MATH 132 FINAL EXAM FALL 2009 This exam should have 20 questions , 5 points each . If you do not have a PENCILto mark your card , please ask to borrow one from your proctor . Write your ( not your SS number ) on the six blank lines on the top ofID NUMBERyour answer card , using one blank for each digit . Shade in the corresponding boxesbelow Print your name at the top of your card . Also . You may bring along a 5 x 7card and the calculator you used for the first three exams. The needed data for powerseries is contained on the last page of this booklet.1) Evaluate cos( ) d .'01#/†BB=38ÐBÑEÑ !FÑ "GÑ/HÑ / ! "IÑ/ ! "#JÑ/ !"##KÑ /1#!"LÑ / ! "1MÑ / ! "#2) Evaluate sin(2x)'!1B.BÞEÑ !FÑÈ##GÑ ! "1HÑ!#1IÑ#1JÑ !È$#1KÑ ! #1LÑ "MÑ1%NÑ# ! "1132f09.4.23) Use partial fractions to evaluate .'12(+1)B " %BB.BEÑ 1.24FÑ 1.38GÑ1.46D) 1.56E) 1.69F) 1.75G) 1.80H) 1.91I) 2.14%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 ."%$' sec ))GÑ = Ð Ñ ."#'ec 2))HÑ % Ð Ñ .'cot#))IÑ - Ð Ñ."##'sc ))JÑ =/- Ð Ñ ! "."%$'))KÑ # +8 Ð Ñ .'t#))LÑ " ! >+8 Ð Ñ .'$))MÑ # " " ÐÑ.'cot#))NÑ " " ÐÑ.'csc$))132f09.4.35) Find the area of the region enclosed by = Ð=Ñ 0ÐBÑ ÐB ! "Ñ + 8.# 0ÐBÑ œ " ! BÞEÑ1%FÑ$%GÑ1HÑ&%IÑ#JÑ16KÑ&'LÑ('MÑ $NÑ&$6) Find the volume of the solid you get by the region enclosed byrevolving the curve = the lines = and = BCß B!C%ßÈabout the -axis .BA) "#Þ' 1FÑ "%Þ%Þ1GÑ "(Þ# 1HÑ "*Þ) 1IÑ &Þ'2 1J Ñ #)Þ% 1KÑ$"Þ 4 1LÑ $$Þ) 1MÑ $&Þ' 1N Ñ $)Þ# 1132f09.4.47) Suppose is the solid whose base in the region bounded by the parabolaS = 4 and the line y = 0 and whose cross sections perpendicular to the y-axisC ! B# are squares. For each y between 0 and 4 for the cross section .find a formula A(y)A) y#B) y%#G%C) HÑ # y#IÑ) ! #CJ " %Ñ y#KÑ %C " #LÑ "# ! y#MÑ #C " %NÑ "'! %C)Ñ œ ß ! Ÿ Ÿ $ ÞCalculate the arc length of the curve y x x #$$Î#A) "#FÑ&$GÑ #HÑ &ÈIÑ #ÈJÑ ($KÑ$#LÑÉ"#MÑ"%$NÑ %132f09.4.59) Determine whether the improper integral converges'081È$)!B.B and if so determine its value .A) 0B) 2C) %$HÑ 4IÑ (È$JÑ'#$KÑ 'LÑ 8MÑ"#NÑ.3@/<1/="!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 JB) 90 JC) 110 JD) 130 JE) 150 JF) 170 JG) 90 J"H) 210 JI) 230 JJ) 250 J132f09.4.611) Find the solution to the initial value differential equation .C.> ##"œ ! CßCÐ!