MATH 132 SOLUTION FOR FINAL EXAM FALL 20091) Evaluate cos( ) d .'01#/†BB=38ÐBÑEÑ ! FÑ " GÑ / HÑ / ! "IÑ/! "JÑ KÑ/ LÑ/ ! "MÑ/! " ‡# !"#/ !"###11Solutions: (u œ=38ÐBÑ.?œ-9=ÐBÑ.BÑ œ / .?œ/ l œ /! "ÐHÑ'!"??"!--------------------------------------------------------------------------------------------2) Evaluate sin(2x)'!1B.BÞEÑ ! FÑ GÑ ! "HÑ IÑ#JÑ ! KÑ ! #LÑ"MÑNÑ#! "ÈÈ#$### %‡!1111111Solutions: ? œ Bß .@ œ =38Ð#BÑ.B ß .? œ .Bß @ œ ! -9=Ð#BÑ.B Þ"#''B.Bœ! B-9=Ð#BÑ " .B œ ! B-9=Ð#BÑ " Þsin(2x)""###%-9=Ð#BÑ =38Ð#BÑW9 B .B œ ! B-9=Ð#BÑ " lœ ÐHÑ'!" !#%#=38Ð#BÑ11sin(2x)1!-----------------------------------------------------------------------------------------------3) Use partial fractions to evaluate .'12(+1)B " %BB.BEÑ FÑ GÑ1.24 1.38 1.46 D) 1.56 E) 1.69 F) 1.75 G) 1.80 H) 1.91 I) 2.14‡Solutions: B " %EFBB B B""BÐB""ÑÐE"FÑB " E(+1)œ " œÊEœ%+8.Fœ! $ÞX2/8 .B œ ! .B œ %68ÐBÑ ! $68ÐB " "Ñl œ ''12(+1)B " %%$BB B B"""##"%68Ð#Ñ ! $68Ð$Ñ " $68Ð#Ñ œ (68Ð#Ñ ! $68Ð$Ñ œ "Þ&&' µ "Þ&' ÐHÑ----------------------------------------------------------------------------------------------%Ñ .B ß Find what becomes of the integral when you make the'x4 + $#B substitution x = 2 tan( ) , ))!## Þ11##‡$ $ #""%#EÑ% >+8 Ð Ñ. FÑ Ð Ñ. GÑ = Ð Ñ. HÑ % Ð Ñ.'''')) )) )) )) sec ec 2 cotIÑ - Ð Ñ. J Ñ =/- Ð Ñ ! ". KÑ# +8ÐÑ.""#%#$ #'' 'sc t)) ) ) ))LÑ "! >+8 Ð Ñ . MÑ # " " ÐÑ. NÑ "" ÐÑ.'''$# $)) )) ))cot cscSolution: dx=2 sec 2 secX2/8 ÐÑ.Þ ÐÑ.##)) )) ''x4 + tan ( )tan ( )$##$B%"%).B œœ))%.œ%>+8ÐÑ.ÐEÑ''tan ( )$) sec2 sec##ÐÑÐÑ)))))$#Þ5) Find the area of the region enclosed by = Ð=Ñ 0ÐBÑ ÐB ! "Ñ +8.# 0ÐBÑ œ " ! BÞEÑFÑGÑHÑIÑ#JÑKÑLÑ MÑ$NÑ116%% % ' ' $$& &(&‡1Solutions: ÐB ! "Ñ œ " ! B 09< B œ ! +8. B œ "Þ S 8 Ò!ß "Óß 638/ 3= +, 9@/Þ#W9 E œ Ð" ! BÑ ! ÐB ! "Ñ .B œ B ! B.Bœ ! lœ ÐJÑ''!!""##BB "#$ '#$"!----------------------------------------------------------------------------------------6) Find the volume of the solid you get by the region enclosed byrevolving the curve = the lines = and = BCß B!C%ßÈabout the -axis .BA) 2 "#Þ' F Ñ "%Þ%Þ GÑ "(Þ# HÑ "*Þ) IÑ &Þ' J Ñ #)Þ%111 1 11‡KÑ$"Þ LÑ $$Þ) MÑ $&Þ' N Ñ $)Þ#4 1111Solution: Region of revolution in x-y plane is enclosed by y œB +8. Cœ%A3>2#! Ÿ B Ÿ #Þ FC .3=- 7/>29. Z œ "' ! B .B œ Ð"'B ! Ñl œ #&Þ' ÐIÑ11 1'!#%B&&#!----------------------------------------------------------------------------------------------7) Suppose is the solid whose base in the region bounded by the parabolaS = 4 and the line y = 0 and whose cross sections perpendicular to the y-axisC ! B# are squares. For each y between 0 and 4 for the cross section .find a formula A(y)A) y B) y ) y y y## # # #%G%CHÑ#IÑ)! #C J " %KÑ%C" #LÑ"#!ÑMÑ #C " %NÑ"'! %C ‡Solution: X2/</1398 2+= 3>= >9: +> >2/ @/<>/B 90 C ! B ß Ð!ß %Ñß + 8. 3>= ,9>>97 = 4 #98 >2/ B ! +B3=ß =9 >2/ 38>/<@+6 3= ! Ÿ C Ÿ %Þ J 9< /+-2 C >2/ =3./ 90 >2/ =;?