 60 miutes 60 secods a b (6 ours) a b a sec our miute b,0,000 ft 60 people 0. a b (,000 mi )9,600,000 people mi Sectio 0. Eercises. Let te lie 6 represet te situatio. Te area uder te lie is te distace traveled, a rectagle, (6)()9 miles.. Let te lie represet te situatio. Te area uder te lie is te umber of gallos pumped, a rectagle, ()(0)0 gallos.. Let te lie 0 represet te situatio. Te area uder te lie is te total umber of cubic feet of water pumped, a rectagle, (0)(600)0,000 ft.. Let te lie 60 represet te situatio. Te area uder te lie is te total populatio, a rectagle, (60)(0),000 people.. Δs Δs Δt # Δt 60 km>. 76 km 6. Δs Δs # Δt mi>a b 6 mi Δt 6 7. a # fk f()+f()+f()+f()+f() k (aswers will var) 8. a # fk f()+f()+f()+f()+f() k (aswers will var) 9. a # fk f(0.)+f(.)+f(.)+f(.)+f(.) k (aswers will var) 0. a # fk f(0.)+f(.)+f(.)+f(.)+f(.) k (aswers will var).. 8 a 0 - i Δ i i square uits 8 a 0 - i Δ i i square uits. c 0,, c, c,, c d, d d, d. c 0,, c, c, c, c,, c, c, c 7, 7,,, d d d, d d d d, d. c,, c, c,, c, c, 7, c 7 d, d d, d d, d 6. c,, c, c,, c, c, 7, c 7, c, 9, c 9 d, d d, d d, d d, d For #7 0, te itervals are of widt, so te area of eac rectagle is f(k)f(k). 7. (a) 8 8 # RRAM: f()+f()+f()+f()

8 Sectio 0. Limits ad Motio: Te Area Problem 9 (c) 8 9. (a) LRAM: f(0)+f()+f()+f() (d) Average: 8. (a) RRAM: f()+f()+f()+f() (c) 0 RRAM: f()+f()+...+f(6) (c) LRAM: f(0)+f()+f()+f() (d) Average: 0 0. (a) LRAM: f(0)+f()+...+f() (d) Average: 8 0 RRAM: f()+f()+f()+8+76

9 9 Capter 0 A Itroductio to Calculus: Limits, Derivatives, ad Itegrals (c) 0. (+) d6. (Trapezoid wit bases ad 7 ad altitude ) LRAM: f(0)+f()+f() (d) Average: 7. d0 (Rectagle wit base ad eigt ) [, 6] b [, ] 6. (-) d6. (Trapezoid wit bases ad 0 ad altitude ) [, 6] b [, ] [, 0] b [, 7]. 6 d0 (Rectagle wit base ad eigt 6) d (Semicircle wit radius ) - [, ] b [ 0., ] [, 0] b [, 7]. d7. (Triagle wit base ad altitude ) d9 (Quarter circle wit radius 6) 0 [, 8] b [ 0., 6] 7 [, 6] b [, 0]. 0. d (Trapezoid wit bases of 0. ad. ad eigt 6) 9. si d (Oe arc of sie curve) 0 [, ] b [, ] [, 8] b [, ]

10 Sectio 0. Limits ad Motio: Te Area Problem 9 0. (si +) d+ (Arc of sie curve plus 0 rectagle wit base ad eigt ). si d (Rectagles i sum are twice as tall, 0 ieldig twice te sum) [, ] b [, ] +. si (-) d (Oe arc of sie curve traslated uits rigt) [, ] b [, ] 6. si a d (Rectagles i sum are twice as b 0 wide, ieldig twice te sum) [, ] b [, ] >. cos d (Oe arc of cosie curve, wic is - > sie curve traslated / uits) [, ] b [, ] 7. si d (Two arces of te sie curve) 0 [, ] b [, ] >. si d (Half-arc of sie curve) 0 [, ] b [, ] > 8. cos d (Two-ad-a-alf arces of te cosie - curve) [, ] b [, ] >. cos d (Half-arc of cosie curve, cogruet 0 to alf-arc of sie curve) [, ] b [, ] 9. Te grap of f()k+ is a lie. If k is a umber betwee 0 ad, te itegral is te area of a trapezoid wit bases of 0k+ ad k+ ad eigt of -0. Te area is + k + k + 6 8k+, so k + d 8k Te grap of f()+ is a lie. Te itegral is te area of a trapezoid wit bases of # 0 + ad k+ ad eigt of k-0k. Te area is k + k + kk + 6k +k, so k [, ] b [, ] + d k + k. 0

