[ver] 4 [sty] [files] [charset] 82 ANSI (Windows, IBM CP 1252) [revisions] 0 [prn] Canon BJ-200ex [port] LPT1: [lang] 1 [fldnames] Field1 Field2 Field3 Field4 Field5 Field6 Field7 Field8 [desc] 834642901 7 833957905 369 8 0 0 0 0 1 [fopts] 0 1 0 0 [lnopts] 2 Body Text 1 [docopts] 5 2 [GramStyle] [tag] Body Text 2 [fnt] Times New Roman 240 0 49152 [algn] 1 1 0 0 0 [spc] 33 273 1 0 0 1 100 [brk] 4 [line] 8 0 1 0 1 1 1 10 10 1 [spec] 0 0 0 1 1 0 0 0 0 [nfmt] 280 1 2 . , $ Body Text 0 0 [tag] Body Single 3 [fnt] Times New Roman 240 0 49152 [algn] 1 1 0 0 0 [spc] 33 273 1 0 0 1 100 [brk] 4 [line] 8 0 1 0 1 1 1 10 10 1 [spec] 0 0 0 1 1 0 0 0 0 [nfmt] 280 1 2 . , $ Body Single 0 0 [tag] Bullet 4 [fnt] Times New Roman 240 0 49152 [algn] 1 1 0 288 288 [spc] 33 273 1 0 0 1 100 [brk] 4 [line] 8 0 1 0 1 1 1 10 10 1 [spec] 0 0 <*0> 360 1 1 0 0 0 0 [nfmt] 272 1 2 . , $ Bullet 0 0 [tag] Bullet 1 5 [fnt] Times New Roman 240 0 49152 [algn] 1 1 288 288 288 [spc] 33 273 1 0 0 1 100 [brk] 4 [line] 8 0 1 0 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0 0 8738 1 0 0 0 0 .28 > 1 7 0 0 0 0 8738 1 0 0 0 0 .40 > 1 8 0 0 0 0 8738 1 0 0 0 0 .54 > 1 9 0 0 0 0 8738 1 0 0 0 0 .68 > 1 10 0 0 0 0 8738 1 0 0 0 0 .83 > 1 11 0 0 0 0 8738 0 0 0 0 0 [e] [tble] [frm] 9 537395392 3378 3184 4192 3516 0 1 3 0 0 0 0 0 0 0 0 16777215 20 5 1938 814 264 [frmname] Frame20 [frmlay] 3516 814 1 0 0 1 3184 0 0 2 0 7 16 0 1 3378 3890 0 [isd] .X20 .tex .X20 1 1 0 0 814 65204 100 0 0 .tex 0 65272 0 [frm] 9 537395392 6016 3184 6623 3516 0 1 3 0 0 0 0 0 0 0 0 16777215 21 6 4576 607 264 [frmname] Frame21 [frmlay] 3516 607 1 0 0 1 3184 0 0 2 0 1 24 0 1 6016 6319 0 [isd] .X21 .tex .X21 1 1 0 0 607 65204 100 0 0 .tex 0 65272 0 [frm] 9 537395392 2155 3848 2313 4180 0 1 3 0 0 0 0 0 0 0 0 16777215 22 7 715 158 264 [frmname] Frame22 [frmlay] 4180 158 1 0 0 1 3848 0 0 2 0 1 56 0 1 2155 2313 0 [isd] .X22 .tex .X22 1 1 0 0 158 65204 100 0 0 .tex 0 65272 0 [frm] 9 537395392 2669 3848 2827 4180 0 1 3 0 0 0 0 0 0 0 0 16777215 23 8 1229 158 264 [frmname] Frame23 [frmlay] 4180 158 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1 792 1440 0 1 0 1 1 0 1 1440 10800 2 2 4680 3 9360 [txt] > [frght] [lyfrm] 1 13248 0 14688 12240 15840 0 1 3 1 0 0 0 0 0 0 0 0 2 [frmlay] 15840 12240 1 1440 792 1 14760 1440 0 1 0 1 1 0 1 1440 10800 2 2 4680 3 9360 [txt] > [elay] [l1] 0 [pg] 8 36 0 40 0 0 0 0 65535 65535 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 42 0 0 32 0 0 0 65534 65535 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 81 0 26 0 1 0 0 65535 4 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 94 0 100 512 0 0 0 65534 65535 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 103 0 49 512 0 0 0 65535 65535 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 142 0 0 0 0 0 0 65535 65535 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 166 0 42 512 0 1 0 65535 2 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 193 97 67 1025 0 0 0 65535 2 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 [edoc] <+B><:s><:#424,9360><:f360,2Photina Casual Black,> <+B><:s><:#424,9360><:f360,2Photina Casual Black,> <+B><:s><:#424,9360><:f360,2Photina Casual Black,> <+B><:s><:#380,9360><:f320,2Photina Casual Black,>Fifth Conference on the Teaching <+B><:s><:#380,9360><:f320,2Photina Casual Black,>of Mathematics <+B><:s><:#332,9360><:f280,2Photina Casual Black,> <+B><:s><:#332,9360><:f280,2Photina Casual Black,>Baltimore, Maryland <+B><:s><:#332,9360><:f280,2Photina Casual Black,>June 21-22, 1996<:f> <+B><:s><:#424,9360><:f360,2Photina Casual Black,> <+B><:s><:#424,9360><:f360,2Photina Casual Black,> <+B><:s><:#424,9360><:f360,2Photina Casual Black,> <+B><:s><:#424,9360><:f360,2Photina Casual Black,> <+B><:s><:#472,9360><:f400,2Photina Casual Black,>Contracting for Calculus: <+B><:s><:#472,9360><:f400,2Photina Casual Black,>Calculus Technology Projects<:f> <+B><:s><:#332,9360><:f280,2Photina Casual Black,> <+B><:s><:#332,9360><:f280,2Photina Casual Black,> <+B><:s><:#332,9360><:f280,2Photina Casual Black,> <+B><:s><:#332,9360><:f280,2Photina Casual