Chapter 5 The Definite Integral

Transcription

1 Sectio. Chpter The Defiite Itegrl Sectio. Estimtig with Fiite Sums (pp. -) Eplortio Which RAM is the Biggest?.. LRAM > MRAM > RRAM, ecuse the heights of the rectgles decrese s ou move towrd the right uder decresig fuctio. Quic Review.. mph i hr mi. mph i hr mi. ft/sec i sec ft/sec mi sec ft/sec i i. mph ft h sec hr ds., m /seci i i i r hr d r. m. ( mph)( h) + ( mph)( h) mi + mi mi mi. gl/mi i h i gl h. ( C/h)( h) + (. C)( h) C sec h. ft /sec i i i d,, ft h d. people/mi i mi, people LRAM > MRAM > RRAM. MRAM > RRAM > LRAM. RRAM > MRAM > LRAM, ecuse the heights of the rectgles icrese s ou move towrd the right uder icresig fuctio. sec. times/sec i i h i., times h Sectio. Eercises. Sice v(t) is strit lie, compute the re uder the curve. () t v() t ( )( ). Sice vt () t+ cretes trpezoid with the -is, compute the re of the curve uder the trpezoid. A h ( + ) t v( ) ( ) + t v( ) ( )+ h A ( + ). Ech rectgle hs se. The height of ech rectgle is Foud usig the poits t (.,.,.,. ) i the equtio v() t t +. The re uder the curve is pproimtel + + +, so the prticle is close to.. Ech rectgle hs se. The height of ech rectgle is foud usig the poits (.,.,.,.,. ) i the equtio vt () t +. The re uder the curve is pproimtel ,so the prticle is close to..

2 Sectio.. () R () Δ + LRAM: [() ( ) ] + [ ( ) ( ) ] +.. () RRAM: + [ () ()] + + [ ( ) ( ) ]. () MRAM: LRAM MRAM RRAM The re is... LRAM MRAM RRAM Estimte the re to e... LRAM MRAM RRAM Estimte the re to e..

3 Sectio... LRAM MRAM RRAM Estimte the re to e.. LRAM MRAM RRAM Estimte the re to e.. Use f () d pproimte the volume usig rh ( ) Δ, so for the MRAM progrm, use ( ) o the itervl,. i MRAM V (). error % error LRAM: Are f( ) i+ f( ) i+ f( ) i+ + f( ) i i ( ). ( mg/l) i sec RRAM: Are f ( ) i + f() i+ f() i+ + f( ) i ( ). ( mg/l) i sec Ptiet's crdic output: mg sec i. L /mi. (mg/l) i sec mi Note tht estimtes for the re m vr.. () LRAM: i ( ) i.. ft () RRAM:i ( ) i.. ft. mi sec () LRAM: i( ) m () RRAM: i( ) m. LRAM: i( ) ft RRAM: i( ) ft ft+ ft Averge ft. () LRAM:.( ). mi RRAM:.( ). mi Averge. mi () The hlfw poit is. mi. The verge of LRAM d RRAM is. t. h d. t. h. Estimte tht it too. h. sec. The cr ws goig mph.. () Use LRAM with ( ). S. S is overestimte ecuse ech rectgle is elow the curve. () V S. % V. () Use RRAM with ( ). S. S is uderestimte ecuse ech rectgle is elow the curve. () V S. % V. () Use LRAM with ( ) o the itervl [, ],. S. m () V S V. %

4 Sectio.. () ( )( )( ), ft () ()( )( ), ft. Use LRAM with o the itervl [, ],. ( ).. Use MRAM with o the itervl [, ] () LRAM : ft/sec () RRAM : ft/sec (c) The upper estimtes for speed re. ft/sec for the first sec,. +.. ft/sec for the secod sec, d ft/sec for the third sec. Therefore, upper estimte for the distce flle is ft.. () ft /sec ( sec )( ft/sec ) ft/sec () Use RRAM with o [, ], ft. () Upper gl Lower gl () Upper gl Lower gl (c),, gl,. h (worst cse),, gl,. h (est cse). () Sice the relese rte of pollutts is icresig, upper estimte is give usig the dt for the ed of ech moth (right rectgles), ssumig tht ew scruers were istlled efore the egiig of Jur. Upper estimte: ( ). tos of pollutts A lower estimte is give usig the dt for the ed of the previous moth (left rectgles). We hve o dt for the egiig of Jur, ut we ow tht pollutts were relesed t the ew-scruer rte of. to/d, so we m use this vlue. Lower Estimte: ( ). tos of pollutts () Usig left rectgles, the mout of pollutts relesed the ed of Octoer is ( ). tos Therefore, totl of tos will hve ee relesed ito the tmosphere the ed of Octoer.. The re of the regio is the totl umer of uits sold, i millios, over the -er period. The re uits re (millios of uits per er)(ers) (millios of uits).. True. Becuse the grph rises from left to right, the lefthd rectgles will ll lie uder the curve.. Flse. For emple, ll three pproimtios re the sme if the fuctio is costt.. E., Use MRAM o the itervl [, ],. ( ). D.. C. si( ) + si si + + si D.. () The digol of the squre hs legth, so the side legth is. Are ( ) () Thi of the octgo s collectio of right trigles with hpoteuse of legth d cute gle mesurig. Are si cos si. (c) Thi of the -go s collectio of right trigles with hpoteuse of legth d cute gle mesurig. Are si cos si. (d) Ech re is less th the re of the circle,. As icreses, the re pproches.. The sttemet is flse. We disprove it presetig couteremple, the fuctio f( ) over the itervl, with. MRAM f(. ). LRAM + RRAM f( ) + f( ) +. MRAM