11 96 Capter 0 A Itroductio to Calculus: Limits, Derivatives, ad Itegrals. Te grap of f()+k is a lie. Te itegral is te area of a trapezoid wit bases of # 0 + k k ad # + k + k ad eigt of -0. Te area is k + + k(+k)+k, so + kd + k. 0. Te grap of f()+ is a lie. Te itegral is te area of a trapezoid wit bases of k+ ad # + 9 ad eigt of -k. Te area is - kk k + k (-k)(+k)-k -k. Sice g() f(), we cosider g to be smmetric wit f about te -ais. For ever value of i te iterval, f() is te distace to te -ais ad similarl, g() is te distace to te -ais; f() ad g() are equidistat from te -ais. As a result, te area uder f() must be eactl equal to te area above g().. Te grap of f() 6 - is te top alf of a circle of radius. Te area of te grap from 0 to is te area of of te etire circle. Tus te desired area is p # 6p p.. Te distace traveled will be te same as te area uder te velocit grap, v(t)t, over te iterval [0, ]. Tat triagular regio as a area of A(/)()(6)6. Te ball falls 6 feet durig te first secods. 6. Te distace traveled will be te same as te area uder te velocit grap, v(t)6t, over te iterval [0, 7]. Tat triagular regio as a area of A(/)(7)()7. Te car travels 7 feet i te first 7 secods. 7. (a) [0, ] b [0, 0] Te ball reaces its maimum eigt we te velocit fuctio is zero; tis is te poit were te ball cages directio ad starts its descet. Solvig for t we 8-t0, we fid t. sec. (c) Te distace te ball as traveled is te area uder te curve, a triagle wit base. ad eigt 8 tus, d0.(.)(8)6 uits. 8. (a) Te rocket reaces its maimum eigt we te velocit fuctio is zero; tis is te poit were te rocket cages directio ad starts its descet. Solvig for t we 70-t0, t. sec. (c) Te distace te rocket as traveled is te area uder te curve, a triagle wit base. ad eigt 70 tus, d (70)(.).6 ft. 9. (a) [0, ] b [ 0, 0] Eac RRAM rectagle will ave widt 0.. Te eigts (usig te absolute value of te velocit) are.0,., 7.6,., 0.6, 7.06, ad.7. Te eigt of te buildig is approimatel 0.[ ].86 feet. 0. Work is defied as force times distace. Te work doe i movig te barrel feet is te area uder te curve created b te give data poits, assumig te barrel weigs approimatel 0 lbs after beig moved feet. I tis case, te area uder te curve is te sum of a rectagle of widt ad eigt 0 ad a triagle of base ad eigt (0-0)700. Te total work performed is ()(0)+ ()(700),00 ft-pouds.. True. Te eact area uder a curve is give b te it as approaces ifiit. Tis is true weter LRAM or RRAM is used.. False. Te statemet f L meas tat f() gets Sq arbitraril close to L as gets arbitraril large.. Sice represets a vertical stretc, b a factor of, of, te area uder te curve betwee 0 ad 9 is doubled. Te aswer is A.. Sice + represets a vertical sift, b uits upward, of, te area is icreased b te cotributio of a 9-b- rectagle a area of square uits. Te aswer is E.. - is sifted rigt uits compared to, but te its of itegratio are sifted rigt uits also, so te area is ucaged. Te aswer is C. 6. represets a orizotal compressio, b a factor of /, ad te iterval of itegratio is sruk i te same wa. So te ew area is / of te old area. Te aswer is D. [0, 8] b [0, 80]

