MAA 102 Calculus for Science Student II [Kalkulus untuk Pelajar Sains II]

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Sharleen Mason
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1 UNIVERSITI SAINS MALAYSIA Secod Semester Eamiatio 04/05 Academic Sessio Jue 05 MAA 0 Calculus for Sciece Studet II [Kalkulus utuk Pelajar Sais II] Duratio : hours [Masa : jam] Please check that this eamiatio paper cosists of SIX pages of prited material before you begi the eamiatio. [Sila pastika bahawa kertas peperiksaa ii megadugi ENAM muka surat yag bercetak sebelum ada memulaka peperiksaa ii.] Istructios: Aswer EIGHT (8) questios. [Araha: Jawab LAPAN (8) soala.] I the evet of ay discrepacies, the Eglish versio shall be used. [Sekiraya terdapat sebarag percaggaha pada soala peperiksaa, versi Bahasa Iggeris hedaklah digua pakai]..../-

2 - - [MAA 0]. Determie whether the followig sequeces coverge or diverge. ( )( + ) + ( ) + [ 0 marks ]. Tetuka sama ada jujuka berikut meumpu atau mecapah. ( )( + ) + ( ) + [ 0 markah ]. Use appropriate test to determie whether the followig series coverge or diverge. l = + ( ) 5 (c) = = ( + ) [ 5 marks ]. Gua ujia yag bersesuaia utuk meetuka sama ada siri-siri berikut meumpu atau mecapah. l = + ( ) 5 (c) = = ( + ) [ 5 markah ].. Fid the iterval ad radius of covergece of ( ) = [ marks ].. Cari selag da jejari peumpua ( ) = [ markah] /-

3 - - [MAA 0] 4. Determie whether the followig improper itegrals coverge or diverge. d 9 + d [ 0 marks ] 4. Tetuka sama ada kamira tak wajar berikut meumpu atau mecapah. + d d 9 [ 0 markah ] 5. A metal plate is heated so that its temperature at a poit (, y) is ( y ) 5 T y, = 50ye +. A bug is placed at the poit (,). ( ) The bug heads toward the directio r = i j. What is the rate of temperature chage i this directio? Is the metal gettig colder or hotter i this directio? (c) I what directio should the bug head i order to warm up at the fastest rate? What is the rate of temperature chage i this directio? I what directio should the bug head i order to cool off at the fastest rate? What is the rate of temperature chage i this directio? [ 4 marks ] 5. Satu plat logam dipaaska sedemikia dega suhu pada titik (, y) diberika ( y ) 5 T y, = 50ye +. Seekor seragga diletakka pada titik (,). oleh ( ) Seragga tersebut meuju ke arah r = i j. Apakah kadar perubaha suhu pada arah ii? Adakah plat logam tersebut semaki sejuk atau paas pada arah ii? (c) Apakah arah tuju seragga tersebut supaya dapat memaaska bada pada kadar yag terpatas? Apakah kadar perubaha suhu pada arah tersebut? Apakah arah tuju seragga tersebut supaya dapat meyejukka bada pada kadar yag terpatas? Apakah kadar perubaha suhu pada arah tersebut? [ 4 markah ] 4/-

4 - 4 - [MAA 0] 6. The legth, width ad height of a rectagular bo are = 5, w =, ad h =, respectively. (i) (ii) (iii) Fid the istataeous rate of chage of the volume of the bo with respect to the legth if width ad height are held costat. Fid the istataeous rate of chage of the volume of the bo with respect to the width if legth ad height are held costat. Fid the istataeous rate of chage of the volume of the bo with respect to the height if legth ad width are held costat. Suppose that the portio of a tree that is usable for lumber is a right circular cylider. If the usable height of a tree icreases ft per year ad the usable diameter icreases 0.5 ft per year, how fast is the volume of usable lumber icreasig the istat the usable height of the tree is 0 ft ad the usable diameter is.5 ft? (Hit: The volume of a right circular cylider is give by π rhwhere r is the radius ad h is the height) [ marks ] 6. Pajag, lebar da tiggi sebuah kotak segi empat tepat adalah masigmasig = 5, w =, ad h =. (i) (ii) (iii) Cari kadar perubaha seketika isipadu kotak terhadap pajag jika lebar da tiggi adalah malar. Cari kadar perubaha seketika isipadu kotak terhadap lebar jika pajag da tiggi adalah malar. Cari kadar perubaha seketika isipadu kotak terhadap tiggi jika pajag da lebar adalah malar. Adaika bahagia pokok yag boleh diguaka utuk papa adalah dalam betuk satu silider tegak. Jika tiggi pokok yag boleh diguaka meigkat kaki setahu da diameter yag boleh diguaka meigkat 0.5 kaki setahu, berapa cepat peigkata isipadu kayu yag boleh diguaka apabila tiggi pokok yag boleh diguaka ialah 0 kaki da diameter yag boleh diguaka ialah.5 kaki? (Petujuk: Isipadu silider tegak diberika oleh π rhdi maa r adalah jejari da h ialah tiggi) [ markah ] 5/-

5 Sketch the regio R eclosed betwee the lies =. The evaluate da over the regio R. R y =, 0 [MAA 0] y =, = ad Use a polar double itegral to fid the area eclosed by the four-petaled rose r = cos θ as follows. r = cos θ [ 4 marks ] 7. Lakarka ratau R yag dibatasi garisa y =, y = 0, = da =. Seterusya ilaika da pada ratau R tersebut. R Guaka kamira polar gada dua utuk mecari luas kawasa yag dibatasi oleh ros empat kelopak r = cos θ seperti berikut. r = cos θ [ 4 markah ] 6/-

6 - 6 - [MAA 0] 8. Suppose that 00 fruit flies are placed i a breedig cotaier that ca support at most 0,000 flies ad that y = y( t) is the umber of fruit flies preset t hours later. Assumig that the populatio grows epoetially at a rate of % per day: Fid a iitial-value problem whose solutio is y( t ). Fid a formula for y( t ). (c) Sketch the graph of populatio growth as time t. (d) How log will it take for the cotaier to reach capacity? [ marks ] 8. Kataka 00 lalat buah diletakka di dalam bekas pembiaka yag boleh meampug palig bayak 0,000 lalat da y = y( t) adalah jumlah lalat buah yag ada pada t jam kemudia. Aggapka bahawa populasi tersebut meigkat dega pesat pada kadar % setiap hari: Cari suatu masalah ilai awal dega peyelesaia y( t ). Cari rumus utuk y( t ). (c) Lakarka graf peigkata populasi apabila masa t. (d) Berapakah lama masa yag diambil utuk bekas tersebut mecapai kapasiti? [ markah ] - ooooooo -

MSG388 Mathematical Algorithms for Computer Graphics [Algoritma Matematik untuk Grafik Komputer]

UNIVERSITI SAINS MALAYSIA Secod Semester Examiatio 015/016 Academic Sessio Jue 016 MSG388 Mathematical Algorithms for Computer Graphics [Algoritma Matematik utuk Grafik Komputer] Duratio : 3 hours [Masa