5 Sectio.. RRAM f ( Δ)[ f( ) + f( ) + + f( ) + f( )] ( Δ)[ f( ) + f( ) + f( ) + + f( )] + ( Δ)[ f( ) f( )] LRAM f + ( Δ)[ f( ) f( )] But f () f () smmetr, so f ( ) f ( ). Therefore, RRAM f LRAM f.. () Ech of the isosceles trigles is mde up of two right trigles hvig hpoteuse d cute gle mesurig. The re of ech isosceles trigle is AT si cos si. () The re of the polgo is AP AT si, so lim AP lim si (c) Multipl ech re r : AT r si AP r si lim A r P Sectio. Defiite Itegrls ( ) Eplortio Fidig Itegrls Siged Ares. +. The sme re s re.) si sits ove rectgle of.. (Ech rectgle i tpicl Riem sum is twice s tll s i si.).. (This is the sme regio s i si, trslted uits to the right.).. (The equl res ove d elow the -is sum to zero.).. (This is the sme re s si, ut elow the -is.).. (Ech rectgle i tpicl Riem sum is twice s wide s i si.).. (The equl res ove d elow the - is sum to zero.).. (The equl res ove d elow the -is sum to zero.).. (This is hlf the re of si.)

6 Sectio... (The equl res ove d elow the -is sum to zero, sice si is odd fuctio.).. Eplortio More Discotiuous Itegrds. The fuctio hs removle discotiuit t.. The thi strip ove hs zero re, so the re uder the curve is the sme s ( + ), which is... + ( + ) K. +. ( ) if is odd.. ( ) if is eve. Sectio. Eercises. lim ( c Δ where is prtitio of [, ].. lim ( c c ) Δ ( ) where is prtitio of [, ].. The grph hs jump discotiuities t ll iteger vlues, ut the Riem sums ted to the re of the shded regio show. The re is the sum of the res of rectgles (oe of them with height ): it( ) Quic Review.. () + ( ) + () + ( ) + (). ( ) [() ] + [() ] + [() ] + [() ] + [() ]. ( j + ) [( ) + ( ) + ( ) + ( ) + ( ) ] j. c. lim Δ where is prtitio of [, ].. lim Δ c [, ]. where is prtitio of. lim c Δ where is prtitio of [, ].. lim (si c ) Δ si where is prtitio of [, ].. [ ( )]. ( ) ( )( ). ( ) dt ( )( ). d θ [ ( )]... ds. [. (. )]... dr ( )

7 Sectio.. Grph the regio uder + for.. Grph the regio uder for. + ()( + ). Grph the regio uder + for. The regio is oe qurter of circle of rdius. (). Grph the regio uder for. / ( + ) ( )( + ) / + ( )( ) ( )( ). Grph the regio uder for.. Grph the regio uder for. This regio is hlf of circle rdius. () ( ) ( )( )

8 Sectio.. Grph the regio uder for.. Grph the regio uder r for r. ( ) ()( ) ()( ) Grph the regio uder + for. ( + ) ( )() + () +. Grph the regio uder θ for θ rdr ( + )( ). ( )( ). ( )( ). sds ( )( + ). tdt ( )( + ) ( ). ( + )( ). ( )( + ) ( ). dt t ( ) ( ) mes il. dt t ( ) ( ) gllos.. dt t clories (. ) ( ). θ d θ ( + )( ).. dt. t... ( ). (. ) liter. NINT +,,,.. + NINT(t,,, ).. NINT(,,. ).. NINT( e,,, ).

9 Sectio.. () The fuctio hs discotiuitt. () +. () The fuctio hs discotiuities t,,,,,,,,,,. () it( ) ( ) + ( ) + ( ) + ( ) + ( ) + ( ) + ( ) + ( ) + ( ) + +. () The fuctio hs discotiuitt. () ( )( ) + ( )( ) +. () The fuctio hs discotiuitt. () ( )( ) ( )( ). Flse. Cosider the fuctio i the grph elow.. True. All the products i the Riem sums re positive.. E. ( f( ) + ) f ( ) + +. D. ( ). C.. A Oserve tht the grph of f( ) is smmetric with respect to the origi. Hece the re ove d elow the -is is equl for. ( re elow -is ) + ( re ove -is). The grph of f( ) + is three uits higher th the grph of g ( ). The etr re is ()(). ( + ) +. Oserve tht the regio uder the grph of f( ) ( ) for is just the regio uder the grph of g ( ) for trslted two uits to the right. ( ). Oserve tht the grph of f( ) is smmetric with respect to the -is d the right hlf is the grph of g ( ).. Oserve from the grph elow tht the regio uder the grph f( ) for cuts out regio R from the squre ideticl to the regio uder the grph of g ( ) for. f() R ( )