12 Sectio 0. More o Limits I te defiitio of te defiate itegral, if f() is egative, te a f( i ) is egative, so te defiite itegral is i egative. For f()si o [0, ], te positive area (from 0 to ) cacels te egative area (from to ), so te defiite itegral is 0. Sice g()- forms a triagle wit area below te -ais o [0, ], (-)d (a) 6 Sectio 0. More o Limits Eploratio. Aswers will var. Possible aswers iclude: Solvig grapicall or algebraicall sows tat 7 we, so we kow tat is te it. [0, ] b [, 0] A table of values also sows tat te value of te fuctio approaces as approaces from eiter directio. 6 Domai: ( q, ) ª (, q) Rage: {} ª (, q) Te area uder f from 0 to is a rectagle of widt ad eigt ad a trapezoid wit bases of ad ad eigt. It does ot reall make a differece tat te fuctio as o value at. 9. True b f()d+ g()d a a B a f( i ) R+ B a g( i ) R B a f( i ) + a g( i ) R Sq a [f( i )+g( i )] Sq i b (f()+g())d a Note: Tere are some subtleties ere, because te i tat are cose for f() ma be differet from te i tat are cose for g(); owever, te result is true, provided te its eist. 60. True, because multiplig te fuctio b 8 will multipl te area b False. Coutereample: Let f(), g(). Te f()g()d but f()d # g()d True, because (area from a to c)+(area from c to b) (area from a to b) 6. False. Itercagig a ad b reverses te sig of b - a, wic reverses te sig of te itegral. a - a 6. True. For a value of, 0. a f()d f( i ) f(a) # a a 0 Sq Sq a i i 00 Sq Sq i i b i Sq i. Aswers will var. Possible aswers iclude: Te graps suggest tat te it eists ad is. Because te grap is a lie wit a discotiuit at 0, tere is o asmptote at 0. A table of values also suggests tat te it is. To sow tat is te it ad.9999 is ot, we ca solve algebraicall. + + S0 S0 + S0 Eploratio. [, ] b [, ] [, ] b [ 0, 60] f 0, Sq f 0 S-q [, ] b [, ]

13 98 Capter 0 A Itroductio to Calculus: Limits, Derivatives, ad Itegrals. Te two orizotal asmptotes are 0 ad 0.. As Sq, - S 0 ad + - S. As S-q, - Sqad + - Sq. Quick Review (a) f( ) , 0 + f(0) 0-6, + (c) f() is udefied. - si -. (a) f( ) si L 0., - si 0 f(0) is udefied. 0 si (c) f() L 0.. (a) Sice -0 ad, te grap of f as vertical asmptotes at ad. Sice f() ad f(), te grap of S-q Sq f as a orizotal asmptote of.. (a) Sice +-0 we ad, te grap of f as vertical asmptotes at ad. Sice f()q ad f() q, te S-q Sq grap of f as o orizotal asmptotes.. Sice, te ed beavior asmptote is Sice, te ed beavior asmptote is (c). 7. (a) [, q) Noe 8. (a) ( q, ) ª (, ) ª (, q), ( 6) /.0 7. (e si()) e si() # 00 S0 8. l a si l()00 b Sp S0 S0 9. a - a - 0. (Sice a +>0 for all a, we do t ave to a + worr about divisio b zero.). (a) divisio b zero S- - 9 S S- 6. (a) divisio b zero S + - S. (a) divisio b zero ( -+) S- + S-. (a) divisio b zero - + ( +) S - S. (a) divisio b zero + - (-) S- + S- 6. (a) divisio b + Ceck left- ad rigt-ad S- + its Rigt: ( +) S- + + S Left: (-) S- - + S- - Sice Z, te it does ot eist. 7. (a) Te square root of egative umbers is ot defied i te real plae. Te it does ot eist. 8. (a) divisio b zero Te it does ot eist. si si si 9. # S0 - S0 S0 - # - S0 (Recall eample ad te product rule) 0. Cotiuous o ( q, ) ª (, q); discotiuous at Sectio 0. Eercises. ( )( ). () [, ] b [, ]