Black,> <+B><:s><:#332,9360><:f280,2Photina Casual Black,> <+B><:s><:#332,9360><:f280,2Photina Casual Black,> <+B><:s><:#332,9360><:f280,2Photina Casual Black,> <+B><:s><:#332,9360><:f280,2Photina Casual Black,> <+B><:s><:#332,9360><:f280,2Photina Casual Black,> <+B><:s><:#332,9360><:f280,2Photina Casual Black,> <+B><:s><:#332,9360><:f280,2Photina Casual Black,> <+B><:s><:#332,9360><:f280,2Photina Casual Black,> <+B><:#332,9360><:f280,2Photina Casual Black,>John Armon <+B><:#332,9360><:f280,2Photina Casual Black,>Robin Hocken <+B><:s><:#332,9360><:f280,2Photina Casual Black,>Patrick Ward <+B><:s><:#332,9360><:f280,2Photina Casual Black,> <+B><:s><:#332,9360><:f280,2Photina Casual Black,>Illinois Central College <+B><:s><:#332,9360><:f280,2Photina Casual Black,>East Peoria, Illinois<:f> <+B><:s><:#332,9360><:f280,2Photina Casual Black,> <+B><:s><:#332,9360><:f280,2Photina Casual Black,> <+B><:s><:#332,9360><:f280,2Photina Casual Black,> <+B><:s><:#332,9360><:f280,2Photina Casual Black,><:P12,0,><:f> <+@><:s><:S+-2><:#426,9360><+!>History/Purpose<-!> <+@><:S+-2><:#3834,9360> In the fall semester of 1995, a collaborative effort to implement and reinforce the use of technology as part of the problem-solving process was instituted in first semester calculus at Illinois Central College. The problem sets, known as Calculus Technol ogy Projects (CTPs), were assigned to the daytime sections of first semester calculus. Many sources were consulted to generate ideas for problems which were challenging, relevant and fun, but primarily which would require some form of technology in the sol ution process. This strategy was designed to increase student participation and interaction, demonstrate the use of technology in solving applications, increase understanding of the material and involve the students in setting individual goals to achieve a desired grade. <+@><:s><:S+-2><:#426,9360> <+@><:s><:S+-2><:#426,9360><+!>Methodology<-!> <+@><:S+-2><:#7242,9360> Four (daytime) sections of Calculus I were offered that fall, each with a different instructor. Approximately every three weeks, a "set" of four projects was offered to the students, who could do as many of the set as they desired. Each of the four instr uctors participated by creating a CTP problem for each set. Once a solution to a CTP had been proposed, the student needed to both submit a written solution and to give an oral presentation to one of the instructors, the author of that CTP. The write-up o f the solution was expected to be done carefully, showing all details and reproducing any graphics utilized in the process. Proper English usage was stressed as well. The presentation portion consisted of the instructor "grilling" the student about differ ent aspects of their proposed solution. Most solutions were accepted after minor revisions and a subsequent scaled-down presentation. Students were well aware from the beginning of the semester that a minimum number of CTPs had to be completed throughout the semester. Coupling this requirement with regular assignments (homework and/or quizzes) and test scores, most students appreciated these stated expectations and were then able to plan ahead and "contract" for their desired grade. The responsibility for reaching the minimum requirements for a grade was left up to the student. In reality, some latitude in the percentage requirements was allowed, but the CTP requirements were the proverbial "bottom line", one of the common threads connecting all four secti ons. <+@><:s><:S+-2><:#426,9360> <+@><:S+-2><:#852,9360> The following is an excerpt from the syllabus given to the students at the beginning of the semester outlining grade requirements and the expectations for the CTPs: <+@><:s><:S+-2><:#384,9360><:f220,2Times New Roman,> <+@><:s><:I720,0,0,720><:S+-1><:#256,9360><:f220,2Times New Roman,><+!>YOUR GRADE...<-!