10 Sectio.. Oserve from the grph of f( ) ( ) for tht there re two regios elow the -is d oe regio ove the is, ech of whose re is equl to the re of the regio uder the grph of g ( ) for..oserve from the grph elow tht the regio etwee the grph of f( ) d the -is for cuts out regio R from the squre ideticl to the regio uder the grph of g ( ) for. R f() ( ) +. Oserve tht the grph of f( ) for is just horizotl stretch of the grph of g ( ) for fctor of. Thus the re uder f( ) for is twice the re uder the grph of g ( ) for.. Oserve tht the grph of f( ) is smmetric with respect to the orgi. Hece the re ove d elow the -is is equl for. ( re elow -is) + ( re ove -is). Oserve from the grph elow tht the regio etwee the grph of f( ) d the -is for cuts out regio R from the squre ideticl to the regio uder the grph of g ( ) for. f() R ( ) +. () As pproches from the right, f () goes to. () Usig right edpoits we hve lim lim lim Note tht d so >, () Δ, () (c) RRAM i i i i i ( + )( + ) ( + )( + ) i (d) lim lim ( + )( + ) i + + lim

11 Sectio.. Cotiued (e) Sice equls the limit of Riem sum over the itervl [, ]s pproches, prt (d) proves tht. Sectio. Defiite Itegrls d Atiderivtives (pp. ) Eplortio How Log is the Averge Chord of Circle? Eplortio Fidig the Derivtive of Itegrl Pictures will vr ccordig to the vlue of chose. (Ideed, this is the poit of the eplortio.) We show tpicl solutio here.. The chord is twice s log s the leg of the right trigle i the first qudrt, which hs legth r the Pthgore Theorem.. We hve chose ritr etwee d.. We hve shded the regio usig verticl lie segmets.. The shded regio c e writte s f() t dt usig the defiitio of the defiite itegrl i Sectio.. We use t s dumm vrile ecuse cot vr etwee d itself.. The re of the shded regio is our vlue of F(). r. Averge vlue r r r. ( ) r r. Averge vlue r r r r i ( re of semicircleof rdius r) i r r r. Although we ol computed the verge legth of chords perpediculr to prticulr dimeter, the sme computtio pplies to dimeter. The verge legth of chord of circle of rdius r is r.. The fuctio r is cotiuous o [ r, r], so the Me Vlue Theorem pplies d there is c i [, ] so tht (c) is the verge vlue r.. We hve drw oe more verticl shdig segmet to represet ΔF.. We hve moved distce of Δ so tht it rests ove the ew shdig segmet.

12 Sectio.. Now the (siged) height of the ewl-dded verticl segmet is f ().. The (siged) re of the segmet is ΔF Δi f( ), so ΔF F () lim f( ) Δ Δ Quic Review.. d si. d cos. d. d. d. d. d. d. d sec t t sec cos cot si sec t + sec sec sec + t + ( + ) + ( + ) e + e i ( ). d + Sectio. Eercises. () g ( ) () g ( ) g ( ) ( + ) (c) f( ) f( ) ( ) + (d) f( ) f( ) f( ) f( ) + f( ) + (e) [ f ( ) g( )] f ( ) g( ) (f) [ f ( ) g( )] f ( ) g( ) f ( ) g( ) (). () f( ) f( ) ( ) () [ f ( ) + h( )] f ( ) + h( ) + (c) [ f ( ) h( )] f ( ) + h( ) (d) f( ) f( ) + f ( ) h( ) () () (e) f( ) f( ) f( ) f ( ) f( ) (f) [ h( ) f ( )] h( ) f ( ). () f( u) du h( ) + f ( ) + () f() z dz f() z dz (c) f() t dt f() t dt (d) [ f( )] f( ). () gt () dt gt () dt () gudu ( ) (c) [ g ( )] g ( ) (d) gr () dr g () r dr +. () f() z dz f() z dz f() z dz f z dz + + () f () z dz + () f() t dt f() t dt f() t dt f() t dt+ f() t dt +. () hr () dr hr () dr+ hr () dr hr () dr+ hr () dr

13 Sectio.. Cotiued () hudu ( ) hudu ( ) hudu ( ) hudu ( ) hudu ( ). m si ( ) si () o [, ] si( ) si ( ) <. m + d mi + o [, ] +. ( ) mi f( ) o [, ] ( )mi f( ) f( ). ( )m f( ) o[, ] f( ) ( )m f( ). A tiderivtive of is F( ). v ( ) F( ) F( ) ( ) Fid ci [, ] such tht c c c ± Sice is i [, ],.. A tiderivtive of is F ( ). v F F [ ( )] ( )] c Fid ci [, ] such tht. c c ± Sice is i [, ],.. A tiderivtive of is F( ). v ( ) F ( ) F ( ) Fid ci [, ] such tht c c c c ± Sice is i [, ],.. A tiderivtive of ( ) is F( ) ( ). v F F + ( ) [ ( ) ( )] Fid ci [, ] such tht ( c ). c ± c or c. Sice oth re i [, ], or.. The regio etwee the grph d the -is is trigle of height d se, so the re of the regio is ()(). v( f ) f ( ).. The regio etwee the grph d the -is is rectgle with hlf circle of rdius cut out. The re of the regio is () (). v( f ) f ( t) dt.. There re equl res ove d elow the -is. v( f ) f ( t) dt i. Sice t θ is odd fuctio, there re equl res ove d elow the -is. / v( f ) f ( ) d / θ θ i /. si cos( ) + cos( ) /. cos si ( ) si /. e e e e /. sec t t.. ( ). () ( ). ( ) ( ). + t ( ) t ( ). si si ( ) /

