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4 Pages

### 242hwsolns0304

Course: MATH 242, Spring 2011
School: Delaware
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Word Count: 592

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solutions Brief for (parts of) Homeworks 3 and 4 Homework 4 /6 /6 3 0 0 = u u3 3 2. I = 1/2 = 0 (1 u2 ) du (1 sin x) cos x dx = cos x dx = 1. sin /6 2 sin 0 1 11 1 =. 2 24 24 sec3 x dx = sec x sec2 x dx. Integrate by parts with u = sec x, dv = sec2 x dx, so du = sec x tan x dx and v = tan x. Then I = sec x tan x sec x tan2 x dx I = sec x tan x sec x(sec2 x 1) dx I = sec x tan x sec3...

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solutions Brief for (parts of) Homeworks 3 and 4 Homework 4 /6 /6 3 0 0 = u u3 3 2. I = 1/2 = 0 (1 u2 ) du (1 sin x) cos x dx = cos x dx = 1. sin /6 2 sin 0 1 11 1 =. 2 24 24 sec3 x dx = sec x sec2 x dx. Integrate by parts with u = sec x, dv = sec2 x dx, so du = sec x tan x dx and v = tan x. Then I = sec x tan x sec x tan2 x dx I = sec x tan x sec x(sec2 x 1) dx I = sec x tan x sec3 x dx + sec x dx I = sec x tan x I + ln | sec x + tan x| 2I = sec x tan x + ln | sec x + tan x| 1 I= sec x tan x + ln | sec x + tan x| + C 2 3. = x3 1000 tan3 dx = 10 sec2 d 2 x2 + 100 100 tan + 100 3 1000 tan 10 sec2 d = 1000 tan3 sec d 10 sec = 1000 tan2 sec tan d = 1000 = 1000 (u2 1) du = 1000 (sec2 1) sec tan d u3 sec3 u = 1000 sec . 3 3 1 x2 + 100 = sec . The nal answer is then 10 1 (x2 + 100)3/2 x2 + 100 1000 +C 3 1000 10 x Since = tan , it follows that 10 which can probably be written in other ways as well. 4. We look for A and B that satisfy x2 x9 A B x9 = = + + 3x 10 (x 2)(x + 5) x2 x+5 which implies x 9 = A(x + 5) + B (x 2). Either expanding and comparing, or plugging in special values like x = 2 and x = 5, we nd that A = 1 and B = 2. Therefore the integral is 2 1 dx = 2 ln |x + 5| ln |x 2| + C. x+5 x2 Homework 3 5 xex dx, try parts with u = x, dv = ex dx. Then du = 1 dx and 1. For 0 v = ex . Then the integral is xex 5 5 0 5 ex dx = 5e5 + ex dx = 5e5 + ex 0 = 5e5 + ex 0 0 0 5 = 5e5 + 1 e5 = 1 6e5 . 5 2. For t sin 2t dt, try parts with u = t, = dv sin 2t dt. Then du = dt and v = (1/2) cos 2t. Then the integral is 1 t 1 t cos 2t dt = cos 2t + cos 2t 2 2 2 2 t 1 = cos 2t + sin 2t + C . 2 4 2 cos 2t dt 3. For ln x dx, try parts with u = ln x, dv = 1 dx. Then du = (1/x) dx and v = x. Then the integral is 1 x ln x x dx = x ln x 1 dx = x ln x x + C. x 4. For arcsin x dx, try parts with u = arcsin x, dv = 1 dx. Then 1 dx and v = x. Then the integral is 1 x2 x 1 2x x arcsin x dx = x arcsin x + dx 2 1 x2 1 x2 1 1 du = x arcsin x + u = x arcsin x + 1 x2 + C . = x arcsin x + 2 u du = ex cos 2x dx, try parts. We can start in either of two ways! 5. For I = Say we choose u = ex and dv = cos 2x dx. Then du = ex dx and v = (1/2) sin 2x. Then we have 1 1 1 1 ex sin 2x dx = ex sin 2x + ex sin 2x dx. I = ex sin 2x 2 2 2 2 Integrate by parts again, using a similar choice as before. Choose u = ex and dv = sin 2x dx. Then du = ex dx and v = (1/2) cos 2x. Then 1 1 I = ex sin 2x + 2 2 1 ex cos 2x 2 1 + ex cos 2x dx 2 1 1 1 I = ex sin 2x ex cos 2x ex cos 2x dx 2 4 4 1 1 5 I = ex sin 2x ex cos 2x 4 2 4 2 x 1 x I = e sin 2x e cos 2x + C . 5 5 /2 1 cos x dx = 2 /2 2 6. 0 (1 + cos 2x) dx = 0 3 1 sin 2x x+ 2 2 /2 = 0 . 4 sin6 x cos3 x dx = 7. = (u6 u8 ) du = /2 cos5 x dx = 0 /2 (cos2 x)2 cos x dx = 0 0 (1 u2 )2 du = x=0 =1 (1 sin2 x)2 cos x dx 1 x=/2 = u6 (1 u2 ) du sin7 x sin9 x u7 u9 = + C. 7 9 7 9 /2 8. sin6 x(1 sin2 x) cos x dx = (1 2u2 + u4 ) du = u 0 21 15 10 3 8 += + =. 35 15 15 15 15 4 2u3 u5 + 3 5 1 0