> <+@><:s><:I720,0,0,720><:S+-1><:#256,9360><:f220,2Times New Roman,>There are three components involved in determining your grade: <+@><:s><:I720,0,0,720><:S+-1><:#256,9360><:f220,2Times New Roman,> I. Five one-hour exams and a cumulative final exam <+@><:s><:I720,0,0,720><:S+-1><:#256,9360><:f220,2Times New Roman,> II. Homework/Quizzes <+@><:I720,0,0,720><:S+-1><:#256,9360><:f220,2Times New Roman,> III. Calculus Technology Projects (CTPs) <+@><:s><:I720,0,0,720><:S+-1><:#256,9360><:f220,2Times New Roman,> <+@><:s><:I720,0,0,720><:S+-1><:#260,9360><:f220,2Times New Roman,><+!>** There is no curve for this class. <-!> <+@><:s><:I720,0,0,720><:S+-1><:#256,9360><+!><:f220,2Times New Roman,> Your grade is determined by the following guidelines:<-!> <+@><:s><:I720,0,0,720><:S+-1><:#256,9360><:f220,2Times New Roman,> <+@><:s><:I720,0,0,720><:S+-1><:#256,9360><:f220,2Times New Roman,>To earn an "A" <+@><:s><:I720,0,0,720><:S+-1><:#256,9360><:f220,2Times New Roman,> Average 90% or higher on all written work <+@><:s><:I720,0,0,720><:S+-1><:#256,9360><:f220,2Times New Roman,> Submit at least 8 acceptable solutions to the Calculus Technology Projects <+@><:s><:I720,0,0,720><:S+-1><:#256,9360><:f220,2Times New Roman,> (with at least one from each set) <+@><:s><:I720,0,0,720><:S+-1><:#256,9360><:f220,2Times New Roman,>To earn a "B" <+@><:s><:I720,0,0,720><:S+-1><:#256,9360><:f220,2Times New Roman,> Average 80% or higher on all written work <+@><:s><:I720,0,0,720><:S+-1><:#256,9360><:f220,2Times New Roman,> Submit at least 5 acceptable solutions to the Calculus Technology Projects <+@><:s><:I720,0,0,720><:S+-1><:#256,9360><:f220,2Times New Roman,> (with at least one from three different sets, including the first set) <+@><:s><:I720,0,0,720><:S+-1><:#256,9360><:f220,2Times New Roman,>To earn a "C" <+@><:s><:I720,0,0,720><:S+-1><:#256,9360><:f220,2Times New Roman,> Average 70% or higher on all written work <+@><:s><:I720,0,0,720><:S+-1><:#256,9360><:f220,2Times New Roman,> Submit at least 3 acceptable solutions to the Calculus Technology Projects <+@><:s><:I720,0,0,720><:S+-1><:#256,9360><:f220,2Times New Roman,> (not all from the same set, with at least one from the first set) <+@><:s><:I720,0,0,720><:S+-1><:#256,9360><:f220,2Times New Roman,>To earn a "D" <+@><:s><:I720,0,0,720><:S+-1><:#256,9360><:f220,2Times New Roman,> Average 60% or higher on all written work <+@><:s><:I720,0,0,720><:S+-1><:#256,9360><:f220,2Times New Roman,> Submit at least 1 acceptable solution to the Calculus Technology Projects <+@><:s><:I720,0,0,720><:S+-1><:#256,9360><:f220,2Times New Roman,> (must come from the first set) <+@><:s><:I720,0,0,720><:S+-1><:#256,9360><:f220,2Times New Roman,> <+@><:I720,0,0,720><:S+-1><:#256,9360><:f220,2Times New Roman,><+!>About the Calculus Technology Projects (CTPs)...<-!> <+@><:s><:I720,0,0,720><:S+-1><:#256,9360><:f220,2Times New Roman,> <+@><:I720,0,0,720><:S+-1><:#2056,9360><:f220,2Times New Roman,> Every four weeks, you will be given a collection of four problems which will require a deeper understanding of the course and/or some extra effort, insight, creativity, along with the use of technology. To earn a grade of A, B, C or D in the course you must complete the specified number of CTPs for that grade as described above. <+!>NOTE: You MUST complete ONE problem from the FIRST SET of CTPs!<-!> Collaboration on these problems is highly encouraged; feel free to work with students in any section of Math 222. Your solution MUST be written in your own words and followed by an o ral presentation to one of the instructors. <+@><:s><:I720,0,0,720><:#256,9360><:f220,2Times New Roman,> <+@><:I720,0,0,720><:S+-1><:#256,9360><+!><:f220,2Times New Roman,>When you believe that you have a completed solution to a CTP:<-!> <+@><:s><:I720,0,0,720><:S+-1><:#256,9360><:f220,2Times New Roman,> <+@><:I720,0,0,720><:S+-1><:#516,9360><+!><:f220,2Times New Roman,>a.<-!> Write out your solution carefully, showing all details, reproducing any charts or <:f><:f220,2Times New Roman,>graphs that you used. Write in complete sentences and use proper vocabulary at all times. <+@><:I720,0,0,720><:S+-1><:#256,9360><:f220,2Times New Roman,> <+@><:I720,0,0,720><:S+-1><:#516,9360><:f220,2Times New Roman,><+!>b.<-!