14 Sectio. e. l e l.. v( f ) si ( cos ( cos ) e. v f e e ( ) (l el e) e e l e. v( f ) sec t t( ). v( f ) t ()t ( ) + ( ). v( f ) + ( ) +. v( f ) sec t sec sec. mi f d m f +. f (. ). + (). + + d Yes, v ( f ) f ( ). This is ecuse v(f) is costt, so v( f ) v f ( ) i v( f ) i v( f ) i ( v ) ( f) ( ) f( ) f( ). () mi () mi mi + h mph mph (c) mi. mph h (d) The verge speed is the totl distce divided the totl time. Algericll, d + d. The driver computed t+ t d d + t t. The two epressios re ot equl. m. Time for first relese mi m /mi m Time for secod relese mi m /mi totl relesed Averge rte m m /mi totl time mi. si. sec + +. Let L() c + d. The the verge vlue of f o [, ] is v( f ) ( c d) + c c + d + d c ( ) + d ( ) c ( + ) + d ( c + d) + ( c + d) L ( ) + L ( ). Flse. For emple, si si, ut the verge vlue of si o [, ] is greter th.. Flse. For emple, ut ( ) ( )

15 Sectio.. A. There is o rule for the multiplictio of fuctios.. D. There is o rule for the egtio of the ouds.. B. v( f ) cos (si si )... C. F ( ) ( ) f( ). () Are h () h + C (c) h h ( ) h +. v( ) ( +) Grph d o grphig clcultor d ( + ) fid the poit of itersectio for >. Thus,.. A tiderivtive of F () is F() d tiderivtive of G () is G(). F ( ) F ( ) F ( ) G ( ) G ( ) G ( ) Sice F ( ) G ( ), F ( ) G ( ), so F ( ) F ( ) G ( ) G ( ). Quic Quiz Sectios... D. ( F( ) + ) B.. C... () f ( ) + f ( ) + + c m f ( ) + ( ) + c c f ( ) ( + + ) f( ) c f ( ) ( ) + ( ) + ( ) + c c f( ) + + () v( f ) ( + + ) ( ) + + Sectio. Fudmetl Theorem of Clculus (pp. ) Eplortio Grphig NINT f. The fuctio t hs verticl smptotes t ll odd multiples of. There re si of these etwee d.. I ttemptig to fid F( ) t( t) dt +, the clcultor must fid limit of Riem sums for the itegrl, usig vlues of t t for t etwee d. The lrge positive d egtive vlues of t t foud er the smptotes cuse the sums to fluctute errticll so tht o limit is pproched. (We will see i Sectio. tht the res er the smptotes re ifiite, lthough NINT is ot desiged to determie this.). t. The domi of this cotiuous fuctio is the ope itervl,.. The domi of F is the sme s the domi of the cotiuous fuctio i step, mel,.

16 Sectio.. We eed to choose closed widow rrower th, to void the smptotes.. The grph of F loos the grph i step. It would e decresig o, d icresig o,, with verticl smptotes t d. Eplortio The Effect of Chgig i. f(t)dt. d. d l. d. d / ( si )( ) (cos )( ) si + cos. d d cos t, si t dt dt d d / dt cost cot t / dt si t. Implicitl differetite: d + () + d d ( ) ( + ) d + +. d / d.. Sice NINT (,,, ), the -itercept is.. Sice NINT (,,, ), the -itercept is. d. Chgig hs o effect o the grph of f t dt (). It will lws e the sme s the grph of f().. Chgig shifts the grph of f() t dt verticll i such w tht is lws the -itercept. If we chge from to, the distce of the verticl shift is f() t dt. Quic Review.. d. d. d cos( ) i cos( ) (si )(cos ) si cos (sec )(sec t ) (t )(sec ) sec t t sec Sectio. Eercises. d. d. d. d. d. d. d. d. d.. d d d t dt si ( ) si ( ) + d t+ t dt cos cos ( ) ( ) d t t dt d t + dt + e d udt t ( ) e t d u e udu e sec sec d + t dt + + t + d si t t dt si + cos + cos d t du dt e e e d tdt du cot cot cot d + u + du u

