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Calc & Analyt Geom II (HONORS)

by: Einar Nitzsche

Calc & Analyt Geom II (HONORS) MATH 2423

Marketplace > University of Oklahoma > Mathematics (M) > MATH 2423 > Calc Analyt Geom II HONORS
Einar Nitzsche
GPA 3.99

J. Albert

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J. Albert
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This 4 page Study Guide was uploaded by Einar Nitzsche on Monday October 26, 2015. The Study Guide belongs to MATH 2423 at University of Oklahoma taught by J. Albert in Fall. Since its upload, it has received 25 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 Exam 2 This exam covers sections 62 63 64 72 73 and 74 of the text as It will include a question asking you to prove one or more of the following formulas e e lnz iii arcsinz or iv arctanz 10 The proof of formula goes like this fee 7 hm dz hao h zeh 7 ex lim hgt0 h h e 71 lim ex hgt0 h h e i ex hm 7 hgt0 h exlem The proofs of formulas ii iii and iv are all similar You can nd all three proofs in your lecture notes or slightly longer versions in the text on pages 411412 456 and 458 Either version is okay for the test The remaining problems on the test will be similar to the homework problems from assignments 4 through 7 You should also review the quiz problems from quizzes 3 and 4 Here is a sectionbysection guide to the material in the text that will be covered on this test 62 This section introduces the formula for volume by slices V 5141 dz You should review the entire section except that you don t need to read Examples 7 8 and 9 since I won t be asking questions like these examples on the exam However it would be a good idea to read one of these examples anyway39 the increased understanding you d gain would be worth it To understand what the formula for volume by slices means you have to visualize a picture like that in Figure 3 on page 355 The object whose volume you are computing is being cut by a slicer and the slices are perpendicular to the zaxis The area of the crosssection of the solid obtained by slicing at location I is If the object is a solid of revolution around the zaxis then the crosssections are either discs as in Figure 6 or washers discs with holes in them as in Figure 8 If the object is not a solid of revolution around the zaxis then the crosssections will not be circles they could be squares as in Example 8 or triangles as in Examples 7 and 9 But as I mentioned above I won t ask questions on the test about such solids39 the only solids you will see on the test will be solids of revolution You should remember that if the slices are perpendicular to the y axis then the volume by slices formula becomes V f0 Ay dy instead In general to make sure you are integrating with respect to the correct variable in your volume formula you should visualize the elements of volume you are using 7 the small pieces you are putting together to make the entire volume 7 and take the variable of integration to measure the thin direction of the element If the elements are thin discs perpendicular to the zaxis the thin direction will be the zdirection 63 There is another formula for volumes of solids of revolution which is quite different in origin from the formula for volume by slices This is the formula for volume by cylindrical shells To review these reread section 63 in its entirety Notice that in all the gures in this section the shells are thin in the zdirection so the gures correspond to integrals with respect to I In example 3 there is an integral with respect to y but the shells are not drawn in the accompanying gure Can you visualize what they would look like 64 There are a variety of different problems about the physical notion of work in this section but the only type of problem I might ask is the type about computing the work required to pump water out of a tank as in Example 5 You should read from the beginning of the section through the end of Example 139 then you can skip straight to Example 5 and read that 72 The rst part of this section is review from precalculus but you should go over it again anyway Especially you should make sure that the laws of exponents in the box labeled Theorem 2 at the bottom of page 394 are second nature to you so you don t have to waste time and effort thinking about them on the exam You can skip the section titled Applications of Exponential Functions on pp 39576 I might mention though that there s been a pretty good math for the masses column running in the New York Times these past few weeks and this weeks had a little bit about exponential functions and logarithms 7 if you want to take a quick break from studying you might check it out at httpopinionatorblogsnytimescom20100328powertools Of course any quick break on the internet runs the risk of turning into an extended break You can also skip the material on pages 396 and 397 including the De nition of e on page 397 This material is designed to lead into the formula for the derivative of 6 but I gave a different proof of this formula in class the one reproduced at the beginning of this review sheet I also gave a different de nition of 6 namely that e limh a 01 h1h You should of course know the formula for the derivative of e on page 398 and its proof given at the beginning of this review sheet You should also