Single-variable Calculus Problems (and some solutions, too!)

1 single -variableCalculus Problems ( and some solutions , too!) Draft of BilaniukDepartment of MathematicsTrent UniversityPeterborough, OntarioCanada K9J is a compilation of a lot of quizzes, tests, and exams, and many of their solutions , fromsome of the Calculus courses taught by the author at Trent University. Typos and othererrors have been preserved for your enjoyment!Most of the content of this compilation can have no claim to originality and should properlybe in the public domain. (It is very unlikely that, say, there is any integral worked out herethat had never been worked out before or that was worked out here in an essentially newway.) Any original content, plus such trivialities as the arrangement of the material, isCopyrightc 2011 Stefan Bilaniukand is licensed under the Creative Commons Attribution Canada view a copy of this license, visit or senda letter to Creative Commons, 444 Castro Street, Suite 900, Mountain View, California94041, (The short version is that you can do whatever you like with material solicensed, so long as you acknowledge where it came from.)

2 This compilation was was assembled with LATEX, using the fullpage, hyperref, and graphicxpackages. The original quizzes, tests, exams, and solutions collected in this work were in-cluded as pdf images, and were originally typeset usingAMS-TEX. Various software, mostfrequently AppleWorks and Maple, was used to create the diagrams and Many Questions4 MATH 110 2001-20025 Quizzes .. 6 Test #1 .. 10 Test #2 .. 11 Final Exam .. 12 MATH 110 A 2002-200314 Quizzes .. 15 Test #1 (page 1 only) .. 21 Test #2 .. 22 Final Exam .. 23 MATH 110 A 2003-200425 Quizzes .. 26 Test #1 .. 32 Test #2 .. 33 Final Exam .. 34 MATH 105H Summer 200836 Quizzes .. 37 Test .. 39 Final Exam .. 40 MATH 1100Y Summer 201042 Quizzes .. 43 Test #1 .. 46 Test #2 .. 47 Final Exam .. 48 MATH 1101Y 2010-201150 Quizzes .. 51 Test #1.

3 531 Test #2 .. 54 Final Exam .. 55 MATH 1100Y Summer 201157 Quizzes .. 58 Test #1 .. 61 Test #2 .. 62 Final Exam .. 63II some Answers65 MATH 110 2001-200266 solutions to the Quizzes .. 67 solutions to Test #1 .. 85 solutions to Test #2 .. 90 MATH 110 A 2002-200394 solutions to the Quizzes .. 95 solutions to Test #1 (page 1 only) .. 121 solutions to Test #2 .. 124 MATH 110 A 2003-2004130 solutions to the Quizzes (#1-5 only) .. 131 solutions to Test #1 .. 138 solutions to Test #2 .. 143 MATH 105H Summer 2008149 solutions to the Quizzes .. 150 solutions to the Test .. 161 MATH 1100Y Summer 2010165 solutions to the Quizzes .. 166 solutions to Test #1 .. 186 solutions to Test #2 .. 191 solutions to the Final Exam .. 197 MATH 1101Y 2010-2011212 solutions to the Quizzes .. 213 solutions to Test #1.

4 227 solutions to Test #2 .. 232 solutions to the Final Exam .. 2392 MATH 1100Y Summer 2011251 solutions to the Quizzes .. 252 solutions to Test #1 .. 269 solutions to Test #2 .. 274 solutions to the Final Exam .. 2813 Part IMany Questions4 Mathematics 110 Calculus I: Calculus of one variable2001-20025 Mathematics110 CalculusofonevariableTrentUniversity2001 -2002 QuizzesQuiz# ,21 September,2001.[15minutes] (x)withdomain( 1,2)suchthatlimx 2f(x)=1butlimx 1f(x)doesnotexist.[4] ! definitionoflimitstoverifythatlimx 3=3.[6]Quiz# ,28 September,2001.[15minutes]Evaluatethefol lowinglimits, 1x+1x2 1[5] 1x+1x2 1[5]Quiz# ,5 October,2001.[20minutes] (x)= x2 6x+9x 3x"=30x=3continuousatx=3?[5] 13cx2+41exist?[5]Quiz#3.(Lateversion.)Fr iday,5 October,2001.[20minutes] (x)={cexx 0c xx<3continuousatx=0?[5] (x+13)22x2+1x,ifitexists.[5]Quiz# ,12 October,2001.}