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Delaware - MATH - 242
Math 242 Review Part 1In this course, we covered most of the following four chapters. Chapter 6 (Applications of Integration) Chapter 7 (Techniques of Integration) Chapter 11 (Innite Series) Chapter 10 (Parametric and Polar Curves)There were three a
Delaware - MATH - 242
Math 242 Review Part 2In addition to (most of) Chapters 6, 7, 11, and 10, we studied the followingthree miscellaneous topics: Newtons Method (4.8) LHopitals Rule (4.4) Arc Length (8.1)1. Use Newtons method with the specied initial approximation x1 t
Delaware - MATH - 242
Math 242Suggested lab/discussion problems, Sept 6 and 81. Evaluate lim x1/x .x12. Evaluate xlim ( x2 + x x).23. Evaluate(3x 2)20 dx.34. Evaluatesin t dt.4
Delaware - MATH - 242
Math 242Suggested lab/discussion problems, Sept 13 and 151. Find the area bounded by the curves y = x2 and y =1x22.+12. Find the area bounded by the curves y = |x| and y = x2 2.23. Let R be the region bounded by the curves y = 1 x2 and y = 0. Fi
Delaware - MATH - 242
Math 242Suggested lab/discussion problems, Sept 20 and 221. Let A be the region bounded by y = ex , y = 1, and x = 2. Find thevolume obtained by revolving A around the line y = 2.12. Let A be the region bounded by y = 4x x2 and y = 3. Find the volume
Delaware - MATH - 242
Math 242Suggested lab/discussion problems, Sept 27 and 291. Evaluatesin4 x dx.12. Evaluatecos5 x dx.2173. Evaluatexex dx.03 /24. Evaluatesin3 x cos11 x dx.04
Delaware - MATH - 242
Math 242Suggested lab/discussion problems, Oct 4 and 6101. Evaluate4x26dx.911dx.2 + 2xxHINT: Start by completing the square.2. Evaluate213. Evaluatex x2 + 4 dx.034a. Evaluatexdx.x2 + 14b. Evaluatex2dx.x2 + 14
Delaware - MATH - 242
Math 242Suggested lab/discussion problems, Oct 11 and 1321. Use the Trapezoidal Rule and Simpsons rule to approximate0using n = 4 subintervals.11dx1 + x42. Evaluate0(x2xdx.+ 2)223. Find a good approximate root of x3 x 1 = 0 by starting wi
Delaware - MATH - 242
Math 242Suggested lab/discussion problems, Oct 18 and 201. By hand, nd an approximation to 71/3 . Choose an appropriate functionf (x), an appropriate rst approximation x1 , and nd x2 .12. Find the length of the curve y = ln(sec x) between x = 0 and x
Delaware - MATH - 242
Math 242Suggested lab/discussion problems, Oct 25 and 27This is a longer list of problems than usual. It can also serve as practice forTest 2. /21a. Evaluatecos3 x dx.0 /21b. Evaluatecos2 x dx.012. Evaluatesec3 x dx.223. Evaluate1x2 1d
Delaware - MATH - 242
Math 242Suggested lab/discussion problems, Nov 1 and 3D1. Does the series converge or diverge?1nn=1 n + 3D2. Does the series converge absolutely, converge conditionally, or diverge?(1)n+1n=2 n ln n1D3. Does the series converge absolutely, conver
Delaware - MATH - 242
Math 242Suggested lab/discussion problems, Nov 8 and 10D1. Does the series converge or diverge?n2 132n=1 n + 2n + 5xnD2. Find the interval of convergence of the power series.n=0 n!1D3. Find the interval of convergence of the power series(2)n
