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 of alculus. a ( b ( f si t dt f dt l t. Evaluat d 8 si t d dt d d d d d t d. Evaluat th itgral. d c ( a d 8 d d ( ( si d d cos f si ( cos ( d g d h d i d 8 8 9 j d k ta sc d l cos d ( m arcsi d cos d o p d q d r ( d ( l l d s d t sc ta d sc u d 9 v d w d csc cot d y ( cos d z d aa l d
Math Rviw Erciss. It is stimatd that t moths from ow th populatio of a crtai tow will b chagig at a dp rat of dt t t popl pr moth. Th currt populatio is. What will b th populatio 9 moths from ow? dq. Aftr t hours o th job, a factory workr ca produc uits at a rat of l( dt t t uits pr hour. How may uits dos a work who arrivs at 8: A.M. produc durig th first hours?. Fid th avrag valu of ach fuctio ovr th giv itrval. a f ( l, b f ( c f ( [ ] [ 8, ] [, ] 8. Suppos th umbr of itms a w workr o a assmbly li producs daily aftr t days I t l t. Fid th avrag umbr of itms producd daily o th job is giv by ( ( by this mploy aftr days. 9. Dtrmi whthr th followig impropr itgral covrgs or divrgs. a d b d c ( d d ( 9. Estimat ( d 9 l d to dcimal placs with usig a right dpoit approimatio b lft dpoit approimatio c midpoit rul d trapzoidal rul Simpso s rul f Fid th maimum rror of stimat for th midpoit rul. g Fid th maimum rror of stimat for th trapzoidal rul. h Fid th maimum rror of stimat for th Simpso s rul. 9 i Fid th act valu of ( l d. Witr 9 Marti Huard
Math Rviw Erciss. Fid th ara of th rgio R if y ad y a R is boudd by th curvs (. b R is boudd by th curvs y, th -ais, ad. c R is boudd by th curvs d R is boudd by th curv y ad y. y ad y.. osidr th rgio R boudd by y, y ad. a Fid th volum of th solid obtaid by rotatig th rgio about th -ais. Sktch th rgio ad a typical disk, washr of shll. b Fid th volum of th solid obtaid by rotatig th rgio about th li. Sktch th rgio ad a typical disk, washr of shll. c Fid th volum of th solid obtaid by rotatig th rgio about th li y 9. Sktch th rgio ad a typical disk, washr of shll. d Fid th volum of th solid whos cross-sctios prpdicular to th -ais ar squars. Fid th volum of th solid whos cross-sctios prpdicular to th -ais ar quilatral triagls.. osidr th rgio R i th first quadrat boudd by th curvs y, y. a Sktch th rgio R ad fid th ara. b Fid th volum obtaid by rotatig R about th -ais. Sktch th rgio ad a typical disk, washr of shll. c Fid th volum obtaid by rotatig R about th y-ais. Sktch th rgio ad a typical disk, washr of shll.. osidr th rgio R boudd by y ad y. Fid th volum of th solid if cross-sctios prpdicular to th -ais ar a Fid th volum of th solid obtaid by rotatig th rgio about th li. Sktch th rgio ad a typical disk, washr of shll. b Fid th volum of th solid obtaid by rotatig th rgio about th li y 9. Sktch th rgio ad a typical disk, washr of shll. c Fid th volum of th solid whos bas is R ad cross-sctios prpdicular to th -ais ar squars d Fid th volum of th solid whos bas is R ad cross-sctios prpdicular to th -ais ar smi-circls. Fid th volum of th solid whos bas is R ad cross-sctios prpdicular to th -ais ar isoscls right triagls with th hypotus as th bas. Witr 9 Marti Huard
Math Rviw Erciss. osidr th rgio R boudd by y ad y. a Fid th volum obtaid by rotatig R about th li y. Sktch th rgio ad a typical disk, washr of shll. b Fid th volum obtaid by rotatig R about th li. Sktch th rgio ad a typical disk, washr of shll. c Fid th volum of th solid whos bas is R ad if cross-sctios prpdicular to th -ais ar quartr circls, whos radius is o th bas.. osidr th rgio R boudd by y y ad y. a Fid th volum obtaid by rotatig R about th li y. Sktch th rgio ad a typical disk, washr of shll. b Fid th volum obtaid by rotatig R about th