+</2+= 6/81>2 #Bß A2/</ B 3= >2/ B ! - 99<. 38+>/ 90 + :938> ÐBß CÑ 98 C œ % ! BÞ#X 2/</09</ EÐCÑ œ %B œ %Ð% ! CÑ œ "' ! %C ÐNÑ#--------------------------------------------------------------------------------------------------)Ñ œ ß ! Ÿ Ÿ $ ÞCalculate the arc length of the curve y x x #$$Î#A) "& ($ " "%#$ $# # $‡FÑ GÑ # HÑ & I Ñ # JÑ KÑ LÑ M Ñ NÑ %ÈÈÉSolution: .C .C.B .D#!$œB ß Ð Ñ œBß =96/81>2 3= 6 œ " " B.B"#'ÈÐ? œ " " Bß .? œ .BÑ œ ? .? œ ? l œ Ð) ! "Ñ œ ÐMÑ'"%## "%$$ $"$##%"$Þ*.B) Determine whether the improper integral converges and if so'081È$)!Bdetermine its value .A) 0 B) 2 C) 4 8%"$#‡HÑ IÑ (JÑ' KÑ'LÑ MÑ NÑ.3@/<1/=È$#$Solution: 'Ð) ! BÑ .B œ ! Ð) ! BÑ Ê!"Î$ #Î$$#'081È$)!B.B œ ! Ð) ! BÑ l œ ! ! Ð ! )Ñœ'ÐKÑ$$###Î$ #Î$)!------------------------------------------------------------------------------------"!Ñ A force of 20 is required to hold a spring stretched 5 beyond its naturalNm length is done in stretching it from 5 to 10 beyond its. How much work m m natural length ?A) 50 J B) 90 J C) 110 J D) 130 J E) 150 J F) 170 J G) 90 J H) 210 J‡"I) 230 J J) 250 JSolution: K3@/8 #! œ &5 Ê 5 œ % +8. J ÐBÑ œ %BÞ X 2/<09</ [ œ %B .B œ'&"!#B l œ "&! N ÐIÑ#"!&-----------------------------------------------------------------------------------------------11) Find the solution to the initial value differential equation .C.> ##"œ ! CßCÐ!Ñœ Þ‡ !" !##""##EÑ C œ Ð> " #Ñ FÑ C œ > " GÑ C œ Ð>" #Ñ HÑC œ Ð>" ÑÈÈIÑC œ Ð> " "Ñ ! JÑC œ Ð>" #Ñ ! KÑ C œ Ð> " #Ñ !# !" !#"" "## #LÑCœ/ ÑCœ=38Ð#>Ñ" NÑC œ -9=Ð>Ñ!>!68Ð>Ñ""## I Solution : ''! .C œ .> " GÞ X2/8 œ > " GÞ CÐ!Ñ œ Ê G œ #Þ"""CC##W9A/1/> C œ Ð> " #Ñ ÐEÑ!"------------------------------------------------------------------------------------------12) Find the sum of the eometric series 1 $ !"!"!ÞÞÞÞÞÞÞ'"# #% %)&#&"#&'#&EÑ FÑ GÑ HÑ IÑ J Ñ KÑ LÑ MÑ N Ñ"& "& "$ * "" #& #& "#& "#& #&% ( &&#&"#$ #($"(‡ Solution: = 3( 1$ !"!"!ÞÞÞÞÞÞÞ !" ! "ÞÞÞÞÑÞ'"# #% %) # % )& #& "#& '#& & #& "#&[/2+@/1/97/><3- =/<3/= A3>2 - œ $ +8. < œ ! Þ X 2 /</09</ 3> - 98@/<1/=#&>9$"&""(#&œÐFÑ4.13) If cos(2x) = a ' , find a!8œ!_88 %B Ð 3> = Q+-6+?<38 =/<3/= Ñ ÞEÑ FÑ ! GÑ HÑ ! IÑ JÑ ! KÑ LÑ ! MÑ N Ñ !""# #" "" "& &# # $ $ % % "# "# "# "#‡ Solution: , so cos(2x) -9=ÐBÑ œ " !""ÞÞÞÞÞ œ " !" "ÞÞÞÞÞ ÞB B %B "'B#x %x #x %x#% # %X23=13@/= ?= + œ œ ÐGÑ%"' #%x $--------------------------------------------------------------------------------------------------14) Find the function whose Maclaurin series is" " $B " *B " #(B " )"B " ÞÞ ÞÞÞÞÞ ß !#B # Þ#$%""$$EÑ =38Ð$BÑ Ñ -9=Ð*BÑ GÑ 68Ð" ! $BÑ HÑ / IÑ J Ñ B *B ‡"""!$B "!*BKÑ LÑ MÑ >+8 Ð$BÑ NÑ >+8 Ð*BÑ""""$B ""*B!" !" Solution : x By substitution we get""! B#$%œ"" B " B ""B " ÞÞÞÞÞÞÞÞÞÞ Þ ""! $Bœ"" $B " *B " #(B " )"B " ÞÞ ÞÞÞÞÞ
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