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Calc & Analyt Geom II (HONORS)

by: Mason Larson DDS

Calc & Analyt Geom II (HONORS) MATH 2423

Marketplace > University of Oklahoma > Mathematics (M) > MATH 2423 > Calc Analyt Geom II HONORS
Mason Larson DDS
GPA 3.82

J. Albert

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J. Albert
Class Notes
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This 5 page Class Notes was uploaded by Mason Larson DDS on Monday October 26, 2015. The Class Notes belongs to MATH 2423 at University of Oklahoma taught by J. Albert in Fall. Since its upload, it has received 10 views. For similar materials see /class/229286/math-2423-university-of-oklahoma in Mathematics (M) at University of Oklahoma.


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Date Created: 10/26/15
Review for Final Exam The nal exam is comprehensive covering the material in all the sections of the text from which problems were assigned 7 with the exceptions of sections 75 and 93 Thus the nal covers sections 52 53 54 55 61 62 63 64 72 73 74 76 78 81 82 83 84 88 91 and 92 You can refer to the review sheets for Exams 1 2 and 3 to see which portions to review of the sections up through section 83 Notice though that I added a few problems from section 83 to assignment l2 and such problems might appear on the nal see below Like Exam 2 the nal might include a question asking you to prove one or more of the following 39ixixti fli7 39 1 71 formulas 1 die 7 e 11 dag lnz 7 E 111 d1 arcs1nz 7 or 1v dag arctanz 7 my See the 171 review sheet for Exam 2 to brush up on these proofs This would also be a good time to make sure you have the basic differentiation and integration formulas memorized see List of Integrals on the review sheet for Exam 3 Remember not to get derivatives and integrals mixed up For example the derivative of lnz is i but the integral of lnz is more complicated you can nd it using integration by parts Here is a sectionbysection guide to the material in the text that was not on the rst three exams but could appear on the nal exam 83 This section was already covered on the third exam but there were also a few problems assigned in section 83 on assignment 12 numbers 23 24 and 35 that you might not have reviewed for the third exam so you might go over them again now They involve completing the square an example is given in Example 7 on pp 5077508 Completing the square is also occasionally useful when doing integration by partial fractions see Example 6 on p 515 in section 84 84 You should review from the beginning of the section through Example 6 including the Note which appears at the bottom of p 515 after Example 6 You can skip rest of the material in the section I will not ask any questions in which the denominator contains a repeated irreducible quadratic factor or problems which require a rationalizing substitution That is you can skip Examples 7 8 an 9 85 l didnlt assign any problems from this section but you might get some bene t from glancing through it It contains hints for deciding which method to use on a given integral What these hints boil down to is if algebraic simpli cation doesn7t help and there s no obvious simplifying substitution to make then if the function to be integrated is a quotient of polynomials use partial fractions ii if it contains the square root of a quadratic use inverse trig substitution and iii if it contains a mixture of algebraic power functions and transcendental exponential logarithmic trig or inverse trig functions try integration by parts 88 Read from the beginning of the section through Example 8 You can skip the remainder of the section A comparison test for improper integrals 91 You should remember formulas 3 and 4 for arc length in the boxes on pages 562 and 563 Also read Examples 1 and 2 from this section You can skip the rest of the section except that reading the rst paragraph at the top of p 565 along with the accompanying diagram might help you to remember the arc length formulas 92 You can skip the rst part of the section in which an integral formula for surface area of a solid of revolution is derived and start reading from about halfway down the page on p 570 where it says Therefore in the case where f is positive through to the end of the section As I was saying in class there are two basic formulas in this section the ones given in the red boxes numbered 7 and 8 The rst is for a solid revolved around the zaxis and the second is for a solid revolved around the y axis The diagrams in l Figure 5 should help in keeping these formulas straighti In each of these two forrnulas7 d3 can be interpreted in two ways7 as explained just below box 8 so each formula can be done either as an integral with respect to z or as an integral with respect to y 93 There will be no material from section 93 on the nal exarni Trigonometry Review Although you probably know enough about trigonometry to get by in this class already it might make the class a lot easier for you if you take a little time to brush up on the following basic facts They re taken from Appendix D of the text but you don7t need to memorize everything that7s in Appendix D 7 only what s mentioned belowi Radian measure of angles We don t use degree measures for angles at all in calculus if we can help it the reason being that the formulas that you7ve learned for derivatives and integrals of trigonometric functions are actually incorrect when degree measures are used If we de ne a function by setting equal to the sine of the angle whose measure is 1 degrees then the derivative of this function is not the cosine of the angle whose measure is 1 degrees So try to get used to expressing all angles in radian measurel When you want to convert from degrees to radians remember that 360 degrees is 27f radians 7 so you have to multiply degrees by 27r360 or 7T180 to get radiansi I still use degree measure often because it s just easier to refer to angles that way But as soon as derivatives or integrals enter the picture itls imperative to switch to radian measurel De nition and values of trigonometric functions For angles 6 between 0 and 90 degrees you can de ne the trigonometric functions by drawing a right triangle with angle 9 in it and letting sint9 be the opposite side over the hypotenuse and 3086 be the adjacent side over the hypotenuse see gure 6 on page A26 in Appendix D Then you can gure out the values of the other four trig functions by using the facts that sin 9 l 0086 secgi cscgiml l tant9 You should memorize the side lengths of the 454590 triangles and 306090 triangles Figure 8 on page A26 With those memorized you then automatically know the values of all the trigonometric functions at 9 7r6 7r4 and TrSl To nd the sines and cosines of angles 