li. Sktch th rgio ad a typical disk, washr of shll.. Fid th volum of a frustum of a pyramid with squar bas of sid b, squar top of sid a, ad hight h. 8. Fid th lgth of th curv giv by ( 9. Fid th lgth of th curv giv by y y btw ad. btw (, ad ( y. Fid th arc lgth fuctio for th curv y l(. Th dmad fuctio for a Ipho is p D( ( ( 8,. ( P, l. startig at th poit ( ad th supply fuctio is p S whr i th umbr of uits (i thousads ad p th pric pr uit (i $. Fid th cosums ad producrs surplus at th quilibrium poit.. A compay stimats that th rvu producd by a machi at tim t will b f ( t thousad dollars pr yar. Fid th total rvu, th prst valu ad th futur valu of th machi ovr th t yars, if moy is worth % compoudd cotiuously. Also, fid th capital valu.. Fid th Gii id for th Lorz curv L( arcta(. t. Solv th followig diffrtial quatios. dy ( ( y a d c y d l y dy dy b sc y arcta d dy d y si y d ( Witr 9 Marti Huard
Math Rviw Erciss. If a w product bcoms kow both by word of mouth ad by advrtisig, th p, th proportio of popl who hav hard of th w product aftr t wks, satisfis a diffrtial quatio ad iitial coditio of th form dp.( p.p( p dt If iitially % of th populatio hav hard of th w product, what will b th proportio of hav hard of it t wks from ow?. Dtrmi if th followig squcs ar mootoic, boudd ad covrgt. a c a (! b ( d a l (. Dtrmi if th followig sris covrg. If so, fid th sum. a b... 9 8 8 c d (! f!!! 8. Dtrmi whthr th followig sris ar covrgt or divrgt. a d b ( ( h g ( ( ( ( c f arcta 9 (! 9. Dtrmi whthr th sris covrgs absolutly, covrgs coditioally or divrgs. ( l a b (. Fid th radius ad itrval of covrgc for th followig powr sris. ( a b d ( ( c ( ( f ( ( (!( (! Witr 9 Marti Huard
Math Rviw Erciss. Fid th Maclauri Sris for f ( usig th dfiitio. a f ( b f ( cos (. Fid th Taylor sris for f ( at a. a f ( si a c f ( a c 8 b f ( (. Us th kow Maclauri sris for, si, cos or th giv fuctio. Giv th radius of covrgc. cos a f ( si b f ( a to obtai th Maclauri sris for c f (. Us multiplicatio or divisio of powr sris to fid th first thr ozro trms i th Maclauri sris for ach fuctio. a f ( si cos b f (. Approimat f by a Taylor polyomial with dgr at th umbr a. Us this approimatio to stimat th giv umbr, ad stimat th accuracy of that approimatio. a f ( a 9. b f ( sc. Usig th Maclauri sris for withi fiv dcimal placs. a ( si a d b sc(, si, cos or d c to approimat th dfiit itgral to. osidr th fuctio f ( l(. a Fid th Maclauri sris for f ( usig th dfiitio. b Us diffrtiatio, as wll as your aswr i (a to fid th Maclauri sris for c Us th aswr from (a to approimat th dfiit itgral l( d withi fiv dcimal placs. d to Witr 9 Marti Huard
Math Rviw Erciss Aswrs. a No b 9 Ys c No d No si8 si b.. a ( l l. a b l 8 8 c d ( l arcta( cos f cos cos g l l h 9 l l 8 i l l ( j l k ta ta l cos si m arcsi cos si o q s 9 p arcta l l l l r ( l t sc u l arcta v l arcta w 8l l l cot cot z l 8 y aa.. 9 uits. b l l 8.. itms 9. a 8 c b Div c Div d. a. b.98 c. d..8 f g h i l l 8.9. a b c 9 d 8. a ( 8 b ( c ( 8 d. a 8 8 b c Witr 9 Marti Huard
Math Rviw Erciss. a 8 b c. a 8 b 88 c. a b 8. h ( a ab b 8.. s( l l l d 8 9. 8. S $ 8 PS $. TR $ P $9 9 A $9 9 V $ 8... a arcta y l l c (. p( t b si arcta l ( l y d y t t y si. a dcrasig, a, cov to b omootoic,, cov to a 9 a c omootoic,, cov to d icrasig, a ot boudd abov, div. a 9 f Divrgs 8. a ovrgs b ovrgs c Divrgs d ovrgs ovrgs f ovrgs g ovrgs h Divrgs 9. a ovrgs coditioally b ovrgs coditioally. a R, I (, b R, I (, c R, I [,] d R, I [, R, I R f R, I { } ( 8 (. a ( b ( c (!! ( (. a b ( ( (! ( 9 ( ( c (. a. a! 8 ( (, R b! b. a ( ( ( 8 (, R c ( (! 9.., R ( b sc ( ( sc(. (. a. b.98 c.. a ( b ( c.99 R <. R <. Witr 9 Marti Huard 8