14 Sectio 0. More o Limits S0 si (Recall eample ad te product rule). [, ] b [, ] si si. S0 S0 0 # 0 [, ] b [, ] si # S0 S0 S0 # + si. S0 S0 # + # + S0 si si # si si S0 # si S0 9. (a) f() f() (c) f() 0. (a) f() S - S + S S - S + f() (c) Z, so te it does ot eist.. (a) True True (c) False (d) False (e) False (f) False (g) False () True (i) False (j) True. (a) True False (c) False (d) True (e) False (f) False (g) False () False (Te it does ot eist at 0.) (i) True. [, ] b [, ] I Eercises # 6, te fuctio is defied ad cotiuous at te value approaced b, ad so te it is simpl te fuctio evaluated at tat value.... S0 si - cos si 0 - cos 0 S0 si + cos si 0 + cos l S > si l si > S7 log log (a) f() f() (c) Z, so te it does ot eist. 8. (a) f() e - log + S - S + S - S + f() e0-0 log 0 + l (c) Z, so te it does ot eist. > l. (a) f().7 f().7 (c) f().7 (a) [, ] b [, ] S0 - S0 + S0 [, ] b [, ] S0 - S0 + f().6 f().6 (c) f().6 S0

15 00 Capter 0 A Itroductio to Calculus: Limits, Derivatives, ad Itegrals. (a) (g()+)+6 (sum rule) S # f() # f()( ) S S S (product rule) (c) g () g # g() # 6 S S S (product rule) g g S (d) f S f - S S (quotiet rule) 6. (a) (f()+g()) f + g Sa Sa Sa - - (sum rule) (f() # g()) f # g Sa Sa Sa - -6 (product rule) (c) (g()+) g + Sa Sa Sa (costat multiple ad sum rules) (d) 7. (a) f()0, f()0 (c) 8. (a) Sa S + f()0 S f(), f() S + f g 9 f Sa g - - Sa S - S - (quotiet rule) (c) Limit does ot eist because f() Z f(). S + S - 9. (a) 8 f()0, f() (c) Limit does ot eist because f() Z f(). S0 + S0-0. (a) f() 8, f() (c) Limit does ot eist because f() Z f(). S- + S- - For Eercises #, use Figure 0... it. it. it 0. it (). 6. S + S - S> - S (a) 8. (a) + si S-q (a) (+ )q 0. S0 - S0 + S- + Sq S-q Sq Sq S-q Sq S-q - - S0 - cos + 0 cos 0 + si S0 - Sq + -q + + S- + S- - + si + 0 Sq S-q + si S-q. (a) + si q Sq + si -q S -q. (a) is udefied, because e goes to S q e- + si zero but si oscillates. S -q e- + si q. (a) is udefied, because si oscillates S q -e si betwee positive ad egative values. S -q -e si 0

16 Sectio 0. More o Limits 0. (a) S q e- cos 0 is udefied, because cos oscillates S -q e- cos betwee positive ad egative values.. S - - -q; 60. S - is udefied; ad - 6. S + [, 9] b [, ] - q; S0 S0 + + S0 ( ++) S0 [, ] b [, ] 7. S- + [, 8] b [ 0, 0] q; S0 S0 [ 7, ] b [, 8] S0 a - + b - 9 # S0 - # + 8. S- - [ 7, ] b [, ] q; - - [ 6, ] b [, ] ta 6. S0 S0 # si cos si # S0 S0 cos 9. [ 7, ] b [, ] - q; S [, ] b [, ] [0, 9.] b [.,.]