9 of more than 90 degrees or less than 0 degrees you have to use the de nitions 2 cost9 E sint9 at T T Here I and y are the z and y coordinates of a point P which lies on the line through the origin which makes an angle 9 with the positive zaxis and T is the distance from P to the origin ltls often easiest to take T 1 so P is on the unit circle Once you know the sine and cosine you can nd the values of the other trigonometric functions using the formulas in equations 1 above The procedure of how to use these formulas is illustrated in Example 3 on page You can use the above de nition and your knowledge of the 454590 and 306090 triangles to easily gure out the sine and cosine of any angle which is a multiple of 30 or 45 degrees including negative anglesl You don7t need to memorize tables like the one in blue on page A2 Quite often you have to nd the angles at which the sine and cosine take the values 0 l and ill The easiest way to do this is by using the formulas in For example to nd the values of 9 for which cos 9 71 you think of the point P on the unit circle whose z coordinate is 71 that is the point P 710 This is on a line which makes an angle of 180 degrees or 7r radians with the positive zaxis so the cosine of 7r is zero But the same line also makes other angles with the positive zaxis namely 7r i 2 7r i 4 etc So the angles whose cosine are zero are 7r plus or minus any multiple of 27L In other words cost9 0 if 9 is in the set i i i 777T 757T 737T 7W7T37T57T 77r i i Trigonometric identities You should know the basic trigonometric identity 3 sin2 9 cos2 9 ll 1 Another identity that comes up a lot is 4 tan2 9 l sec2 9 but one bit of good news is that you donlt have remember 4 separately since you can always derive it from 3 when you need it simply by dividing 3 by cos2 9 and using the rst two de nitions in There are a number of other trigonometric identities in Appendix D all of which are occasionally useful but the only other ones I recommend you memorize are sin7t9 7 sin 9 cos7t9 cost9 sin2t9 2 sin 9 cos 9 cos2t9 cos2 9 7 sin2 9 Any other identities you might need on the exams llll supply to you on the exams themselvesi Graphs of the trigonometric functions You should memorize the graphs of the trigonometric functions Figure 13 of Appendix D That completes the review of what you need to remember of trigonometry from precalculusi As an added bonus here7s a summary of what you will need to remember of trigonometry from calculus Derivatives and antiderivatives of trigonometric functions You should memorize the formulas for the derivatives of the six trigonometric functions d d Es1nz cosz csc z 7 csczcotz d d d7cos z 7sinz d7sec z secztanz z z d d d7tan z sec2 z d7cot z 7 csc2 z z z Another bit of good news is that you don7t have to remember the corresponding antiderivative formulas separately 7 theylre just the reverse of the above formulasi For example Since ditan z sec2 z then sec2 z dz tanz O z One thing to watch out for is that we don7t yet have formulas for the antiderivatives of tanz and sec z So you have to be careful not to inadvertently indulge in wishful thinking and say that the antiderivative of tanz is sec2 z or something of the sort We will be getting formulas for the antiderivatives of tanz and secz later in chapter 7 after we7ve introduced logarithmsi These won7t be covered on the rst exam of course but if you want a sneak preview the formulas are tanz dz lnlseczlC secz dz lnlsecztanzlCi Review sheet for 3rd exam The third exam will cover sections 74 76 78 81 82 and 83 You should review the problems from Assignments 8 9 10 and 1139 and Quizzes 5 and 6 I recommend trying to do similar problems you can easily recognize which problems in the text are similar to the ones I assigned without help from anyone as a check on whether you re ready for the test There was a problem from section 75 on one of the assignments but this type of problem will not appear on this exam Here is a guide to which portions of these sections will and will not be covered on the exam 74 Derivatives of logarithmic functions Most of this section was already covered on the second exam Of course you should still know what is in this section for the third exam since otherwise you won t be able to do many of the problems from later sections But more speci cally the exam might contain a problem involving logarithmic differentiation see pages 4177418 76 Inverse trigonometric functions It s worth reviewing the entire section but you can skip Examples 2 and 6 if you like Example 3 is especially important for what comes later in section 83 You should memorize the formulas for the derivative of the arcsine and arctangent functions see the box at the top of page 459 and the corresponding formulas for integrals see boxes 12 and 13 on page 460 You will not be asked to prove the formulas for the derivative of the arcsine and arctangent functions on this exam 78 Indeterminate forms and L hopital s rule You should review the entire section except skip Cauchy s Mean Value Theorem p 477 and its proof I m not too likely to ask questions like Examples 9 and 10 but I think it pays to read these examples anyway List of Integrals On p 488 there is a convenient list of all the integrals you should know for the exam Actually you can omit from this list the integrals of cscz csczcotz and cotz as well as those of the hyperbolic functions sinhz and coshz we haven t covered hyperbolic functions in this class 81 Integration by parts Review the entire section except you can skip Example 6 You should memorize the integration by parts formula fu d1 uv 7 f1 du 82 Trigonometric integrals There are a lot of rules in this section in red boxes but I don t recommend memorizing any of them Instead just read the examples carefully you can skip Example 9 2 If the formula for the integral of sec 1 or the formulas cos2 I l cos 21 or sin I l 7 cos 21 are needed for the exam I ll supply them to you on the exam but it certainly doesn t hurt to memorize them 83 Trigonometric substitution This section would be better titled Inverse 39I rigonometric Substi tution to re ect the fact that the substitutions here do not involve setting the new variable u equal to a trigonometric function of the old variable I eg u sin I but the other way around the old variable I is set equal to a trigonometric function of the new variable 9 eg z sin 9 You should review the entire section or better yet try doing some of the problems at the end of the section which weren t assigned Remember that the answers to the problems should be functions of the original variable 139 to change from the new variable 9 back to I it is often useful to draw a diagram like the ones in gures I 3 or 4


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