5 [10minutes] (x)iff(x)=57x.[10]Quiz# ,19 October,2001.[17minutes] +1x2[3] (cos(x))[3] (x+1)5e 5x[4]Quiz# ,2 November,2001.[20minutes] +xy+x=13.[4] [3] [3]6 Quiz# ,9 November,2001.[13minutes] (x)=x2e xon( , )anddeterminewhichareabsolute.[10]Quiz# ,23 November,2001.[15minutes] [10][ThevolumeofasphereofradiusrisV=43 r3.]Quiz# ,30 November,2001.[20minutes] 0(2x2+1)dx.[6][Youmayneedtoknowthatn i=1i2=n(n+1)(2n+1)6.] 0(3x+1)dxcorrespondingtothepartitionx0=0 ,x1=23,x2=43,x3=2,withx 1=13,x 2=1,andx 3=53.[4]Quiz#9.(Lateversion.)Friday,30 November,2001.[20minutes] 1(x+1)dx.[6][Youmayneedtoknowthatn i=1i=n(n+1)2.] 0x2dxcorrespondingtothepartitionx0=0,x1= 1,x2=2,x3=3,x4=4,withx 1=0,x 2=2,x 3=2,andx 4=4.[4]Quiz# ,7 December,2001.[20minutes]Giventhat4 1xdx= 1x2dx=21, 1(x+1)2dx.[5] 1x3/2dx.[5]Quiz#10.(Lateversion.)Friday, 7 December,2001.[20minutes] ,putthefollowingdefiniteintegralsinorder ,fromsmallesttolargest.

6 [5] 20 x2+1dx 10xdx 10 x2+1dx 10x3dx 20(x+3) (youneednotevaluateit)adefiniteintegral( s)representingtheareaoftheregionboundedb ythecurvesy=x x3andy=x3 x.[5]7 Quiz# ,11 January,2002.[15minutes] (x2+x+1)3(4x+2)dx.[5] (x)=sin(x)cos(x)for0 x 2.[5]Quiz# ,18 January,2002.[15minutes] 1ln(x2)xdx.[5] xandy=x x3.[5]Quiz#12.(Lateversion.)Friday,18 January,2002.[15minutes] (2) 1exe2x+1dx.[5] 1andabovebythecurvey=cos( 2x),where 1 x 1.[5]Quiz# ,25 January,2002.[19minutes] ,y=x,andx=2aboutthex-axis.[10]Quiz# ,1 February,2002.[17minutes] [10]Quiz# ,15 February,2002.[25minutes] /40tan2(x)dx[4]2. x2+4x+5dx[6]Quiz# ,1 March,2002.[25minutes] : x2 2x 6(x2+2x+5)(x 1)dxQuiz# ,8 March,2002.[25minutes] : 21x(x 1)2dxQuiz# ,15 March,2002.[18minutes] n=0[1n+1+3n3n+1][4]2. n=02533n+1[6] ,18 March,2002.[15minutes]Computeanytwoof1 te t[5]2. 0te tdt[5]3.

7 N=01n2+3n+2[5]Quiz# ,22 March,2002.[20minutes] n=11nn[5]2. n=04n+12n2+6n+13[5]Quiz#19.(Alternatever sion.)Friday,22 March,2002.[20minutes]Considertheseries n=0arctan(n+1)n2+2n+2 [5] [5]Quiz# ,2 April,2002.[10minutes] n=0( 1)n+cos(n )n+1convergesabsolutely,convergesconditi onally,ordiverges.[10]Quiz#20.(Alternate version.)Tuesday,2 April,2002.[10minutes] n=0( 1)n+1+sin(n + 2)n+1convergesabsolutely,con-vergescondi tionally,ordiverges.[10]Quiz# ,5 April,2002.[10minutes] ,whenitconverges,equalsf(x)=3x2(1 x3)2.[10]Quiz#21.(Alternateversion.)Frid ay,5 April,2002.[10minutes] n=0( 1)nxn+1n!(whentheseriesconverges).[10]9 Mathematics 110 Calculus of one variableTrent University, 2001-2002 Test #1 Friday, 16 November, 2001 Time: 50 anytwoofa-c.[10 = 2 5 ea.] the definition of limits to verify that limx 2(5 x) = the definition of the derivative to verify thatf (2) = 5 iff(x) = 2x2 3x+ whetherf(x) ={x 1x2 1x6= 141x= 1is continuous ata= 1 or anythreeofa-d.}