> Make an appointment to discuss your solution(s) with the instructor indicated on the project sheet. There will be a sign-up sheet posted outside each instructor's office. <:f><+!><:f220,2Times New Roman,> <+@><:I720,0,0,720><:S+-1><:#256,9360><+!><:f220,2Times New Roman,> <+@><:I720,0,0,720><:S+-1><:#256,9360><+!><:f220,2Times New Roman,> <+@><:I720,0,0,720><:S+-1><:#1028,9360><+!><:f220,2Times New Roman,>c.<-!> If it is determined that your solution is excellent, it will be accepted with no revision. If there are any concerns and the instructor suggests some revisions, you may resubmit your solution at a later time. DO NOT exp ect all of your solutions to be accepted the first time; we are looking for excellence. <+@><:s><:I720,0,0,720><:S+-1><:#256,9360><:f220,2Times New Roman,> <+@><:s><:I720,0,0,720><:S+-1><:#768,9360><:f220,2Times New Roman,>There will be 16 problems given overall, and you must submit acceptable answers to at least three of them to earn a C. You will be given 3 - 4 weeks to work on each set of problems. No late solutions will be accepted!! <+@><:s><:I720,0,0,720><:S+-1><:#256,9360><:f220,2Times New Roman,> <+@><:I720,0,0,720><:S+-1><:#1536,9360><:f220,2Times New Roman,>Our goals for the CTPs include having you use new technologies along with technologies you may already be familiar with whenever possible. We also wish to improve your abilities to communicate your ideas in a clear, concise manner. The technologies currently available include graphing<:f><:f220,2Times New Roman,> calculators and computer software such as <+">Mathematica<-"> which are available for your use in the Math Lab (room 235A). Not all CTPs will involve technology, but you may find the work MUCH less tedious with it!<:f> <+@><:s><:S+-2><:#426,9360> <+@><:s><:S+-2><:#426,9360><+!>Shortcomings<-!> <+@><:S+-2><:#2982,9360> Every endeavor has drawbacks, and this of course was no exception. From our point of view as instructors, it was a very time-consuming course. As a group, we had weekly meetings to plan the CTPs, compare progress in the sections, collaborate on writing e xams, and evaluate how the projects were working. Individually, we also had to create the projects, learn various types of technology/software and listen to presentations. During the first set of CTPs, when each student was required to complete at least o ne, each instructor heard between 60 and 80 student presentations, and most required corrections and re-presentations. <+@><:s><:S+-2><:#2556,9360> Students took liberty with our encouragement to "work together" and some blatant copying and pirating of computer programs was done. We sought to minimize this by requiring an individual defense of each solution; if a student did not understand their writ ten solution, it became obvious in a matter of minutes. The students were given three to four weeks to complete each set. Predictably, many procrastinated until the last few days before the due date which produced a lot of stress on us as instructors as wel l as the students. <+@><:s><:S+-2><:#426,9360> <+@><:s><:S+-2><:#426,9360><+!>Achievements<-!> <+@><:S+-2><:#1278,9360> The CTPs did result in many benefits for both instructors and students. As instructors, we were forced to look very closely at application problems and the implementation of technology which positively affected our teaching and examples in the classroom. We enjoyed <+@><:s><:S+-2>working together to create challenging problems for the students and learning different styles <:p<* >>and approaches from one another. We also had the opportunity to interact with students from different sections as they presented their solutions to the original problem's author. It allowed us to work and learn with a wider variety of students. <+@><:s><:S+-2><:#3834,9360> The majority of students seemed to benefit in many ways. They learned to work together on a much bigger scale than typical classroom group assignments. They had to take the initiative to discuss and share ideas on the challenging problems. They interact ed (and commiserated) with students