17 Sectio.. d d + si u u du + si ( ) + cos + cos ( ) d d d. ( + t ) dt ( + t ) dt l l l +. d. d. d. d. d. d. d ( ) d d t t dt t t dt d t d t dt dt cos t cos c os du t cos + d t + + t+ d dt t + du t t+ dt t d d ( r ) dr ( r ) dr si si si si du ( ) ( + ) d d + p dp p l l dp l( du + ) l( + p ) d tdt du du cos + cos cos cos + cos d cos t dt du cos si si si cos cos si. si t dt t. e tt dt. E s Es. cost dt+. cos t dt t. e dt+ du. I / / ( l) l l+ l ll l.. l l. l /. ( + ) + + ( + ) / /. ( ). /. /.. si cos ( ). ( + cos ) + si ( + ) ( + ). / /. sec θ d θ t θ / ( ). /. csc θ d θ cot θ / / / ( ).. csc cot csc ( ) ( ) / / / /. sec t sec / ( ). r+ dr r ( ) ( + )

18 Sectio.. ( ) u / du u du u. Grph. / u u ( ) ( ) Over [, ]: ( ) Over [, ]: ( ) Totl re +. Grph. Over [, ]: ( ) ( ) Over [, ]: ( ) ( ) Over [, ]: ( ) Totl re + +. Grph. Over [, ]: ( ) + + Over [, ]: ( ) + + Totl re +. Grph. Over [, ]: ( ) ( ) Over [, ]: ( ) Totl re +. First, fid the re uder the grph of. Net fid the re uder the grph of. ( ) Are of the shded regio +. First fid the re uder the grph of. / / Net fid the re uder the grph of. Are of the shded regio +. First, fid the re uder the grph of cos. ( + cos ) + si The re of the rectgle is. Are of the shded regio.. First, fid the re of the regio etwee si d the -is for,. / si cos / / / The re of the rectgle is si Are of the shded regio. NINT + si,,,.

19 Sectio.. NINT,,.,... NINT( cos,,,. ). etwee d NINT(,,, ). t. Plot NINT( e t ),,,,. i [, ] [, ] widow, the use the itersect fuctio to fid... Whe,. NINT(,,, ).. f () t dt K f () t dt + K f() t dt+ f() t dt f() t dt+ f() t dt f () t dt K ( t t+ ) dt t t + t + + ( ). To fid tiderivtive of si, recll from trigoometr tht cos si, so si cos. K si t dt cos ( ) si ( ) si cos si cos si cos.. () H( ) f( t) dt d () H ( ) f( t) dt f( ) H ( ) > whe f( ) >. H is icresig o [, ]. (c) H is cocve up o the ope itervl where H ( ) f ( ) >. f ( ) > whe <. H is cocve up o (, ). (d) H( ) f( t) dt > ecuse there is more re ove the -is th elow the -is. H() is positive. (e) H ( ) f( ) t d. Sice H ( ) f( ) > o [, ), the vlues of H re icresig to the left of, d sice H ( ) f( ) < o (, ], the vlues of H re decresig to the right of. H chieves its mimum vlue t. (f) H ( ) > o (, ]. Sice H( ), H chieves its miimum vlue t.. () s () t f().the t velocit t t is f() uits/sec. () s () t f () t < t t sice the grph is decresig, so ccelertio t t is egtive. (c) s() f() ()(). uits (d) s hs its lrgest vlue t t sec sice s () f() d s () f () <. (e) The ccelertio is zero whe s () t f () t. This occurs whe t sec d t sec. (f) Sice s( ) d s () t f() t > o (, ), the prticle moves w from the origi i the positive directio o (, ). The prticle the moves i the egtive directio, towrds the origi, o (, ) sice s () t f() t < o (, ) d the re elow the -is is smller th the re ove the -is. (g) The prticle is o the positive side sice s() f() > (the re elow the -is is smller th the re ove the -is).. () s () f() uits / sec () s () f () > so ccelertio is positive. (c) s() f() ( )() uits (d) s() f() ( )() + ()(), so the prticle psses through the origi t t sec. (e) s () t f () t t t sec (f) The prticle is movig w from the origi i the egtive directio o (, ) sice s( ) d s () t < o (, ). The prticle is movig towrd the origi o (, ) sice s () t > o (, ) d s(). The prticle moves w from the origi i the positive directio for t > sice s ( t) >.

20 Sectio.. Cotiued (g) The prticle is o the positive side sice s() f() > (the re elow the -is is smller th the re ove the -is). d d. f( ) f( t) dt ( + ) d. f ( ) + + t dt + f ( ) f ( ) + t dt + L ( ) + d. f( ) f( t) dt d ( ) cos ( si )+ i cos si + cos f ( ) si + cos. Oe rch of si is from to. / / Are si cos ( ). () () The verte is t ( ) ( ). (Recll tht the verte of prol + + cis t. ), so the height is. (c) The se is ( ). (se)(height) (). True. The Fudmetl Theorem of Clculus gurtees tht F is differetile o I, so it must e cotiuous o I.. Flse. I fct, e is rel umer, so its derivtive is lws.. D.. D. See the Fudmetl Theorem of Clculus.. E. f( ) + f ( )( ) f ( ) f ( ) ( ). E. si( t) si( t). () f (t) is eve fuctio so dt dt. t t si( t) Si( ) dt t si( t) dt t si( t) dt Si( ) t si t () Si( ) dt t (c) Si ( ) f( t) whe t, ozero iteger. (d) [, ] [,, ]. () c ( ) dc c ( ) or $ () c ( ) dc c ( ) or $. ( + ) + + ( ) +. thousd The comp should epect $.. () drums () ( drums)($. per drum ) $. () True, ecuse h ( ) f( )d therefore h ( ) f ( ). () True ecuse h d h re oth differetile prt (). (c) True, ecuse h () f().