Delaware - MATH - 242
Math 242Suggested lab/discussion problems, Nov 15 and 17D1. Find a power series representation of the function f (x) =is the radius of convergence of your power series?1. What1 + x2D2. Find a power series representation of the function f (x) = arct
Delaware - MATH - 242
Math 242Suggested lab/discussion problems, Nov 221. Evaluate the integralcos x (1 + sin2 x) dx.2. Evaluate the integralsin3 xdx.cos x113. Evaluate the integral0earctan xdx.1 + x24. Does the series2n n!converge or diverge?n=1 (n + 2)!25
Delaware - MATH - 242
Math 242Suggested lab/discussion problems, Nov 29 and Dec 1converge or diverge? If it converges,nn=1does it converge absolutely or conditionally?(1)n sinD1. Does the seriesnD2. Does the seriesn=123nconverge or diverge?1D3. Does the series
Delaware - MATH - 242
Math 242, U of D, Spring 2011Analytic Geometry and Calculus BInstructor: Idris Mercer, Ewing 529, idmercer@math.udel.eduIn emails, please write Math 242 and your section number.Oce hours: Mondays 5:006:30, Thursdays 9:0011:00, or by appointmentCourse
Delaware - MATH - 242
M ath 242 Section 010: Test #1NAME: _L/CPlease circle your l ab/discussion:020021022This test has 6 questions on 6 pages.The p oints per page are 5, 5, 5, 5, C, 6.Show your &quot;work. A correct answer w ith l ittle or no reasoning may reeeivea low s
Delaware - MATH - 242
M ath 242 Section O il: Test #1NAME: _frPlease circle your lab/discussion:030031032This test has 6 questions on 6 pages.The points per page are 5, 5, 5, 5, G, 6.Show your work. A correct answer with little or no reasoning may receivea low score.
Delaware - MATH - 242
Math 242 Section 012: Tost #1NAME:KEYPlease c ircle y our l ab/discussion: 040 ( 8:00) 041 (12:30) 042 ( 1:30)This t est, has 6 questions on 6 pages.The, p oints per page are 5, 5. 5, .r&gt;, G , fi.Show your work. A correct answer with little or no r
Delaware - MATH - 242
Math 2 42 S uction 0 14: TostN AME:KEYPlease c ircle your l ab/discussion: 060 ( 12:00) 061 (11:00) 062 ( 2:30)This test has 6 questions on 6 pages.The points per page are 5. 5, 5, 5, 6. 6.Show your work. A correct, a nswer with l ittle or no r easo
Delaware - MATH - 242
M ath 242 Section 050: Testi/c:NAME: _(^f^This test has G q uestions on 6 pages.The p oints p er pa^e are 5, 5, 5, 5, ( i, 0.Show y our w ork, A c orrect answer w ith l ittle o r no r easoning m ay r eceivea. low score. (However, there is no need to
Delaware - MATH - 242
Math 242 Section 010: Test #2N AME:Please circle your lab/discussion:020021022This test has 6 questions on 6 pages.The points per page are 5, 5, 5, 5, 6, 6.Show your work. A correct answer w ith little or no reasoning may receivea low score. (How
Delaware - MATH - 242
Math 242 Section 014: Test #2NAME:Please circle your lab/discussion:060061062This test has 6 questions on 6 pages.The points per page are 5, 5, 5, 5, 6, 6.Show your work. A correct answer w ith little or no reasoning may receivea low score. (Howe
Delaware - MATH - 242
Math 242 Section 012: Test #2NAME:Please circle your lab/discussion: 040 (8:00) 041 (12:30) 042 (1:30)There are 6 questions on 6 pages. Points per page: 5, 5, 5, 5, 6, 6.Show your work. A correct answer with little or no reasoning may receivea low sc