Ñœ ÞEÑ C œ Ð> " #Ñ!"FÑ C œ > ""#GÑC œ Ð> " #ÑÈ!#HÑ C œ Ð> " Ñ"##ÈIÑC œ Ð> " "Ñ !#"#JÑC œ Ð>" #Ñ !!""#KÑ C œ Ð> " #Ñ !!#"#LÑ C œ />!68Ð>ÑMÑ C œ =38Ð#>Ñ ""#NÑC œ -9=Ð>Ñ!"#12) Find the sum of the eometric series 1 $ !"!"!ÞÞÞÞÞÞÞ'"# #% %)&#&"#&'#&EÑ"&%FÑ"&(GÑ"$&HÑ*&IÑ""#&JÑ#&"#KÑ#&$LÑ"#&#(MÑ"#&$NÑ#&"(132f09.4.713) If cos(2x) = a , find a!8œ!_88 %B Ð 3>= Q +-6+?<38 =/<3/= Ñ ÞEÑ"#FÑ !"#GÑ#$HÑ !#$IÑ"%JÑ !"%KÑ""#LÑ !""#MÑ&"#NÑ !&"#14) Find the function whose Maclaurin series is" " $B " *B " #(B " )"B " ÞÞ ÞÞÞÞÞ ß !#B # Þ#$%""$$EÑ =38Ð$BÑBÑ-9=Ð*BÑGÑ68Ð" ! $BÑHÑ /*BIÑ""!$BJÑ""!*BKÑ"""$BLÑ"""*BMÑ >+8 Ð$BÑ!"NÑ>+8 Ð*BÑ!"132f09.4.815 If a ( x is the Taylor series of the function f(x) = cos(x)Ñ ! Ñ!8œ!_8816 at c = , then find the value of a the coeficient of the term a ( x1 1'$$ ß$! ÑÞ6A) 0B) "'GÑ !"'HÑ""#IÑ !""#JÑÈ#'KÑ !È#'LÑ $ÈMÑÈ$#NÑ !"#16) Find the of convergence of the power series interval!8œ!_Ð$B!#Ñ#88ÞEÑ !#B ###$$FÑ ! ŸB###$$ GÑ !#BŸ##$$ HÑ !#B #%%$$E Ñ ! ŸB#44$$F Ñ !#BŸ44$$G 0Ñ # B #4$H 0 ÑŸB#4$I 0 Ñ # BŸ4$J) ! _ # B # _132f09.4.917) Find the 4 term in the for the function f(x) = (1>2#Bbinomial series ! ÑÞ"$EÑ&"' B$FÑ !&"' B$GÑ"%)" B$HÑ !"%)" B$IÑ&""# B$JÑ !&""# B$KÑ%!)" B$LÑ !%!)" B$MÑ"##" B$NÑ !"##" B$")Ñ Using the error formula for alternating series, what error do you make by approximating S = by S (i.e. the sum of the first three terms) ?!8œ"_Ð!"Ñ8Ð8"#ÑÐ8"$Ñ$8EÑ""!B) "*!C) ""!!D) ""')E) "#%F) "$&'G) ""!!!H) ""!!!!I)"&!!J) "$&!132f09.4.10"* B Þ) Find the term a in the Maclaurin series of f( ) = #$B!=38ÐBÑBA) 0B) 1C) -1D) "#E) -"#F) "#%KÑ - "#%LÑ""#!I) - ""#!NÑ"%)#!Ñ ÞFind the function whose Maclaurin series is !8œ!_Ð!"Ñ B$Ð#8""Ñ8#8""#8""EÑ =38Ð$BÑFÑ -9=Ð$BÑGÑ""!$BHÑ =38Ð ÑB$IÑ-9=Ð ÑB$JÑ /$BKÑ /B$LÑ >+8 Ð Ñ!"B$MÑ"""$BNÑ68Ð"! $BÑ132f09.4.11 Frequently used Maclaurin Series""! B#88œ!_œ"" B " B " ÞÞÞÞÞÞÞÞÞÞ œ B ß lBl # "!/œ"" B """""ÞÞÞÞÞ œ ß lBl # _BBBBB B#x $x %x &x 8x8œ!_#$%& 8 !=38ÐBÑ œ B !""ÞÞÞÞÞ œ ß lBl # _BB$x &x Ð#8""Ñx8œ!_Ð!"Ñ B$&8#8""!-9=ÐBÑ œ " !""ÞÞÞÞÞ œ ß lBl # _BB#x %x Ð#8Ñx8œ!_Ð!"Ñ B#%8#8!arctan(x) œB!"!ÞÞÞÞÞÞ œ ß ! " # B # "BB8œ!_Ð!"Ñ B8352+135 2+1!88Ð"" BÑ œ "" B " B " B " ÞÞÞÞ œ B ß lBl # "a naa aa anaanÐ !"Ñ Ð !"ÑÐ !#Ñ#x $x#$œ!_!Œ !68Ð" " BÑ œ B !"!"ÞÞÞÞ œ ß lBl # "BBB#$% 88œ"_Ð!"Ñ
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