Chapter 6 Test December 9, 2008 Name

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1 Chapr 6 Ts Dcmbr 9, 8 Nam. Evalua - ÄÄ ÄÄ ÄÄ ÄÄ ÄÄ - + H - L u - and du u - du A - u - D ÄÄÄÄ 6 6. Evalua i j + z i j + z i 7 j ÄÄÄ + z 9 + ÄÄÄ ÄÄÄ 9 ÄÄÄ + C 6 ÄÄÄ + ÄÄÄ 9 ÄÄÄ + C 9. Solv h inial valu problm. Suppor our answr b ovrlaing our luion on a slop fild for h diffrnial quaion. ÄÄÄ, HL ÄÄÄ ÄÄÄ ÄÄÄ ln»» + C»» + C»» C ± C or A and if HL, A A and. Evalua sc an Ä + sc

2 sc an Ä + sc u sc and du sc an or an - Hsc L + C ÄÄÄ + u du an- u + C. Solv h following diffrnial quaion b h chniqu of sparaion of variabls : è!!! sc è!!!! ÄÄÄ Ä Ä, HL. ÄÄÄ è!!!! sc è!!! Ä Ä - ÄÄÄ cos i j z ÄÄÄ - ÄÄÄ cos i j z ÄÄÄ ÄÄÄ Ä u and du ÄÄÄ - ÄÄÄ - or du cos u du sin u ln À À + C or sin "### ln»» + C Now, w plug in and sin "#### ln»» + C C and sin "### ln»» or sin "### ln "#### sin - i j ln z or i jsin - i j ln z z 6. Evalua Hln L Hln L w ln w ln or w dw w w dw and, using abular ingraion, ln ln w w dw u dv w w - w w + w + C or w w Hln L - Hln L + + C and for h dfini ingral, w w A Hln L - Hln L + D H - + L - H - + L w - w 7. Evalua sin H L

3 u v - cos HL du dv sin HL sin H L - u v sin HL du dv cos HL A - cos HL + i j or A - - cos HL + i j sin HL - A z A - cos HL + sin HL or A cos HL + cos HL sin HL - sin HL z cos HL + - sin HL - A cos HL + sin HL + C 8. Evalua cos I M cos I M cos I MHL w and dw or w cos w dw u dv w cos w sin w - cos w Hw sin w + cos w + CL dw sin I M + cos I M + D 9. Evalua ÄÄÄÄ cos H L ÄÄÄÄ cos H u du Hln L or L Ä ln sc u + an u + C Ä ln sc H L + ln ln Ä ln an H L du Ä ln sc u du + C. Evalua co - i j z u co - i j z v co - i j z + - i du j z + ÄÄÄ ÄÄÄ + - ÄÄÄ + dv co- i j z + ln I + M + C. Th "half lif" of our calculus Flash Card nowldg is das. This mans ha, in das, ou will forg half of h cards ha ou now. Assuming ponnial dca, and ha ou currnl now of h cards, find a formula for h numbr of cards ha ou now. How long wil i a for ou o now onl of h cards?

4 i j ÄÄÄ z or ln das ln I M i j ÄÄÄ z i j ÄÄÄ z or i j ÄÄÄ z ln - ln ÄÄÄ Hln - ln L ln Ä ln ÄÄÄ. Can ou bliv i?! A rumor has sard ha DRuir' s rcn absnc is du o a rip h mad o Chicago o b considrd as an advir o Prsidn - lc Obama. Th sprad of h rumor is modld b h diffrnial quaion. P i j - P ÄÄ z, wih P a H in dasl rprsning h numbr of sudns 6 saring h rumor. If M ÄÄ P HM - PL and P Ä ÄÄÄÄ M + A -, A M - P ÄÄÄ, P HaL Find a luion o his diffrnial quaion. HbL Find h numbr of sudns who hav hard h rumor afr das. HaL W r o mach h diffrnial quaion ÄÄ Ä. Ä P H - PL Ä. Ä P H - PL 6 HL HbL afr das P M and P P ÄÄ ÄÄ + ÄÄÄ -. HL ÄÄ P HM - PL M Ä 6 P ÄÄÄÄ H - PL ÄÄ Ä + ÄÄ - - ÄÄ Ä ÄÄ ÄÄÄ sudns P i j - P ÄÄ z 6 Ä P ÄÄÄ H - PL ÄÄ ÄÄÄÄ + ÄÄÄ -.. I' s cold ousid! Jus bfor coming o class, h lmonad ha ou wr holding was a a mpraur of F. Th mpraur of h classroom is 7 F. Afr minus, h lmonad has warmd up o F. How much longr will i a for h lmonad o warm up o 6 F? i j Ths ma hlp : dt ÄÄÄ - HT - T s L and T - T s HT - T s L - z Using T - T s HT - T s L - w hav - 7 H - 7LI - ÄÄÄ Ä M - ÄÄÄ 8 7 T 7 - i j 8 ÄÄÄ z 7 ln - ln Hln 8 - ln 7L i ln j Ä ÄÄÄ ln 7 - ln 8 - z hours mor i j 8 ÄÄÄ z 7 ln Ä ÄÄÄ hours ln 7 - ln 8 I - M i j 8 ÄÄÄ z 7. Us Eulr s Mhod o numricall lv h iniial valu problm +, H-L, on h inrval -, saring a - and wih.

5 Eulr s Mhod n + n + f H n, n L D and + H + H-LL HL H-, L + H - L HL - H, -L - + H- + L HL - H, -L

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