review Examples 2 and 5 and all the material on pages 400 and 401 73 Section 73 is actually review from precalculus but in my experience a lot of calculus students need to review this material In particular the laws of logarithms in the box at the bottom of p 405 should be second nature just like the laws of exponents Also remember that if a property of logarithms ISN T in this box then it probably isn t true For example there is NO law of logarithms concerning the expressions lnz y or ln 1 74 In this course we cover all of section 74 but for the second exam you won t need to know the material in the last two subsections titled General Logarithmic and Exponential Functions and Logarithmic Differentiation That material won t be covered till the third exam So for the second exam you can just review pages 411 to 415 Review for Test 1 This test is over the material in Sections 51 52 53 54 55 and 61 of the text which was covered in Assignments 1 2 and 3 In addition to going over the homework problems from these assignments to make sure you understand how they were done l7d recommend reading from the text following the outline below Its also a good idea to try some extra problems from the text similar to those assigned especially in section 55 This may be a good place to comment on strategy for doing homework When working problems always make a sustained effort to do the problem by yourself You7ll find that sometimes you7ll be able to do the problem yourself after being stuck for a while But if you keep on being stuck and not making progress go on to the next problem and get help later There are solutions manuals available for this text but I recommend not getting one or if you have one throwing it away 7 its too dif cult to resist the temptation to look into it at the first sign of dif culty You may miss points by not handing in a correct problem that you struggled with but I promise that you7ll more than make up for them on the quizzes and tests with the increased understanding you gained in the process 51 Areas and distances You should read this section carefully to make sure you understand the meaning of and the motivation for the definition of definite integral in the next section In particular you should understand what Figures 8 and 9 on page 292 are getting at and what the meaning is of the formulas for A at the bottom of page 294 You can skip the computations in the solution of Example 2 although you should understand what the statement lm R7 13 means and how its related to figures 8 and 9 You can also skip the material in the last pages of the section titled The Distance Problem 52 The de nite integral You should re read and be able to understand all the material in this section I might ask you to give the definition of the definite integral see the red box on page 300 You do not have to repeat it word for word but your definition should explain the meanings of all the symbols used See the definition I gave in class for a somewhat shorter version The properties of the integral listed in the red boxes on pages 307 and 308 are as important for what is NOT there as for what actually is there Notice that there is no property concerning the integral of the product of two functions fab fxgac dxl 53 The fundamental theorem of calculus Be familiar with the statements of the Fundamental Theorem of Calculus parts 1 red box on page 315 and 2 red box on page 318 and be able to use them to do problems as in examples 2 4 5 6 and 7 in this section 54 Inde nite integrals and the net change theorem Re read pages 324 326 and make sure you know all the formulas in the red box on page 325 Again notice that there is no formula in this box concerning the integral of the product of two functions f facgac dx We did not cover the Net Change Theorem77 in class you need not read the material on pages 327 to 329 55 The substitution rule This is one of the most important sections in the book in the sense that you Will need to use the substitution rule over and over again in this and future math and engineering courses Read the entire section carefully The final part titled Symmetry on pages 337 to 338 is not absolutely necessary for the exam but the material it presents is often useful 61 Areas between curves This section is the first of several in Chapter 6 in Which integrals are used to compute things other than the area under a curve Read the entire section except that you can skip Example 4 if you like Try to pay particular attention to the explanation on page 347 leading up to the formula in the red box labelled 2 The goal of this section and in fact all of chapter 6 is not to present a bunch of formulas to be memorized but rather to explain the process of representing something an area a volume work as a limit of Riemann sums Which can then be rewritten as an integral You have mastered the subject of integration When you don7t need to remember any formulas at all but rather in any given situation can gure out the correct integral on your own You might pause to look ahead to Chapter 9 at this point Chapter 9 is not covered until Calculus lll but actually it contains exactly the same sort of examples as Chapter 6 Also When you are taking Calculus Ill and get to chapter 9 you7ll profit by looking back to Chapter 6


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