Calc & Analyt Geom II (HONORS)
Calc & Analyt Geom II (HONORS) MATH 2423
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This 4 page Class Notes was uploaded by Einar Nitzsche on Monday October 26, 2015. The Class Notes belongs to MATH 2423 at University of Oklahoma taught by Kevin Grasse in Fall. Since its upload, it has received 36 views. For similar materials see /class/229289/math-2423-university-of-oklahoma in Mathematics (M) at University of Oklahoma.
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Date Created: 10/26/15
Information About Hour Exam I MATH 2423020 Spring 2012 General Information Hour Exam I will take place on Monday February 20 the date has been changed from the preceding Friday because of the University s partial closure on Monday February 13 and covers Sections 41 through 45 and Sections 51 through 53 of our textbook be advised that although I call it an hour exam you will in fact have only 55 minutes to complete it You may use a graphing calculator with features comparable to the TI 84 but you may not use a more powerful calculator such as the TI8992 TI Inspire or HP48 basically I prohibit any calculator that has a symbolic capability with which one could evaluate derivatives or inde nite integrals if you have any doubts about the appropriateness of your calculator you should contact me before the exam date No other references are allowed and you are prohibited from storing any class related formulas or other material facts in your calculator s memory to do otherwise will be viewed as academic misconduct and prosecuted as such My exams typically are four pages long and contain the exam questions interspersed with space to show work and answers Virtually all of the questions will be of the show your work and produce an answer variety The multiplechoice andor truefalse question formats will be used rarely in this course if at all A sample exam is posted on the Exam Information D2L module for our course to give you some idea of the structure of my hour examsl The single best way to study for this exam and the subsequent exams is to work through all of the assigned homework problems until you thoroughly understand them and can do them on your own without reference to the textbook instructorprepared solutions or a clever study partnerl If time permits you should work additional odd numbered problems in the text to gain further practice with the material Note 1 Be sure to work the problems on Recommended HW5 since you are not turning in that assignment for grading Solutions to these and all other assigned problems are posted on the Homework Information D2L module for our course You will also have an opportunity to ask questions about the material on HW5 which deals with Sections 52 and 53 at your discussion section meetings on Feb 15 and 16 Note 2 On all exams for this course I will assume that you are fully familiar with the basic rules and formulas for computing derivatives that you learned in Calculus It Calculus II builds on Calculus I in an essential way and there is simply no way to separate the two courses Here are some other speci c things that you should concentrate on 1 Be able to compute approximations to areas under graphs of given functions on given intervals with a speci ed number of rectangles using left or right endpoints or midpoints such as Probsl 4 5 on p 293 and Probl 9 on p 307 2 Be sure you understand the more theoretical problems such as Probl 20 on p 295 and Probsl 17 18 21 and 23 on pl 307 However note that formulas such as n2n n 2n33n2n z T and 1 22 f n 391 will be provided to you if needed so you don t have to memorize themi Know the basic properties of the integral listed in Sect 42 these are summarized at the end of this handout for your convenience Also review problems 34 35 and 37 on ppl 3077308 which emphasize the area interpretation of the de nite integraL 4 I will ask you to state the version of the Fundamental Theorem of Calculus that appears on p 317 of the textbook so memorize it The statement statement is repeated here for your convenience A on V The Flmdamental Theorem of Calculus Suppose f is continuous on abi ll Ifgz dt then g z 2 f dz Fb 7 Fa where F is any antiderivative of f that is F Memorizing this will assure you of getting a few points on the examl Know memorize the basic Table of Inde nite Integrals given on p 322 with two exceptions you don t need to memorize the