from all four sections rather than being confined to their own section. Many students commented that they were able to see applications of the calculus in a clearer way than previous to the course and were able to apprec iate the power (and even the beauty!) of it more fully. Even being "forced" to learn to use technology was appreciated by certain students. Many students became very adept at using various forms of technology and in several cases taught us quite a few int eresting things too! <+@><:s><:S+-2><:#2556,9360> Another gain we observed in students was an increased cognizance of the efforts required to understand the material and achieve a desired grade in a course. We were able to observe students who went on to Calculus II and those who had to repeat Calculus I in the spring semester. These students had better study habits, were willing to put more time into the course and were able to communicate their ideas and abstract concepts more readily than the average student we had encountered. <+@><:s><:S+-2><:#426,9360> <+@><:s><:S+-2><:#426,9360><+!>Summary<-!> <+@><:s><:S+-2><:#3408,9360> Overall, we feel that the benefits of this approach to the course far outweighed the drawbacks. This strategy allowed both instructors and students to personalize the course to their interests and enabled us to get to know the students on a deeper level t han in a traditional classroom situation. We believe the students emerged from the course with an increased grasp of the concepts involved and will continue to benefit from the exposure to the hands-on technology they experienced. We also feel this approa ch has potential for use in other classes and at various levels. Several ideas from this approach have been adapted and implemented into the other courses we teach. <+@><:s><:S+-2><:#426,9360> <+@><:s><:S+-1><:#284,9360>We would appreciate any comments or suggestions the reader may have. <+@><:s><:S+-1><:#284,9360>You may contact us at Illinois Central College, phone 309/694-5369 or 309/694-5707. <+@><:S+-1>Included is a three problem sample of CTPs. <:p<* >> <+B><:s><:#372,9360><:f320,,><-!><-#><+!>Resource Distribution<-!><:f> <+@><:s><:#284,9360><+!> <+@><:s><:#1024,9360> <:f220,,>Whether a resource is distributed evenly among members of a population is often an important political or economic question. How can we measure this? How can we decide which country has the most equitable income distribution? These are question s economists sometimes face. This problem looks at one way of approaching this. <+@><:#1292,9360><:f220,,> Suppose the resource is distributed evenly. Then any 20% of the population will have 20% of the resource. Similarly, any 30% will have 30% of the resource and so on. If, however, the resource is not distributed evenly, the poorest <+!>P<-!>% of the population (in terms of this resource) will not have <+!>P<-!>% of the goods. Suppose <+!><+">F(x)<-"><-!> represents the fraction of the resources owned by the poorest fraction, <+!><+">x<-"><-!>, of the population. Thus, <+"><+!>F(0.<-!><-"><+!><+">4)<-!><-"> <+!>= <+">0.1<-"><-!> means that the poorest 40% of the population owns 10% of the resource.<:f> <+@><:s><:#284,9360> <+@><:s><:#260,9360> <:f220,,><+!>(a)<-!> What would the function <+!><+">F<-"><-!> be if the resource were distributed evenly? <+@><:s><:#256,9360><:f220,,> <+@><:s><:#260,9360><:f220,,> <+!>(b) <-!> For a general function, what must <+"><+!>F(0)<-!><-"> be? <+"><+!>F(1)<-!><-">? <+@><:s><:#260,9360><:f220,,> Do you think <+"><+!>F<-!><-"> is increasing or decreasing? Concave up or down? <+@><:s><:#256,9360><:f220,,> Be prepared to defend your ideas! <+@><:s><:#256,9360><:f220,,> <+@><:#260,9360><:f220,,> <+!>(c)<-!> <+">Gini'<-">s index of inequality, <+!>G<-!>, is one way to measure how evenly the resource is <+@><:s><:#256,9360><:f220,,> distributed. It is defined by <+@><:s><:#256,9360><:f220,,> <+@><:s><:#260,9360><:f220,,> Show graphically what <+!>G<-!> represents (using the <+!><+">F<-"><-!> you described in part (b) ). <+@><:s><:#256,9360><:f220,,> <+@><:#260,9360><:f220,,> <+!