Delaware - MATH - 242
M ath 242 Section 014: Test #2NAME:&lt;^ ^ VPlease circle your lab/discussion: 060 (12:00) 061 (11:00) 062 (2:30)There are 6 questions on 6 pages. Points per page: 5, 5, 5, 5, 6, 6.Show your work. A correct answer with little or no reasoning may receive
Delaware - MATH - 242
Math 242 Section 050: Test #2NAME:There are 6 questions on 6 pages. Points per page: 5, 5, 5, 5, 6, 6.Show your work. A correct answer with little or no reasoning may receivea low score. (However, there is no need to be wordy.)The 7th page has some f
Delaware - MATH - 242
Math 242 Section 012: Test #3N AME:KEYPloa.sc c ircle your l ab/discussion: 040 (8:00) 041 (12:30) 042 (1:30)There are 6 questions on 6 pages. Points per page: 5, 5, 5, 0, 5, 6.The 7tli page has some formulas you can refer to if needed.[5 points] Qu
Delaware - MATH - 242
Math 242 Section 014: Test #3N AME:'Please circle your lab/discussion: 060 (12:00) 061 (11:00) 062 (2:30)There are 6 questions on 6 pages. Points per page: 5, 5, 5, C, 5, 6.The 7th page has some f ormulas you eari refer to if n eeded.[5 p oints] Que
Delaware - MATH - 242
M ath 242 Section 050: Test #3NAME:_There are 6 questions on 6 pages. P ointy per page: 5, 5, 5, 0, 5, 6.The 7th page has some f ormulas you can refer to if needed.[5 points] Question 1. D etermine whether the i nfinite seriesis absolutely convergen
Delaware - MATH - 242
TO S OMfcdx.8. Evaluate-x-K/0010-0.6f)0rh.9. Evaluate / ye~-6y dy.JoU'-O.b-0.6Jo10. Evaluate6'~0 -r /2 cos o1 + sinXV~Jo,rffl.U11. Evaluateu=It +=?323UJ:,12. Evaluate /_1/1.\.il:i2shr 0 cos-0 &lt;/0.7ofNTT/2-00U
Delaware - MATH - 242
IDCHf\Pm 7In problems 23 to 28, evaluate the integral or show t hat it is d ivergent.,00,PML23.i'&quot;%'3L\J324.,dxMM25. /f l/audx13r&gt;\?rovooH&gt;-4--V\MvoXr.X\MotocX^-t&lt;xcto4-P\IO4-14:tou]OS
Delaware - MATH - 242
1. Sketch t he curve. Indicate the direction in which the curve is traversed ast increases. If possible, eliminate the parameter to find a Cartesian equationof the curve.la.x = 3i - 5,y = 2t + la+5&quot; - s-tM&quot;3fH1l-1-lb.3.&gt;x = t2 - 2,3(/ =
Delaware - MATH - 242
iitIii questions 1 to 28, d etermine w hether t he series converges or diverges.iooV^ / o-n3-&quot; +1.3/2p-sn?;rsOO-12. V tan-j^vi n1&lt;:.os KIntA-M^1fw^ ^-^ . ^ ~^)3 c osy?,3.77/V2 _ 2q3'^5 AKI ^-2^choose c, ^V)J|c.*I^/ ^/HfCCO
Delaware - MATH - 242
Iii questions 29 to 36, d etermine whether the scries converges absolutely, converges conditionally, ordiverges.n29.3n0&gt;30.J'IJnofisi/In=l5011In questions 37 to 42, find the interval of convergence of the power scries.3. - 2 )/^a-Z
Delaware - MATH - 242
(?ay71/-7CXTOoT7T007X=Y)p~l7._vi /3/3/3-/Wt /i i^.^/ ^3n3&quot;- Ji lt*i^*&quot;-oi&quot;^tA -^oOi&lt;~)~y^,tTi 3 + 5'&quot;)'.-IK00f-i' n5An+1n3/2 - 2rartA/|AI&quot;/6^J=.^_/-x16. V ^n)*17.n siTi1n^W00O Tl/aL,
Delaware - MATH - 242
In questions 1 to 28, determine whether the series converges or diverges.11.3n + 3/2nn=1tan2.n=11n3.3 cos n3/2 2n=1 n14.n=15.n=1132n1+n1ln(n + 1)6.1n=2 n(ln n)27.1n=2 n ln n8.12n=2 n(ln n)9.12n=2 (ln n)310.1n=
Delaware - MATH - 242
The last page of your test will contain exactly these formulas in exactly thisformat.sin2 x =11 cos 2x2cos2 x =11 + cos 2x21dx = ?1 + x2sec x dx = ln | sec x + tan x| + C1+x2 x3x+++ = ?1!2!3!xx 3 x5 x7++ = ?3!5!7!1x 2 x4 x6