inde nite integral formulas for csc21 and csczcotz for nowl These formulas A 01 V 1 A 00 V A to V are all direct analogs of corresponding differentiation formulas that you should already knowi For your convenience a copy of this table is included at the end of this handouti Work as many problems as you can over and above the assigned homework from Sect 45 on the substitution rule77 or u substitutions This topic requires lots of practice In Sect 51 the basic principle is that the area between the graphs of two functions 9 and y 91 over an interval a g 1 g b is given by b A Vltzgt7gltzgtlda but to actually evaluate the integral we must get rid of the absolute value around 7 lf 2 91 on a g 1 g b then we can simply write 791l 791 while if91 Z on a g 1 g b then 7 91 7 in both bases 7 can be interpreted as upper curve minus lower curve In more complicated situations where the graphs of f and 9 cross in the interval 971 the integral from a to I must be split up accordingly such as in Probi 25 on p 349 In Sect 52 deals with the computation ofvolumes of solids of revolution Here the basic principle is that to obtain the volume you integrate 7r gtlt radius2 which is the cross sectional area over the pertinent values of the variable So if we take the region bounded by the graph of y and the 1axis from 1 a to 1 b and revolve this region about the 1axisy then the volume of the generated solid is v 7 bwfz2dz you need to know this and all subsequent formulas concerning volumesi More generallyy if you consider the region bounded by the graph of y and some general horizontal line 9 5 instead of just the 1axisy y 0y and if you revolve that region about 9 c then the generated solid has volume Vab7rf17c2d1i Finally if the region bounded between two curves 2 91 2 0 on a g 1 g b is rotated about the 1axisy t en 1 V 7rfr27yr2ldr7 so we integrate 7r gtlt outer radius2 7 7r gtlt inner radius2 the cross section looks like a washer Corresponding formulas hold if we rotate about the y axis or a line 1 d and our functions are speci ed as 1 1 99 7 just replace 1 by y and compute integrals with respect to 9 Section 53 introduces the shell77 method for nding volumes of solids of revolution This is fundamen tally different from the principle of integrating the crosssectional area which leads to the washer77 methody but it works better than the washer method in some types of examples The basic principle in the shell method is that you approximate the solid by shells instead of washers and you integrate 27f gtlt shell radius gtlt shell height over the pertinent values of the variable The easiest case is when the have a function f satisfying Z 0 for a g 1 g b where a gt 0V and we take the region between the graph 9 and the 1axis from 1 a to 1 b and revolve it about the y axis Then the volume of the generated solid is b V 27TIfI d1 so 1 shell radius and shell heighti If 2 91 on a g 1 g b and we revolve the region between the graphs of f and 9 about the y axisy then the shell height becomes 7 91 and the resulting formula is b V 2W1 7 d1 2 H 53 H 0 CAD g 01 on I 5 0 to If we revolve around a line z d where d lt a instead ofjust the y axisy z 0y then the shell radius is z 7 dy instead ofjust zy and the above formulas become I b V 27rz 7 dz and V 27rz 7d 7 dzi We can also use the shell method for computing volumes of solids obtained by revolving regions about the zaxis In such cases our functions are usually speci ed as z and the appropriate formulas are obtained as above by replacing z by y in the integrals Howevery in all cases we are still integrating 27f gtlt shell radius gtlt shell height when we use the shell methodi Properties of the Integral b cdz 01 7 a c constant a Abmz gzl dz 7 b fzdz bgltzgtdz i ab dz cab dz 0 constant a bum 7gltzgt1 dz 7 b mm 7 bgltzgtdz 274 are called the linean39ty properties of the integral 0 b C i ddquot39 b d d d a 1t1v1tydlt lt0 gt Ab z z zb z ifz20fora z b 5 fzdz20 b b ifzZgzford z b 5 fzdzZgzdz b m fz Mford z b 5 mb7d fzdz Mb7d 678 are called the comparison properties of the integral i zero area All dz 0 a b backwards area dz 7 dz b a A Short Table of Inde nite Integrals also known as antiderivatives kdz kz C k and C are constants szz sinzdz7coszC7 coszdzsinzc Ir1 Tl C 7 is any real number 7l sec2zdztanzC7 secztanzdzseczC dzkfz dz 6 isaconstant fzgz dzfzdzgz dz 3 you can skip the formulas for the inde nite integrals of csc2 z and cscz cot z for now