>(d) <-!> Find the value of <+">Gini<-">'s index for <-"> <+@><:s><:f220,,> <:A3> <:A2> <+@><:s><:f220,,> <:A1> <:A0> <+@><:#256,9360><:f220,,> Generally, what do you think <+">Gini<-">'s index says about the inequality of resource <+@><:s><:#256,9360><:f220,,> distribution? <+@><:s><:#256,9360><:f220,,> <+@><:#260,9360><:f220,,> <+!>(e) <-!> Suppose <+!><+">F(x)<-"><-!> were described in the following data table: <+@><:s><:f220,,><:t4> <+@><:#520,9360> <:f220,,><:f><:f220,,>You can use <+!>PtOn<-!> in the TI-85<+!> <:f220,,>CATALOG<-!> to get a look at <+"><+!>F(x)<-!><-"> by plotting each point. Use the form <+!>PtOn(x,y)<-!> where x and y are the coordinates of the point to be graphed. <+@><:#520,9360><+!><:f220,,>NOTE!!<-!> If you want to graph <+"><+!>y = x<-!><-"> and <+!><+">F(x)<-"><-!> together, graph the line <+!><+">y = x<-"><-!> in the <:f220,,><+!>GRAPH<-!> menu <+#>first<-#> - otherwise, going to the <+!>GRAPH<-!> menu erases all your plotted points! <+@><:s><:#256,9360><:f220,,> <+@><:#260,9360><:f220,,> Use an approximation technique to find <+!>G<-!> for this<+"><+!> F(x)<-!><-">. <+@><:s><:#256,9360><:f220,,> <+@><:#260,9360><:f220,,> <+!>(f) <-!> For <+#>both<-#> the <+!><+">F(x)<-"><-!> given in the table, and for <+"><+!>F(x) = x<+&>2<-!><-"><-&>, create a table showing the <+@><:#260,9360><:f220,,> function, <+"><+!>H(x)<-!><-">, where <+!><+">H<-"><-!> is the fraction of the resources owned by the richest fraction, <+!><+">x<-"><-!>, <+@><:s><:#256,9360><:f220,,> of the population. <+@><:s><:#256,9360><:f220,,> <+@><:#260,9360><:f220,,> <+!>(g) <-!> Is <+!><+">H(x)<-"><-!> the inverse of <+"><+!>F(x)<-!><-">? Justify your answer! <+@><:s><:#256,9360><:f220,,> <+@><:s><:#260,9360><:f220,,> <+!>(h)<-!> Could you create an index to measure the inequality of resource distribution based <+@><:#260,9360><:f220,,> on <+"><+!>H(x)<-!><-"> instead of on <+!><+">F(x)<-"><-!>? (Call it the <+">Hocken<-"> index!) What might it look like? <+@><:s><:#256,9360><:f220,,> Justify graphically and algebraically.<:f> <+@><:s><:#284,9360> <+B><:s><:#424,9360><:f360,,><+!> <+B><:s><:#424,9360><+!><:f360,,>The Graphing<-!><:f><+!><:f360,,> Calculator: it slices, it dices,....<-!><:f> <+B><:s><:#328,9360><:f280,,><+!>(sorry, no free pen sets or julienne fries...)<-!><:f> <+@><:s><:#284,9360> <+@><:s><:#284,9360> <+@><:s><:f280,,>Given a function <:A5> and an interval <:A6>, write a mini-program for a graphing calculator (or computer) that will approximate the volume of the solid (from <:A7> to <:A8>) formed by rotating <:A9>:<:f> <+@><:s><:#284,9360> <+@> <:f280,,>(a<:f><:f280,,>) around the x-axis (discs) <:A10> slicing <+@><:f280,,> (b<:f><:f280,,>) around the y-axis (shells) <:A11> coring/peeling<:f> <+@><:s><:#332,9360><:f280,,> <+@><:s><:#332,9360><:f280,,>** Also, tell what problems may develop using this program<:f><:f280,,> <+@><:s><:#332,9360><:f280,,> <+@><:s><:#332,9360><:f280,,> <+@><:s><:#332,9360><:f280,,> <+@><:#336,9360><:f280,,><+!>HINTS:<-!> (for the TI-85) <+@><:s><:#332,9360><:f280,,> <+@><:f280,,> InpSt<:f><:f280,,> <:A12> <:B<*1>><:f><:f280,,>lets user input an equation as a sentence <+@><:s><:f280,,> and stores <:f><:f280,,>it to <:A13> <+@><:s><:#332,9360><:f280,,> <+@><:f280,,> St<:B<*(>>Eq (<:A14>) <:B<*1>><:f><:f280,,>stores <:A15> as an equation into <:A16> <+@><:s><:#332,9360><:f280,,> <+@><:#332,9360><:f280,,> Prompt A,B <:B<*1>><:f><:f280,,>lets user input values into A and B <+@><:s><:#332,9360><:f280,,> <+@><:f280,,> fnInt(<:A17>A,B) <:B<*1>> <:A18><:p<* >> <+B><:s><:#328,9360><:f280,,><+!>Newton's Method and Complex-Valued Functions<-!