Delaware - MATH - 242
Review for beginning of Math 242This document covers some topics that you should have learned in Math 241or whatever your equivalent course was.If x = 3, evaluate x2 , x2 , 2x2 , and 2x2 .If x = 3, evaluate x2 , x2 , 2x2 , and 2x2 .Expand and simplif
Delaware - MATH - 242
Strategy for testing seriesWe skipped Section 11.7, which is very short and contains some strategyfor testing series. Heres my version of that strategy.STEP 0. Does the nth term an approach 0? If it doesnt, then the seriesdiverges. (I think of this as
Delaware - MATH - 242
INFINITE SERIES (Chapter 11)General facts:A sequence is nothing but an innite list. A series is a sum of innitely many numbers.An innite series may either converge (i.e. reach a nite total) or diverge (i.e. not reach a nite total).1A famous example o
Delaware - MATH - 242
This page has some formulas that will NOT be given on your test.You will probably want to memorize these formulas, and also understandwhen and how they are used.This is an INCOMPLETE list of SOME things you will need to know. R2 r2 dxVolume by washer
Delaware - MATH - 242
This page is not part of your test.It contains some formulas you may nd useful.Note that some formulas contain blanks.You should know how to ll in the blanks, but you are not required to llthem in explicitly.sin cos tan 0/6/4/3/2dsin x =dx
Delaware - MATH - 242
This page is not part of your test.It contains some formulas you may nd useful.Note that some formulas contain blanks.You should know how to ll in the blanks, but you are not required to llthem in explicitly.sin cos tan 0/6/4/3/2sin2 x =11
Delaware - MATH - 242
Math 242Suggested lab/discussion problems, Oct 25 and 27This is a longer list of problems t han u sual. It can also serve as pract.iec forTest 2.r1'2 cosla. Evaluate /,rr/21x dx.7O\$ 2 / X COS ^o^6(A ' S'^ ^/-COSIb. Evaluate /o,cos x dx
Delaware - MATH - 242
This page is not part of your test.It contains some formulas you may nd useful.Note that some formulas contain blanks.You should know how to ll in the blanks, but you are not required to llthem in explicitly.sin2 x =11 cos 2x2cos2 x =11 + cos 2
Delaware - MATH - 242
Math 242 Section 014: Test #1NAME:Please circle your lab/discussion: 060 (12:00) 061 (11:00) 062 (2:30)This test has 6 questions on 6 pages.The points per page are 5, 5, 5, 5, 6, 6.Show your work. A correct answer with little or no reasoning may rece
Delaware - MATH - 242
Math 242 Section 012: Test #1NAME:Please circle your lab/discussion: 040 (8:00) 041 (12:30) 042 (1:30)This test has 6 questions on 6 pages.The points per page are 5, 5, 5, 5, 6, 6.Show your work. A correct answer with little or no reasoning may recei
Delaware - MATH - 242
Math 242 Section 050: Test #1NAME:This test has 6 questions on 6 pages.The points per page are 5, 5, 5, 5, 6, 6.Show your work. A correct answer with little or no reasoning may receivea low score. (However, there is no need to be wordy.)The 7th page
Delaware - MATH - 242
Math 242 Section 014: Test #2NAME:Please circle your lab/discussion: 060 (12:00) 061 (11:00) 062 (2:30)There are 6 questions on 6 pages. Points per page: 5, 5, 5, 5, 6, 6.Show your work. A correct answer with little or no reasoning may receivea low s