21 Sectio.. Cotiued (d) True, ecue h () f() d h () f () <. (e) Flse, ecuse h () f () <. (f) Flse, ecuse h () f () (g) True, ecuse h ( ) f( ),d f is decresig fuctio tht icludes the poit (,).. Sice f(t)is odd, f() t dt f() t dt ecuse the re etwee the curve d the -is from to is the opposite of the re etwee the curve d the -is from to, ut it is o the opposite side of the -is. f() t dt f() t dt f() t dt f() t dt Thus f() t dt is eve.. Sice f(t) is eve, f() t dt f() t dt ecuse the re etwee the curve d the -is from to is the sme s the re etwee the curve d the -is from to. Thus f() t dt f() t dt f() t dt f() t dt is odd.. If f is eve cotiuous fuctio, the f() t dt is odd, d ut f() t dt f( ). Therefore, f is the derivtive of the odd cotiuous fuctio f() t dt. Similrl, if f is odd cotiuous fuctio, the f is the derivtive of the eve cotiuous fuctio f() t dt.. Solvig NINT si t, t,, t grphicll, the solutio is..we ow rgue tht there re o other solutios, usig the fuctios Si() d f(t) s defied i Eercise. Sice d si Si( ) f( ), Si ( ) is icresig o ech itervl,( + ) d decresig o ech itervl ( + ),( + ), where K is oegtive iteger. Thus, for >, Si( ) hs its locl miim t, where is positive iteger. Furthermore, ech rch of f( )is smller i height th the ( + ) ( + ) previous oe, so f( ) > f( ). This ( + ) ( + ) mes tht Si( + ) ) Si( ) f( ) >, so ech successive miimum vlue is greter th the previous si oe. Sice f ( ) NINT, o,,. ( ) d Si is cotiuous for >, this mes Si ( ) > (d. hece Si( ) ) for. Now, Si( ) hs ectl oe solutio i the itervl [, ] ecuse Si() is icresig o this itervl d. is solutio. Furthermore, Si( ) hs o solutio o the itervl [, ] ecuse Si() is decresig o this itervl d Si( ). >. Thus, Si( ) hs ectl oe solutio i the itervl [, ). Also, there is o solutio i the itervl (, ] ecuse Si() is odd Eercise (or ), which mes tht Si( ) for ( sice Si( ) for ). Sectio. Trpezoidl Rule (pp. ) Eplortio Are Uder Prolic Arc. Let f( ) A + B+ C The f( h) Ah Bh+ C, f( ) A( ) + B( ) + C C, d f( h) Ah + Bh+ C Ah Bh + C + C + Ah + Bh + C h Ah + C.. Ap ( A + B+ C) h A + B h + C h A h + B h + Ch A h + B h Ch A h + Ch h Ah + C ( ). Sustitute the epressio i step for the pretheticll eclosed epressio i step : h Ap Ah C ( + ) h ( + + ). Quic Review.. si cos < o [, ], so the curve is cocve dow o [, ].. > o [, ], so the curve is cocve up o [, ].. < o [, ], so the curve is cocve dow o [, ].

22 Sectio.. cos si o [, ], so the curve is cocve dow o [, ].. e > o [, ], so the curve is cocve up o [, ].. e < o [, ], so the curve is cocve dow o [, ].. > o [, ], so the curve is cocve up o [, ].. csc cot ( csc )( csc ) + (csc cot )(cot ) csc + csc cot > o [, ], so the curve is cocve up o [, ].. < o [, ], so the curve is cocve dow o [, ].. cos + si si + cos < o [, ], so the curve is cocve dow. Sectio. Eercises. () f( ), h f() T () () f ( ), f ( ) The pproimtio is ect. + (c). () f( ), h f() T (). () f ( ), f ( ) > o [, ] The proimtio is overestimte. (c). () f( ), h f() T (). () f ( ), f ( ) > o [, ] The proimtio is overestimte. (c). () f ( ), h f() T () f ( ), f ( ) > o [, ] The pproimtio is overestimte. (c) l l.