> <+@><:s><:#284,9360><:f> <+@><:f240,,>Newton's method is used to help us approximate the root(s) of functions with the help of the tangent lines to the graph of the original function, <:A19><:f><:f240,,> If a function has no real roots, such as <:A20>, Newton's method will not converge to a solution. But as a polynomial equation, <:A21> has two solutions in the complex<:f><:f240,,> numbers: <:A22> and <:A23>. These two solutions can be found on the complex plane, which has, instead of the <:A24> and <:A25> axes, the real and imaginary axes. So, in the above example, the points <:A26> and <:A27> represent the complex numbers <:A28> and <:A29> in the complex plane. Newton's method is easily adapted to complex-valued functions of the form <:A30>, where <:A31> is a complex number of the form <:A32>: simply replace <:A33> with <:A34> and compute <:A35> so the formula <+@><:s><:f240,,> <:A36> <+@><:s><:#284,9360><:f240,,>will apply to complex-valued functions as well. <+@><:s><:#284,9360><:f240,,> <+@><:s><:f240,,>1) Let <:A37> be a complex-valued function. <+@><:s><:f240,,> a) By using the definition <:A38>, replace <:A39> in the function with <:A40> <+@><:s><:R1,12,1,720,1,1080,1,2160,1,2880,1,3600,1,4320,1,5040,1,5760,1,6480,1,7200,1,7920,1,8640,><:f240,,> and write each part as a function of two variables, <:A41> and <:A42>. <+@><:s><:R1,12,1,720,1,1080,1,2160,1,2880,1,3600,1,4320,1,5040,1,5760,1,6480,1,7200,1,7920,1,8640,><:f240,,> b) Substitute your initial guess: the complex number <:A43> <+@><:s><:R1,12,1,720,1,1080,1,2160,1,2880,1,3600,1,4320,1,5040,1,5760,1,6480,1,7200,1,7920,1,8640,><:f240,,> (<:A44> and <:A45>) into the formula for Newton's method and simplify <+@><:s><:R1,12,1,720,1,1080,1,2160,1,2880,1,3600,1,4320,1,5040,1,5760,1,6480,1,7200,1,7920,1,8640,><:f240,,> to find <:A46> <+@><:s><:R1,12,1,720,1,1080,1,2160,1,2880,1,3600,1,4320,1,5040,1,5760,1,6480,1,7200,1,7920,1,8640,><:f240,,> c) Use <:A47> to calculate <:A48> and continue the process until your answer converges <+@><:s><:R1,12,1,720,1,1080,1,2160,1,2880,1,3600,1,4320,1,5040,1,5760,1,6480,1,7200,1,7920,1,8640,><:f240,,> to <:A49>: it should take four iterations. Plot your points on the complex plane. <+@><:s><:R1,12,1,720,1,1080,1,2160,1,2880,1,3600,1,4320,1,5040,1,5760,1,6480,1,7200,1,7920,1,8640,><:f240,,> d) Use an initial guess of <:A50> to find the second answer of <:A51> again, <+@><:s><:R1,12,1,720,1,1080,1,2160,1,2880,1,3600,1,4320,1,5040,1,5760,1,6480,1,7200,1,7920,1,8640,><:#284,9360><:f240,,> it should take four iterations. Plot your points on the complex plane. <+@><:s><:R1,12,1,720,1,1080,1,2160,1,2880,1,3600,1,4320,1,5040,1,5760,1,6480,1,7200,1,7920,1,8640,><:f240,,> e) What do you suppose would happen if your initial guess was <:A52> which is <+@><:s><:R1,12,1,720,1,1080,1,2160,1,2880,1,3600,1,4320,1,5040,1,5760,1,6480,1,7200,1,7920,1,8640,><:#284,9360><:f240,,> a real number? Try a few iterations to justify your claim. <+@><:s><:R1,12,1,720,1,1080,1,2160,1,2880,1,3600,1,4320,1,5040,1,5760,1,6480,1,7200,1,7920,1,8640,><:#284,9360><:f240,,> <+@><:s><:R1,12,1,720,1,1080,1,2160,1,2880,1,3600,1,4320,1,5040,1,5760,1,6480,1,7200,1,7920,1,8640,><:f240,,>2) Let <:A53> <+@><:s><:R1,12,1,720,1,1080,1,2160,1,2880,1,3600,1,4320,1,5040,1,5760,1,6480,1,7200,1,7920,1,8640,><:f240,,> a) Find all three solutions to the equation <:A54> <+@><:s><:R1,12,1,720,1,1080,1,2160,1,2880,1,3600,1,4320,1,5040,1,5760,1,6480,1,7200,1,7920,1,8640,><:#284,9360><:f240,,> b) Approximate the real-valued solution using Newton's method. <+@><:s><:R1,12,1,720,1,1080,1,2160,1,2880,1,3600,1,4320,1,5040,1,5760,1,6480,1,7200,1,7920,1,8640,><:#284,9360><:f240,,> c) Approximate one of the complex-valued solutions using Newton's method for <+@><:s><:R1,12,1,720,1,1080,1,2160,1,2880,1,3600,1,4320,1,5040,1,5760,1,6480,1,7200,1,7920,1,8640,><:#284,9360><:f240,,> complex numbers: make an initial guess and plot your points on the complex plane. <+@><:s><:R1,12,1,720,1,1080,1,2160,1,2880,1,3600,1,4320,1,5040,1,5760,1,6480,1,7200,1,7920,1,8640,><:#284,9360><:f240,,> <+@><:s><:R1,12,1,720,1,1080,1,2160,1,2880,1,3600,1,4320,1,5040,1,5760,1,6480,1,7200,1,7920,1,8640,><:#284,9360><:f240,,> <+@><:R1,12,1,720,1,1080,1,2160,1,2880,1,3600,1,4320,1,5040,1,5760,1,6480,1,7200,1,7920,1,8640,><:#568,9360><:f240,,>On pages 180 and 181 in your text, you will find a wonderful application of solutions to equations like you have just solved - to generate fractal patterns. > Times New Roman,18,14,0,0,0,0,0 $F(x)=x^5$SS>>``.Times New Roman,18,14,0,0,0,0,0 $F(x)=x^4$SS?