17 0 Capter 0 A Itroductio to Calculus: Limits, Derivatives, ad Itegrals - 6. is udefied. S (a) π [, ] b [, ] S0 S0 0 S0 ( p, 0) ª (0, p) (c) p (d) p 80. (a) 67. c si a because S 0 S0 bd 0 ad - si a. b 68. cos a S7 b cos a 7 b L is udefied, sice + S - - -q S - ad + S + - q Sq Sq 7. Sq l l l l 7. False. S Sq Sq Sq l l l l 0 f f f - S S + 7. False. For eample, if f()si(/) ad g(), te f does ot eist but f # g 0. S0 S0 ( p, 0) ª (0, p) (c) 0 (d) p 8. (a) π S - S - +. Te aswer is B. S q, q. S - - S + - Te aswer is A q, -q. S - - S + - Te aswer is C S - S Te aswer is D. S (, 0) ª (0, ) (c) (d) 8. (a) 6 ( q, ) ª (, q) (c) Noe (d) Noe

18 Sectio 0. More o Limits 0 8. (a) f() were te umber of + 6.9e -0.9, mots f() 7.7 (c) Te rabbit populatio will stabilize at a little less ta 8,000. (d) Oe possible aswer: As populatios burgeo, resources suc as food, water, ad safe aves from predators become more scarce ad te populatio teds to stabilize based o te resources available to it tis is wat is ofte call a maimum sustaiable populatio [, ] b [0, 60] 0 Sq (a) ; 0; # q S0 ; 0; is ot defied q S0 (c) ; (-) 0; - q S S - - S S + - (-)0 S S S (-)0 S - Tus, fg0 S S0 S0 (d) ; (-) 0; - q S S q S S 8 (e) Oe possible aswer: Notig ca reall be said te it ma be udefied, q, or a umber. 89. (a) For a 8-sided polgo, we ave 8 isosceles triagles of area b. Tus, A 8 # bb. Similarl, for a -sided polgo, we ave triagles of area b. Tus A # # b b. S Cosider a -sided polgo iscribed i a circle of radius r. Sice a circle alwas is 60, we see tat eac agle etedig from te ceter of te circle to two 60 cosecutive vertices is a agle of. Droppig a perpedicular from te ceter of te circle to te midpoit of te base of te triagle (wic is also oe of 60 te sides) results i a agle of. Sice ta we ave, ad fiall b ta a 60 b. a 60 b b> b ta a 60 b (c) Sice A b ad b ta a 60, we ave b A a ta a 60 bb ta a 60 b.

19 0 Capter 0 A Itroductio to Calculus: Limits, Derivatives, ad Itegrals p p (d) A ,000.6, , ,000.6 Yes, A S p as S q. (e) A , , , , Yes, S q, A S 9p. (f) Oe possible aswer: A ta a 80 ta a 80 b b Sq Sq Sq pp As te umber of sides of te polgo icreases, te distace betwee ad te edge of te circle becomes progressivel smaller. As S q, S radius of te circle. 90. (a) 0 Sice () -()+-6+, a. 9. (a) a a a 9. (a) (c) g() 9. (a) (c) g() ++ Sectio 0. Numerical Derivatives ad Itegrals Eploratio. RRAM value.6070 ad te NINT value Te ew commad is sum(seq(/(+ K # >00 # >00, K,, 00)). Te calculated value.7, wic is a better approimatio ta for 0 rectagles. p. Te itegral is si d. Te RRAM value 0 is.999 ad te NINT value is.. Te commad is sum(seq( ( + K # >0 # >0, K,, 0)). Te calculated value.766 ad te NINT value is f() + (c) g() + + Quick Review log - log