23 Sectio.. () f( ), h f( ) T ( + ()+ + + ). () / / f ( ), f ( ) < o [, ] The pproimtio is uderestimte. (c) /. () f( ) si, h f () T +. + ()+ +. () f ( ) cos, f ( ) si < o [, ] The pproimtio is uderestimte. (c) si cos h. T ( ) f( ) ( + ( ) + ( ) + ( ) + ( ) + ( ) + ) h. T ( ) f( ) ( + ( ) + ( ) + ( ) + ( ) + ( ) + ). (. (.) (.) (.).)( ) ft. () ( + ( ) + ( ) + ( ) + + ( ) + )( ),, ft () You pl to strt with, fish. You ited to hve (. )(, ), fish to e cught. Sice,., the tow c sell t most liceses.. Sum the trpezoids d multipl to chge secods to hours (. ( ) (.. )( ) (.. )( ) + (..)( + ) + (..)( + ) + (.. ) ( + ) + (.. ) ( + ) + (.. )( + ) + (.. )( + ) + (.. )( + ) + (.. )( + )). mi feet.. Sum the trpezoids d multipl to chge secods to hours. ( + ( ) + ( ) + ( ) + ( ) + ( )+ ( ) + ( ) + ). mi feet.. () / () + () +. () / () + () +. () / () + (). () / () l l l.. () ( + ( ) + ( ) + ( ) + ( )). () / / / ( ) ( )

24 Sectio. /. () si si si ( )+ + + si si + si. () si cos cos ( cos) ( ). () f( ), h f () S ( + ( )+ ( )+ ( )+ ) ( ) () E s (c) For f ( ), M si ce f ( ) ( ). f (d) Simpso s Rule for cuic polomils will lws give ect vlues sice f ( ) for ll cuic polomils.. The verge of the discrete tempertures gives equl weight to the low vlues t the ed.. () ( + i + i + + i + ) v( f ). i () We re pproimtig the re uder the temperture grph. B doulig the edpoits, the error i the first d lst trpezoids icreses.. Setch grph of lie segmets joied t shrp corers. Oe emple:. S., S.. S., S.. S., S. usig. s lower limit S., S. usig. s lower limit. S., S.. () T. () T. T. E T.... T (c) E E T (d), h, M ET ET E T ( ). () S. S S.. () E S (c) E s.. E s (d), h, M E s E s ( ) s T E s

25 Sectio. i.. h i. Estimte the re to e [ + (. ) + ( ) + ( ) + ( ) + (. ) + ]. i. Note tht the t cross-sectio is represeted the shded re, ot the etire wig cross-sectio. Usig Simpso s Rule, estimte the cross-sectio re to e [ ] [. + (.) + (.) + (.) + (.) + (. ) +.]. ft Legth ( l) l / ft. ft. ft. Flse. The Trpezoidl Rule will over estimte the itegrl if it is cocve up.. Flse. For emple, the two pproimtios will e the sme if f is costt o [, ].. A. LRAM < T < RRAM, so RRAM <.. e e e e e. B e + e + e + e /. C. si si + si. C si si si / () + ( + ). () f ( ) cos( ) f ( ) i si( ) + cos( ) si ( ) + cos( ) () [, ] [, ] (c) The grph shows tht f ( ) so f ( ) for. h (d) ET ( ) ( h )( ) h. (e) For < h., ET. <. (f) ( ) h.. () f ( ) i cos( ) si( ) si( ) cos ( ) si ( ) ( ) f ( ) i si ( ) cos ( ) i cos ( ) si( ) ( )si( ) cos ( ) () [, ] [, ] ( (c) The grph shows tht f ) ( ) so ( f ) ( ) for.. T. S h (d) ES ( ) ( h )( ) h. (e) For < h., ES. <. (f) ( ) h. h [ ] h [ ] + h [ ] LRAM + RRAM h [ ] [ h ( ) + ( h)( )] + T MRAM, where h. Quic Quiz Sectios. d.. C. f( ) (( )( + ) + ( )( + ) + ( )( + )) (si ). D. si cos (si ( )) (si ( )) cos cos.

26 Chpter Review d t. C. df ( ) e dt df ( ) ( ) ( ) e.. () + + ( ) (si si(. ) si(. ) + si(. ) + si( )). () F icreses o [, ] d [, ] ecuse si( t ) > d (c) f() t si( t ) dt K K K Chpter Review ( ).. RRAM : LRAM : MRAM : T ( LRAM + RRAM ) +.. ( ). LRAM MRAM RRAM l l l l.. () f( ) f( ) The sttemet is true. - () [ f( ) + g( )] f ( ) + g( ) f( ) + f( ) The sttemet is true. g( )

27 Chpter Review. Cotiued (c) If f( ) g( ) o [, ], the f ( ) g( ), ut this is ot true sice f( ) f( ) + f( ) + d g ( ). The sttemet is flse.. () Volume of oe clider: rh si ( m) Δ Totl volume: V lim mi si ( ) Δ () Use si o[, ]. i NINT ( si,,, ).. () Approimtios m vr. Usig Simpso s Rule, the re uder the curve is pproimtel [ + (.) + () + () + (.) + (.) + (. ) + (. ) + (. ) + ( ) + ]. The od trveled out. m. (). () Positio (m) s Time (sec) The curve is lws icresig ecuse the velocit is lws positive, d the grph is steepest whe the velocit is highest, t t. () si (c) (d) (e) ( ) + ( si ) t i. The grph is ove the -is for < d elow the -is for < Totl re ( ) ( ) [ ] [ ]. The grph is ove the -is for -is for < Totl re / cos / / / si si ( ) ( ) < cos. ( ). / /. cos si. + ( ) + ( ) d elow the. ( s s + ) ds s s + s... ( ) / / d ( ) dt t t / / t dt t ( ) / /. sec θ dθ tθ e. e l. ( + ) ( + ) ( + ) ( )

