DOC PREVIEW
WUSTL MATH 132 - 132f13.4sol

This preview shows page 1-2 out of 6 pages.

Save
View full document
View full document
Premium Document
Do you want full access? Go Premium and unlock all 6 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 6 pages.
Access to all documents
Download any document
Ad free experience
Premium Document
Do you want full access? Go Premium and unlock all 6 pages.
Access to all documents
Download any document
Ad free experience

Unformatted text preview:

MATH 132 FINAL EXAM SOLUTIONS F13M132.4sol.11) Use substitution to evaluate '0ÈÈ$%BB "#.BEÑ ! FÑ " GÑ # HÑ $ IÑ % J Ñ & KÑ ' LÑ ( Ñ ) N Ñ * I solution: (? œ B  "ß .B œ #B .BÑ œ # ? .? œ % ? l œ %Ð#  "Ñ œ %Þ ÐIÑ# "%'" "# #%"-------------------------------------------------------------------------------------------2) Using integration by parts where'B / .B œ 0ÐBÑ  Gß 0ÐBÑ œ À$BEÑ B / FÑ $ B / GÑ B /  / HÑ $ B /  $ / IÑ B /  $ /" " " "$ $ $ $$B $B $B $B $B $B $B $B J Ñ $ B /  / KÑ B /  / LÑ $ B /  * / MÑ B /  /$B $B $B $B $B $B $B $B" " " " "$ $ * * $ N Ñ B /  /" "* $$B $B solution: ( ? œ Bß .? œ .Bß .@ œ / .Bß @ œ / Ñ œ B/  / .B œ$B $B $B $B" " "$ $ $' " "$ *$B $BB/  /  GÞ ÐKÑ------------------------------------------------------------------------------------------------3) Using partial fractions, find a solution to '" " $(3x 1) (3x+1).BÞ ÐB $ ÑA) B) C) D) E) " $B" " $B" " $B" " $B" " $B"# $B" $B" $B" # $B" $B"68Ð Ñ 68Ð Ñ 68Ð Ñ 68Ð Ñ 68Ð Ñ3 6 3F) G) " $B" " "$B" #6 368Ð Ñ 68ÒÐ$B  "ÑÐ$B  "ÑÓ LÑ 68ÒÐ$B  "ÑÐ$B  "ÑÓMÑ 68ÒÐ$B  "ÑÐ$B  "ÑÓ N Ñ / ‡ /"$B" $B"6solution: ' '" " "  ' '"Î# "Î#(3x 1) (3x+1) (3x 1) (3x+1).B œ Ð  Ñ .B œ 68Ð$B  "Ñ  68Ð$B  "Ñ œ" $B"' $B"68Ð Ñ  GÞ ÐGÑ-------------------------------------------------------------------------------------------------4) Find what becomes of the integral when you make the substitution'#(ÐB  *Ñ# #.Bß B œ $ >+8Ð Ñß ! - - Þ) )1#A) B) C) D) E) ' ' ' ' '=38Ð Ñ . -9=Ð Ñ . . . =/- Ð Ñ .) ) ) ) ) ) ) )" "=38Ð Ñ -9=Ð Ñ#) )F) G) H) I) ' ' ' '-=- Ð Ñ . =38 Ð Ñ . -9= Ð Ñ . Ð>+8 Ð Ñ  *Ñ .# # # #) ) ) ) ) ) ) )J) 'Ð=/- Ð Ñ  *Ñ .#) )solution: ( B œ $ >+8Ð Ñß .B œ $=/- Ð Ñ . Ñ œ œ) ) )##(† $=/- Ð Ñ .Ð* >+8 Ð Ñ*Ñ'## #) )) ' ' '#(Ð $=/- Ð Ñ Ñ .=/- Ð Ñ =/- Ð Ñ"###) )) ) 81 4œ . œ -9= Ð Ñ . Þ ÐLÑ) ) )132F1 .4 .$ =96 #5) Find the of the region enclosed by the curve and the linearea C œ BÈ B œ # C Ð C œ BÑ Þ"#EÑ " FÑ GÑ HÑ IÑ J Ñ KÑ LÑ MÑ N Ñ " $ % ( & "" * "% "$# # $ & $ & # $ &solution: Intersection points and is always top, andÐ!ß !Ñ Ð%ß #Ñ Þ C œ B C œ BÈ"# always bottom . E œ Ð B  BÑ .B œ B  B l œ  % œ Þ ÐHÑ'!%" # " "' %# $ % $ $#" $# #%!-----------------------------------------------------------------------------------------6) Find the of the solid obtained by the region enclosed by the curvevolume rotating and the lines 2 and x 0 about the .C œ # B ß C œ ß œÈx-axisEÑ FÑ GÑ HÑ IÑ J Ñ KÑ # LÑ Ñ $ N Ñ1 1 1 1 1 1 1% % % % # % $$ & ( * % I 1 1 19solution: and 2 intersect at and .Using the washerC œ # B C œ Ð!