20 Sectio 0. Numerical Derivatives ad Itegrals si.0 - si l.00 - l e e Eercises 0. I # 0, use NDER o a calculator to fid te umerical derivative of te fuctio at te specific poit I #, use NINT o a calculator to fid te umerical itegral of te fuctio over te specified iterval mi. 6.9 mi. (a) v ave [, 6] b [0, 0] -0-0 ft>sec (c) st L -6.08t + 0.6t (d) v. sec L ft>sec (e) Set s(t) equal to zero ad solve for t usig te quadratic equatio. t sec (Te mius sig was cose to give t 0.) Usig NDER at t.86 sec gives v 79.8 ft/sec.. (a) Te average rate of cage betwee two data poits is foud b eamiig Te average rate of cage. of te gross domestic product from is $.7 billio per ear. Te average rate of cage of te gross domestic product 0,80.8-0,00.8 from is $ billio per ear. Te quadratic regressio model for te data is [, ] b [000, 000] p (c) To estimate te rate of cage of te gross domestic product i 997, we will use te calculator NDER computatio ad evaluate tat we 7 to obtai $.6 billio per ear. To estimate te rate of cage of te gross domestic product i 00, we will use te calculator NDER computatio ad evaluate tat we to obtai $6. billio per ear. (d) Usig te quadratic regressio model to predict te gross domestic product i 007 were 7 ields $,089. billio or $. trillio. Tis aswer is reasoable because te gross domestic product will probabl cotiue to icrease.. (a) Te midpoits of te subitervals will be 0., 0.7,., etc. Te average velocities will be te successive eigt differeces divided b 0. tat is, times. Midpoit s/ t 0. 0 ft/sec

21 06 Capter 0 A Itroductio to Calculus: Limits, Derivatives, ad Itegrals [0, 6] b [ 80, 0].8t+0. (c) Substitutig t. leads to 7.9 ft/sec. Tis is close to te value of 7.88 ft/sec foud i Eercise d. 6. (a) Let 990 be 0. Te first subiterval as a midpoit 0 + of.. O tat iterval, te average rate of cage is Te rest of te midpoits ad values of / are computed similarl ad are sow i te followig table. Midpoit / [0, ] b [0, 60] p (c) Te liear regressio model for te data is Usig tis regressio model to approimate te rate of cage i 997 ields 7.9(7) Usig te same model to approimate te rate of cage i 00 ields 7.9()+6... Tese values are reasoabl close to te results of NDER from Eercise c. 7. Te average velocities, s/ t, for te successive 0.-secod itervals are 8,, 0, 6, ad 7 ft/sec. Multiplig eac b 0. sec ad te summig tem gives te estimated distace: 00 ft. 8. Te average velocities, s/ t, for te successive 0.-secod itervals are as give: Iterval s/ t.09 m/sec Multiplig eac s/ t b 0. sec ad te summig tem gives te estimated distace:.07 m. 9. Te program accepts iputs wic determie te widt ad umber of approimatig rectagles. Tese rectagles are summed ad te result is output to te scree. 0. Te program accepts iputs wic determie te widt ad umber of approimatig rectagles. Tese rectagles are summed ad te result is output to te scree. For #, verif te fuctio is o-egative b grapig it over te iterval.. N LRAM RRAM Average (c) flt gives 7.; at N 00, te average is 7... N LRAM RRAM Average (c) flt gives ; N 00 is ver close at.0.. N LRAM RRAM Average (c) flt gives 9.; at N 00, te average is 9... N LRAM RRAM Average (c) flt gives 0; N 00 as te same result.. N LRAM RRAM Average (c) flt gives 0.; at N 00, te average is N LRAM RRAM Average (c) flt gives 60, ver close to N 00 of N LRAM RRAM Average (c) flt gives 7.9, te same result as N 00.