Delaware - MATH - 242
Math 242 Section 012: Test #2NAME:Please circle your lab/discussion: 040 (8:00) 041 (12:30) 042 (1:30)There are 6 questions on 6 pages. Points per page: 5, 5, 5, 5, 6, 6.Show your work. A correct answer with little or no reasoning may receivea low sc
Delaware - MATH - 242
Math 242 Section 050: Test #2NAME:There are 6 questions on 6 pages. Points per page: 5, 5, 5, 5, 6, 6.Show your work. A correct answer with little or no reasoning may receivea low score. (However, there is no need to be wordy.)The 7th page has some f
Delaware - MATH - 242
Math 242 Section 014: Test #3NAME:Please circle your lab/discussion: 060 (12:00) 061 (11:00) 062 (2:30)There are 6 questions on 6 pages. Points per page: 5, 5, 5, 6, 5, 6.The 7th page has some formulas you can refer to if needed.[5 points] Question 1
Delaware - MATH - 242
Math 242 Section 012: Test #3NAME:Please circle your lab/discussion: 040 (8:00) 041 (12:30) 042 (1:30)There are 6 questions on 6 pages. Points per page: 5, 5, 5, 6, 5, 6.The 7th page has some formulas you can refer to if needed.[5 points] Question 1.
Delaware - MATH - 242
Math 242 Section 050: Test #3NAME:There are 6 questions on 6 pages. Points per page: 5, 5, 5, 6, 5, 6.The 7th page has some formulas you can refer to if needed.[5 points] Question 1. Determine whether the innite seriesn(1)n n4 + 5n=0is absolutely
Delaware - MATH - 242
Math 242 Final Exam, Spring 2011Sections 012, 014, 050NAME:There are 12 questions on 12 pages.The points per page are 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6.The 13th page has some formulas you can refer to if needed.[5 points] Question 1. Evaluate the l
Delaware - MATH - 242
Math 242 Review Chapter 10 HighlightsFor slope or concavity of parametric curves, use the following:dy=dxd2 yd dy==2dxdx dxdThe latter equation comes from(u) =dxdudtdxdtdydtdxdtd dy()dt dxdxdt.For converting between rectangul
Delaware - MATH - 210
Math 210-011 Final Exam, Spring 2011NAME:_There are 12 questions on 12 pages.The points per page are 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6.[2 points] Question la. Construct a t ruth table for P V (P + Q).PQTTFFTFTFTFTTTTTT[2 points] Qu
Delaware - MATH - 210
Math 210 Homework 1Due Friday February 18th1. Classify each of the following statements as true or false. 3(i) 3 is rational or (4)2 = 16(ii) 4 = 2 + 2 7 &lt; 50(iii) 4 = 2 + 2 7 &lt; 50(iv) 4 = 2 + 2 7 &gt; 50(v) 4 = 2 + 2 7 &gt; 502. Rewrite each statement
Delaware - MATH - 210
Math 210 Homework 2Due Wednesday February 23rd1. (This example is due to Lewis Carroll.) Consider the statements: Animals that do not kick are always unexcitable. Donkeys have no horns. A bualo can always toss you over a gate. No animals that kick a
Delaware - MATH - 210
Math 210 Homework 3Due Monday February 28th1. Connectives for statements are similar in some ways to operationson sets. This question will illustrate that.(i) Construct a truth table for (P Q).(ii) Construct a truth table for (P ) Q.Your answers to
Delaware - MATH - 210
(TO0CA&gt;nDo?V 3' ^n^-1 H ~n-i H T; -nHTS Tl -1 HJ^7jT7Tin H -nHnTMjJoj&gt;71 T -M?&quot;o!-oOiCi- D.- g=VA/Jcns-rr^c^c^&gt;cCOOD-sonCtoCo3CneDObc10DrscCoCONCX)COoc*ctoImSCCoCO]octo