ß !Ñ Ð"ß #ÑÈ method we get Z œ Ð%  %BÑ .B œ Ð%B  #B Ñ l œ # Þ ÐKÑ1 1 1'!"#"!---------------------------------------------------------------------------------------7) Find the of the curve , to 2 decimal places.arc length B œ # C ! Ÿ C Ÿ " ß$Î#EÑ FÑ Ñ HÑ IÑ J Ñ KÑ LÑ MÑ1.57 1.94 C 2.27 2.84 3.45 3.92 4.46 4.84 5.17 J) 5.68solution: .B .B.C .C#!"œ $ C Ê Ð Ñ œ *C Ê P œ "  *C .C"#'È Ð? œ "  *Cß .? œ * .CÑ œ ? .? œ ? l œ" " #* * $""!$Î#'"#"!" ##($Î#Ð"!  "Ñ œ #Þ#')$&% µ #Þ#(Þ ÐGÑ--------------------------------------------------------------------------------------------8) Find the value of the improper integral , if it converges.'0111B.BEÑ IÑ J Ñ KÑ LÑ MÑ N Ñ .3@/<1/=0 B) 2 C) 4 D) 6 8 " / " #// % $/ $solution: œ Ð  68Ð"  BÑ Ñ œ  68Ð"  ,Ñ œ ∞ ß .3@/<1/=Þ ÐN Ñlim lim,Ä" ,Ä",!-------------------------------------------------------------------------------------------9) Find the solution to the differential equation with initial condition.C.# tyœ $> / ßCÐ!Ñ œ 68Ð#ÑÞEÑ C œ # / FÑ C œ # / GÑ C œ # / HÑ C œ 68 Ð> Ñ  68Ð#Ñ> Î# $> > Î$ $# # $ IÑ C œ 68Ð>  #Ñ J Ñ C œ # 68Ð>  /Ñ KÑ C œ Ð>  "Ñ  # LÑ C œ Ð>  #Ñ$ $ # MÑ C œ >  N Ñ C œ >  %#"# Èsolution: Initial condition is ' '/ .C œ $> .>  G Ê / œ >  GÞ > œ !ß C œ 68Ð#ÑC # C $so and we get / œ G Ê G œ # / œ >  # Ê C œ 68Ð>  #ÑÞ ÐIÑ68Ð#Ñ C $ $132F1 .4 .$ =96 $1 ) Find the sum of the infinite series ! Ð  ÑÞ 8œ"∞& "# $8 8EÑ FÑ GÑ HÑ IÑ J Ñ KÑ LÑ MÑ N Ñ& & $ * " & # " " ## # & $ # &3 4 2 4 solution: œ 8œ" 8œ"∞ ∞& " " *# $ # #&Î# "Î$" "8 8" "# $ œ  œ &  œ Þ ÐHÑ------------------------------------------------------------------------------------" Ñ Ð  "Ñ1 Using the fewest terms possible, approximate the sum with an 8œ"∞8"88 "# error of less than !Þ#& ÞEÑ "Î$ FÑ #Î$ GÑ "Î% HÑ "Î# IÑ $Î% Ñ "Î& KÑ #Î& LÑ $Î& MÑ %Î& N Ñ " F solution: and , 8œ"∞8"8 " # $ % % $8 " # & "! "( "( "!Ð  "Ñ œ     ÞÞÞ - !Þ#& $ !Þ#&#Þ so approximation will be the first 3 terms, " # $ % ## & "! "! &  œ œ Þ ÐKÑ------------------------------------------------------------------------------------------------12) Find the complete (either absolute or conditional) of theinterval of convergence power series 8œ"∞ÐB#Ñ8†#(-1)8" 88ÞA)  # - B - # FÑ  # Ÿ B - # GÑ  # - B Ÿ # HÑ  # Ÿ B Ÿ #I  % - B - ! J Ñ  % Ÿ B - ! KÑ  % - B Ÿ ! LÑ  % Ÿ B Ÿ !) MÑ Ö!× N Ñ Ö  #× solution: Ratio test lÐB#Ñl lB#lÐ8"Ñ # lÐB#Ñl # 8" #8 # " …


View Full Document

WUSTL MATH 132 - 132f13.4sol

Documents in this Course
shapiro1

shapiro1

11 pages

shapiro

shapiro

11 pages

shapiro1

shapiro1

11 pages

shapiro

shapiro

11 pages

Load more
Download 132f13.4sol
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view 132f13.4sol and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view 132f13.4sol 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?