22 Sectio 0. Numerical Derivatives ad Itegrals N LRAM RRAM Average (c) flt gives.67, te same result as N N LRAM RRAM Average (c) flt gives, te same result as N N LRAM RRAM Average (c) flt0.7, te same result as N 00.. N LRAM RRAM Average (c) flt0.9, te same result as N 00.. N LRAM RRAM Average (c) flt gives.0, wic is te same result as N 00.. True. Te otatio NDER refers to a smmetric differece quotiet usig False. NINT will var te value of util te umerical itegral gets close to a itig value.. NINT will use as ma rectagles as are eeded to obtai a accurate estimate. Te aswer is B. (Note: NDER estimates te derivative, ot te itegral.) 6. Te most accurate estimate is a smmetric differece quotiet wit a small (ad of course wit, ot, i te deomiator). Te aswer is E. 7. Istataeous velocit is te derivative, ot a itegral, of te positio fuctio. Te aswer is C. 8. Area uder a curve tat represets f() is a itegral, ot te derivative, of f(). Te aswer is D. 9. (a) f'() g'() f.00 - f (c) Stadard: f.00 - f.999 Smmetric: (d) Te smmetric metod provides a closer approimatio to f'(). g.00 - g (e) Stadard: L g.00 - g.999 Smmetric: Te smmetric metod provides a closer approimatio to g'(). 0. (a) [0, ] b [0, ].7 (c).7 (d) Aswers will var but te true aswer is e.7. f0 + - f0-0. f'(0) if 0 is approaced from te rigt, ad if 0 is approaced from te left. Tis occurs because calculators ted to take average values for derivatives istead of applig te defiitio. For eample, a calculator ma calculate te derivative of f(0) b takig f f [, ] b [, ] f'(0) does ot eist because f() is discotiuous at 0. Te calculator gives a icorrrect aswer, NDER f(0)000, because it divides b 0.00 istead of lettig S 0.

23 08 Capter 0 A Itroductio to Calculus: Limits, Derivatives, ad Itegrals. (a) Let abs(si ()), wic is f(). Te NINT (Y, X, 0, ) gives. Let abs( --), wic is f(). Te NINT (Y, X, 0, ) gives Some fuctios, suc as ave a sigularit, te - cease to eist at tat poit ad te q. Usig S - our rectagular approimatios, owever, we ca fid te area uder te curve wit successivel smaller widts. Sice eac of tese widts are fiite, we simpl determie ow close our approimatio must be to determie te fiite area uder te curve. Evetuall te rectagle et to becomes so ti as to reder its area close eoug to zero to be igored.. Sice f() g() for all values of o te iterval, A bba b - a N b f a a + kb - a b R NSq a k N -B a b - a N b g a a + kb - a b Rr N a b - a N b Bf a a + kb - a b NSq a k N kb - a - g a a + b R. N If te area of bot curves is alread kow ad f() g() for all values of, te area betwee te curves is simpl te area uder f mius te area uder g. 6. A() (d) Te data seem to support a curve of A(). f + - f + - (e) A'() +. Te two fuctios are eactl te same. 7. A() (c) f() [0, ] b [, 0] [, ] b [, 0] (d) Te eact value of A() for a greater ta zero appears to be. f + - f + - (e) A'() +. Te fuctios are eactl te same. 8. Aswers ma var. (c) [0, ] b [, 0] [, ] b [, ]

24 Capter 0 Review 09 Capter 0 Review. (a) Does ot eist.. (a) Does ot eist.. (a). (a). S [, ] b [, ] si si cos + cos si 6. S0 S0 si cos cos - cos + cos - si - si S0 a si [8 cos -6 cos +( cos -)(- si )] b S0 # [8-6+(-)(-0)] [, ] b [, ] S- + S- si - 9. ta () S0 S0 cos - # S [ 9., 9.] b [., 0.] S + S f S - S - [, ] b [, ] - -q 0. μ udefied - S0 S0 - - q S0 + [, 6] b [, 8] [ 6, 6] b [, ]

