MATH 10B Review SessionInstructionsInstructions continuedFINALS TIME AND PLACEDoes anyone not know where his or her final is?Additional Sources For Help This WeekWhat is the best way to study?Color code!Course Outline (not all inclusive)Question 1aSolution 1aSolution 1a continuedQuestion 1bSolution 1b (for LEFT(4) )Solution 1b (for RIGHT(4) )Solution 1b (for MID(4))Solution 1b (for TRAP(4))Question 1cSolution 1cEstimating IntegralsBefore we move on… These are basic integrals you should know!More basic integrals you should know!The sine and cosine wheel:Slide 24Question 2a,b,c,dSolution 2aSlide 27Slide 28Solution 2a continuedSlide 30Slide 31Slide 32Solution 2bSlide 34Slide 35Slide 36Slide 37Short Solution to 2cShort Solution to 2c continuedQuestion 3a,b,cNotes on Question 3a,b,cMore Notes on Question 3a,b,cQuestion 4Question 5Question 6Question 7Question 8Solution 8Solution 8 continuedSlide 50Slide 51Question 9Solution 9Solution 9 continuedFranklin KenterMarch 15, 2009For this review session we will be doing actual problems so that you know what to study for, and moreover, so that you can identify your weaknesses and focus in those areas tGet into groups of 3 or 4. If possible, make a new friend or two.For each question, I will taking 3 or 4 groups to do the problems on the board. Each group that does so will be rewarded. Then we will go over the problems afterwards.While doing these problems, you reference your notes (even if your group is at the board). However, every time you reference your notes, YOU HAVE TO WRITE WHAT YOU REFERENCED ON YOUR NOTE SHEET!Lastly, this is a work in progress, please look out for and point out any mistakes.TIMEWed. Mar. 18,  7:00pm-10:00pm PLACE depends upon your lecture:John Eggers (9am Lecture) MANDE AUD Michael Volpato (12pm Lecture) WLH 2005 Shengli Kong (1pm Lecture) Last names A – L: PCYNH 109Last names M – Z: PCYNH 106 Evelyn Lunasin (4pm Lecture) Last names A – O: LEDDN AUD Last names P – Z: HSS 1330John Eggers (9am Lecture) MANDE AUD Michael Volpato (12pm Lecture) WLH 2005 Shengli Kong (1pm Lecture) Last names A – L: PCYNH 109Last names M – Z: PCYNH 106 Evelyn Lunasin (4pm Lecture) Last names A – O: LEDDN AUD Last names P – Z: HSS 1330I will stay here for a short time after.All of this information (including this slideshow) can be found at math.ucsd.edu/~fkenter/Dan Minsky has office hours: Monday and Tuesday from 4-5pm in APM 6446.Jacob Hughes will be holding another review session: Tuesday 8-9:30pm in CENTER 119DO PRACTICE PROBLEMS- as many as you can until you know it inside and outSpeaking or which:THERE ARE ALSO OTHER PRACTICE FINALS ON MY WEBSITE: math.ucsd.edu/~fkenter/ANSWERS will be in a green glowing boxIMPORTANT CONCEPTS will be in an orange glowing box.Guess what my favorite colors are?Chapters 5-6: The concept of an integralChapters 5 & 7.5: Estimating integralsChapter 7: Solving Integrals Algebraically- with the tools of substitution, by parts, andpartial fractions and trig substitutionChapter 7.7-8: Improper IntegralsChapter 8: Applications of integrals: specifically- Volumes of solids andValue of money over timeChapter 11: Differential EquationsHere is our first question. It’s easy, but has a small little trick:Estimate using LEFT(1)312dxxEstimate using LEFT(1) This is the number of subdivisions NOT the size of Δx.This means that there is 1 subdivision. Δx = = 312dxxnab 2113Intervalsf(x), on the left * Δx = Area For That Subdisvision(1,3) 1 2 2 2That might have been too easy, let us try this:Estimate using LEFT(4), RIGHT(4), MID(4), and TRAP(4)312dxxIntervalsf(x), on the left * Δx = Area For That Subdisvision(1,1.5) 1 0.5 0.5(1.5,2) 2.25 0.5 1.125(2,2.5) 4 0.5 2(2.5,3) 6.25 0.5 3.125nab 5.04136.75Intervalsf(x), on the right * Δx = Area For That Subdisvision(1,1.5) 2.25 0.5 1.125(1.5,2) 4 0.5 2(2,2.5) 6.25 0.5 3.125(2.5,3) 9 0.5 4.5nab 5.041310.75Intervalsf(x), on the left * Δx = Area For That Subdisvision(1,1.5)Mid = 1.251.5625 0.5 1.125(1.5,2)Mid = 1.753.0625 0.5 2(2,2.5)Mid = 2.255.0625 0.5 3.125(2.5,3)Mid = 2.757.5625 0.5 4.5nab 5.04138.625You *could* do a table for this one, but tell you what, it is much easier to take advantage of the fact that:TRAP(n) = ½ *(LEFT(n) + RIGHT(n)) So TRAP(4) = ½ *(LEFT(4) + RIGHT(4)) =½ *(6.75 + 10.25) = 8.5That might have been too easy, let us try this:Referring to Which ones of LEFT(4), RIGHT(4), MID(4), and TRAP(4) are overestimates? Underestimates?Why?312dxxNote the following rules:LEFT is an underestimate if f(x) is increasing, and an overestimate if f(x) is decreasing.Conversely, RIGHT is an overestimate if f(x) is increasing and an underestimate if f(x) is decreasing.Likewise, TRAP is an underestimate if f(x) is concave down, and an overestimate if f(x) is concave up.MID is an overestimate if f(x) is concave down, and an underestimate if f(x) is concave up.f(x) = , is concave up and increasing on (1,3).Apply the above rules to determine the answer.2xA few notes: the rules for Question 1c only work if the integral is *strictly* increasing (or decreasing, concave up, or concave down) on the interval.Cxxdx ||ln/1CedxexxCnxdxxnn)1/(1Except for n=-1Cxdxx )sin()cos(Cxdxx )cos()sin()sin(x)sin(x)cos(x)cos(xClockwise toDifferentiate:dxd)sin(x)sin(x)cos(x)cos(xCounterclockwise toIntegrate:dxCompute the following integrals: dxxx7)ln(dxexx)cos(dxxex2dxxx )1)(1(12What tool do we use? Why?dxxx7)ln(What tool do we use? Why?By parts- namely because substitution will not work (i.e. no part is the derivative of the other)dxxx7)ln(What tool do we use? Why?By parts- namely because substitution will not work (i.e. no part is the derivative of the other)The By Parts Formula: vuuvuv ''dxxx7)ln(But which one is u and which on is v'?dxxx7)ln(But which one is u and which on is v'?We will make u = ln(x). Why?dxxx7)ln(Because is “easier” to integrate! A ranking of functions to choose for u is (in general) given by:LogInverse Trig Polynomials Sine and Cosine7xdxxx7)ln(dxxx7)ln(xuxu/1')ln(78'8/xvxv dxxxxxdxxx 8/*/18/)ln()ln(887 vuuvuv ''Cxxx  64/8/)ln(88Don’t forget +CWhat tool do we use?