@``._^jF^ |Times New Roman,18,14,0,0,0,0,0 $F(x)=x^3$SS>>``.Times New Roman,18,14,0,0,0,0,0 $F(x)=x^2$SS?@``.%BVTimes New Roman,18,14,0,0,0,0,0 $y=f(x)$SS12``.Times New Roman,18,14,0,0,0,0,0 $[a,b]$SS%&``.Dnui.ndTimes New Roman,18,14,0,0,0,0,0 $a$SS  ``.\v 0SBTimes New Roman,18,14,0,0,0,0,0 $b$SS  ``.Times New Roman,18,14,0,0,0,0,0 $y=f(x)$SS22``.Times New Roman,18,14,0,0,0,0,0 $\leftarrow $SS``.Times New Roman,18,14,0,0,0,0,0 $\leftarrow $SS``.Times New Roman,18,14,0,0,0,0,0 $z$SS``.Times New Roman,18,14,0,0,0,0,0 $z$SS``.Times New Roman,18,14,0,0,0,0,0 $z,y1$SS``.Times New Roman,18,14,0,0,0,0,0 $z$SS``.Times New Roman,18,14,0,0,0,0,0 $y1$SS``.Times New Roman,18,14,0,0,0,0,0 $f(x),x,$SS..``.Times New Roman,18,14,0,0,0,0,0 $\approx \dint_A^Bf(x)dx$SSIJ``.C%f""x?gf "M3qTimes New Roman,18,14,0,0,0,0,0 $f(x).$SS``.Times New Roman,18,14,0,0,0,0,0 $f(x)=x^2+1$SSRR``.Times New Roman,18,14,0,0,0,0,0 $x^2+1=0$SSCD``.Times New Roman,18,14,0,0,0,0,0 $x=i$SS ``.WV SFFTimes New Roman,18,14,0,0,0,0,0 $x=-i$SS((``.Times New Roman,18,14,0,0,0,0,0 $x$SS  ``.iE@isTimes New Roman,18,14,0,0,0,0,0 $y$SS``.Times New Roman,18,14,0,0,0,0,0 $(0,1)$SS$$``.Times New Roman,18,14,0,0,0,0,0 $(0,-1)$SS-.``.Times New Roman,18,14,0,0,0,0,0 $0+1i$SS'(``.Times New Roman,18,14,0,0,0,0,0 $0-1i$SS&&``.Times New Roman,18,14,0,0,0,0,0 $f(z)$SS``.Times New Roman,18,14,0,0,0,0,0 $z$SS``.Times New Roman,18,14,0,0,0,0,0 $a+bi$SS'(``.Times New Roman,18,14,0,0,0,0,0 $x$SS``.Times New Roman,18,14,0,0,0,0,0 $z$SS``.Times New Roman,18,14,0,0,0,0,0 $f^{\prime }(z)$SS``.Times New Roman,18,14,0,0,0,0,0 $z_{n+1}=z_n-\dfrac{f(z_n)}{f^{\prime }(z_n)}$SSq(r``.Times New Roman,18,14,0,0,0,0,0 $f(z)=z^2+1$SSPP``.Times New Roman,18,14,0,0,0,0,0 $z=x+yi$SS<<``.Times New Roman,18,14,0,0,0,0,0 $z$SS``.Times New Roman,18,14,0,0,0,0,0 $x+yi$SS%&``.Times New Roman,18,14,0,0,0,0,0 $x$SS``.Times New Roman,18,14,0,0,0,0,0 $y$SS``.Times New Roman,18,14,0,0,0,0,0 $z_0=1+.5i$SSKL``.keurnklthsdrd- arycTimes New Roman,18,14,0,0,0,0,0 $x=1$SS""``.Times New Roman,18,14,0,0,0,0,0 $y=.5$SS'(``.^ Times New Roman,18,14,0,0,0,0,0 $z_1.$SS``.Times New Roman,18,14,0,0,0,0,0 $z_1$SS``.Times New Roman,18,14,0,0,0,0,0 $z_2$SS``.4Times New Roman,18,14,0,0,0,0,0 $0+1i$SS'(``.Times New Roman,18,14,0,0,0,0,0 $.5-1i$SS+,``.peTimes New Roman,18,14,0,0,0,0,0 $0+-1i:$SS88``.Times New Roman,18,14,0,0,0,0,0 $.5+0i$SS+,``.\m] \ple cULe!=RTT::Times New Roman,18,14,0,0,0,0,0 $f(z)=z^3-1$SSPP``.Times New Roman,18,14,0,0,0,0,0 $f(z)=0.$SS66``. [Embedded] 18 .tex 53480 43 53523 1258 17 .tex 54781 43 54824 1298 16 .tex 56122 43 56165 1258 15 .tex 57423 43 57466 1298 20 .tex 58764 41 58805 1018 21 .tex 59823 40 59863 778 22 .tex 60641 36 60677 218 23 .tex 60895 36 60931 218 24 .tex 61149 41 61190 1018 25 .tex 62208 46 62254 378 26 .tex 62632 46 62678 378 27 .tex 63056 36 63092 186 28 .tex 63278 36 63314 178 29 .tex 63492 39 63531 618 30 .tex 64149 36 64185 178 31 .tex 64363 37 64400 378 32 .tex 64778 42 64820 938 33 .tex 65758 58 65816 2312 34 .tex 68128 40 68168 618 35 .tex 68786 45 68831 1658 36 .tex 70489 42 70531 1378 37 .tex 71909 38 71947 658 38 .tex 72605 39 72644 818 39 .tex 73462 36 73498 218 40 .tex 73716 36 73752 178 41 .tex 73930 40 73970 738 42 .tex 74708 41 74749 938 43 .tex 75687 39 75726 858 44 .tex 76584 39 76623 816 45 .tex 77439 39 77478 498 46 .tex 77976 36 78012 178 47 .tex 78190 39 78229 818 48 .tex 79047 36 79083 178 49 .tex 79261 36 79297 178 50 .tex 79475 49 79524 618 51 .tex 80142 79 80221 4578 52 .tex 84799 45 84844 1618 53 .tex 86462 41 86503 1218 54 .tex 87721 36 87757 178 55 .tex 87935 39 87974 778 56 .tex 88752 36 88788 186 57 .tex 88974 36 89010 186 58 .tex 89196 44 89240 1538 59 .tex 90778 38 90816 698 60 .tex 91514 39 91553 818 61 .tex 92371 39 92410 418 62 .tex 92828 38 92866 338 63 .tex 93204 38 93242 338 64 .tex 93580 39 93619 818 65 .tex 94437 40 94477 898 66 .tex 95375 41 95416 1138 67 .tex 96554 40 96594 898 68 .tex 97492 45 97537 1618 69 .tex 99155 42 99197 1152 00100351