28 Chpter Review. + ( + ) ( ). sec t sec / /. si( ) cos( ). d + + l l l. Grph o [, ]. The regio uder the curve is qurter of circle of rdius. ( ). Grph o [, ]. The regio uder the curve cosists of two trigles. ( + )( ) ( )( ). Grph o [, ].. () Note tht ech itervl is d hours Upper estimte: ( ). L Lower estimte: ( ). L () [..).).) + ( + ( + + ( +.]. L. () Upper estimte: ( ). ft Lower estimte: ( ). ft () [..).).) ] + ( + ( + + ( +. ft. Oe possile swer: The is importt ecuse it correspods to the ctul phsicl qutit Δ i Riem sum. Without the Δ, our itegrl pproimtios would e w off.. f( ) f( ) + f( ) ( ) + + [ ] +. Let f( ) + si m f sice m si mi f sice mi si (mi f )( ) + si (m f )( ) < + si /. () v( ) The regio uder the curve rdius. is hlf circle of ( )... / / () v( ) d d d + cos d + cos ( ) i ( ) + cos ( ) d dt + t +

29 Chpter Review d d. dt t + t + i ( ) c ( ) t dt + / dt + t c( ) + The totl cost for pritig ewsletters is $.. v() I ( + t) dt [ t+ t Rich s verge dil ivetor is cses. ct (). It () + t v() c ( + t) dt + t t Rich s verge dil holdig cost is $. We could lso s (.).. ( t - t + ) dt t -t +t Usig grphig clcultor,. or... () True, ecuse g ( ) f( ). () True, ecuse g is differetile. (c) True, ecuse g ( ) f(). (d) Flse, ecuse g () f () >. (e) True, ecuse g ( ) f() d g () f () >. (f) Flse, ecuse g () f (). (g) True, ecuse g ( ) f( ), d f is icresig fuctio which icludes the poit (, ).. + F() F( ) si t. ( ) dt + t. + Thus, it stisfies coditio i. () + t dt () + + Thus, it stisfies coditio ii.. Grph (). t dt + t + + ( ) +. () Ech itervl is mi h. [. + (. ) + (. ) + + (. ) +. ]. gl () ( mi/h) h/gl. mi/gl. () Usig the freefll equtio s gt from Sectio., the distce A flls i secods is ( )( ) ft. Whe her prchute opes, her ltitude is ft. () The distce B flls i secods is ( )( ) ft. Whe her prchute opes, her ltitude is ft. (c) Let t represet the umer of secods fter A jumps. For t sec, A s positio is give SA () t ( t ) t, so A lds t t sec. For t + sec, B s positio is give SB () t ( t ) t, so B lds t t. sec. B lds first.

30 Chpter Review. () Are of the trpezoid ( + + h)( ) h( ) Are of the rectgle ( h ) h h ( + ) + ( h) h ( + + ) () Let h. h S [ ] [ h ( + + ) + h ( + + ) + + h ( + + )] Sice ech epressio of the form h ( i + i + i) is equl to twice the re of the ith of rectgles plus the re of the ith of i MRAM + T trpezoids, S.. () g() f() t dt () g() f() t dt ()() (c) g( ) f() t dt f() t dt ( ) (d) g ( ) f( ); Sice f( )> for < < d f( )< for < <, g( ) hs reltive mimum t. (e) g ( ) f( ) The equtio of the tget lie is ( ) ( + ) or + (f) g ( ) f ( ), f ( ) t d f ( )is ot defied t. The iflectio poits re t d. Note tht g ( ) f ( )is udefied t s well, ut sice g ( ) f ( ) is egtive o oth sides of, is ot iflectio poit. (g) Note tht the solute mimum is g( ) d the solute miimum is g( ) f() t dt f() t dt ( ). The rge of g is,. /. () NINT( e,,, ). / NINT( e,,, ). () The re is.. First estimte the surfce re of the swmp. [ + ( ) + ( ) + ( ) + ( ) + ( ) + ] ft d ( ft)( ft )i d ft. () V ( v ) si ( t) m Usig NINT: v( V ) Vm si t dt ( ) ( ) ( Vm) ( Vm) si ( t) dt ( Vm) V rms V m ( ) V m () V m. volts. () Rtdt () (. ( ) +. )., which is the totl umer of gllos of wter tht flowed through the pipe durig the hour period. () Yes, ecuse R() R(), the Me Vlue Theorem gurtees tht there is umer c etwee d such tht R () c. (c) Qt ( ). ( + ( ) ( ) ). gl / hr. f () () + () f ( ) + f () () + + ( + ) f ( ) f( ) + c f ( ) c + ( + c) c f ( ) +. () g ( ) ( ( ) ( ( ))) g ( ) ( ( + )) () g( ) f( ) (c) The miimum vlue is g ( ) (d) g hs poit of iflectio t. It is the ol plce where the slope goes from positive to egtive.