8 [12 = 3 4 ea.] xcos(x) sin(cos( x)) (xy) = anytwoofa-c.[8 = 2 4 ea.] the equation of the tangent line toy= arctan(x) at the point(1, 4). many functionsh(x) are there such thath (x) =h(x)? the graph of a functionf(x) which is continuous on the domain (0, ) suchthat limx 0f(x) = and with a local maximum atx= 5 as its only critical point,or explain why there cannot be such a the intercepts, maxima and minima, inflection points, and vertical and horizontalasymptotes off(x) =xx2+ 1and sketch the graph off(x) based on this information.[10]10 Mathematics 110 Calculus of one variableTrent University, 2001-2002 Test #2 Friday, 8 February, 2002 Time: 50 anythreeof the integralsa-e.[12 = 3 4 ea.]a. /2 /2cos3(x)dxb. x2ln(x)dxc. 10(ex)2dxd. e2xln(e2x+ 1)e2x+ 1dxe. e1(ln(x)) anytwoofa-c.[8 = 2 4 ea.] 0(2x+ 3)dxusing the Right-hand 0 tdt(wherex 0) without evaluating the 1 1 x2dxby interpreting it as an is poured at a rate of 1m3/mininto a conical tank (set up point down) 2mhigh and with radius 1mat the top.

9 How quickly is the water rising in the tank atthe instant that it is 1mdeep over the tip of the cone?[8](The volume of a cone of heighthand radiusris13 r2h.) the region in the first quadrant with upper boundaryy=x2and lowerboundaryy=x3, and also the solid obtained by rotating this region about the region and find its area.[4] the solid and find its volume.[7] is the average area of either a washer or a shell (your pick!) for the solid?[1]11 Mathematics 110 Calculus of one variableFinal ExaminationTrent University, 24 April, 2002 Time:3hoursBrought to you byStefan :Show all your work and justify all your in doubt aboutsomething,ask!Aids:Calculator; 11 aid sheet or the pamphletFormula for Success; all four of1 anythreeofa e.[9 = 3 3ea.] (x) (x)x2+ (y)=e 0xcos(t2) sec2(arctan(x)) anythreeof the integralsa e.

10 [12 = 3 4ea.]a. 2x+3 1 x2dxb. /4 /4arctan(x)dxc. cos2(t)dtd. 11x3+xdxe. 1x2+2x+ anythreeof the limitsa e.[9 = 3 3ea.] 1x2+2x+1x+ 2n4n+ 0arctan(x) n2+2n+22n+ sin(x)x whether the given series converges absolutely, converges conditionally, ordiverges in anythreeofa e.[12 = 3 4ea.]a. n=22nln (n2)b. n=0( 1)n(n+1)n2+3n+9c. n=1( 1)4n2n+3d. n=0( 2)nn!e. n=0arctan( n)5n112 Part the domain, all maximum, minimum, and inflection points,andallverticalandhorizontal asymptotes off(x)=xln(x), and sketch its graph.[14] the region in the first quadrant bounded byy=sin(x)andy=2 the region.[2] the solid obtained by revolving the region about they-axis.[3] the volume of the solid.[7]Part the MacLaurin series of sin(x)anddetermineitsradiusofconvergence .[12] a function which is equal to 2x+3x2+8x3+15x4+32x5+63x6+128x7+ ,at least when this power series converges.