25 0 Capter 0 A Itroductio to Calculus: Limits, Derivatives, ad Itegrals S + S - f() p 0 p (f) p p ,00 f() p.6 0 p f() , q - -q S-q Sq Sq S-q f()0 f() f() -q f() -q # S0 S0 + - # - S S0 S0 +6+ S0-9. f() so f as vertical asmptotes at + +, ad. Sice f()0 ad Sq S-q f()0, f also as a orizotal asmptote at f() so f as a vertical asmptote at. -, Sice f()q ad f() q, f as o Sq S-q orizotal asmptotes S - S S - S. S0 # S0 - S. - si - # S cos si - - si - S - si - a S - (-0 # ) a cos - 6 b - -. S (-)0 S - + S0 si - 6 cos - 6 # + - # si - si S ( ++) S - S - Z - F() S - S Z - F() # cos - 6 b

26 Capter 0 Review f + - f 9. f'() ( -9) 9 f + - f 0. f'() f.0 - f (a) f + - f f.0 - f. (a).0 - f + - f # f + - f. (a) m (, f())(, 0) so te equatio of te taget lie at is -. f (a) m (7, f(7))(7, ) so te equatio of te taget lie at 7 is. 6-6 # L f + - f f + - f For #7 8, verif te fuctio is o-egative troug grapical or umerical aalsis. 7. LRAM:.976 RRAM: Average: LRAM: 9. RRAM: Average: (a) Usig 0 for 990, te scatter plot of te data is: [, ] b [0, 00] Te average rate of cage for 990 to 99 is foud b eamiig cets per ear. - 0 Te average rate of cage for 997 to 998 is foud b eamiig cets per ear. - 0 (c) Te average rate eibits te greatest icrease from oe ear to te et cosecutive ear i te iterval from (d) Te average rate eibits te greatest decrease from oe ear to te et cosecutive ear i te iterval from

27 Capter 0 A Itroductio to Calculus: Limits, Derivatives, ad Itegrals (e) Te liear regressio model for te data is: [, ] b [0, 00] (f) Te cubic regressio model for te data is: f()si [0, 7] b [.,.] [0, 7] b [.,.] Most of te data poits touc te regressio curve, wic would suggest tat tis is a fairl good model. Oe possible aswer: Te cubic regressio model is te best. It sows a larger rate of icrease i fuel prices, wic is wat is appeig i toda s market. (g) Te cubic regressio model for te data is: Usig NDER wit tis regressio model ields te followig istataeous rates of cage for: 997. cets per gallo 998. cets per gallo 999. cets per gallo cets per gallo Te cubic regressio model is sowig a rate of icrease. Te true rate of cage, owever, ma be iger ta wat te model idicates. () If we use te give cubic regressio model, te average price of a gallo of regular gas i 007 will be 0. cets per gallo. Based o wat is appeig wit te curret fuel prices, tis predictio ma ot be ig eoug. 0. (a) [, ] b [0, 00] 0.6 A() (c) f'()cos, te fuctio beig itegrated. (d) Te derivative of NINT(f(t), t, 0, ) gives f(t). Capter 0 Project. Te scatter plot of te populatio data for Clark Cout, NV, is as follows. Te ear 970 is represeted b t0. [, ] b [0, ]. Te average populatio growt rate from is,7,7-77,0,8,07,97 people/ear Te average populatio growt rate from is,7,7-6,087,,0,77 people/ear Te average populatio growt rate from is,7,7-770,80 9,07 67,0 people/ear Te average populatio growt rate from is,7,7 -,0, 69,90 7, people/ear Te epoetial regressio model for te populatio data is 7,66.87 # t.. If we use NDER wit te give epoetial regressio model, te istataeous populatio growt rate i 00 is 9,9 people/ear. Te average growt rate from most closel matces te istataeous growt rate of 00.. Usig te epoetial regressio model 7,66.87 # t to predict te populatio of Clark Cout for te ears 00, 00, ad 00 ields: 7,66.87 # ,8,0 people 7,66.87 # ,099, people 7,66.87 # ,0,86 people Te web site predictios are probabl more reasoable, sice te give data ad scatter plot suggest tat growt i recet ears as bee fairl liear.