Winter 2008 - North Carolina Teachers of Advanced Placement
Contents
1. G H Hardy A Course of Pure Mathematics 10 th edition 1993 http www amazon com gp product 052 1092272 Michael Spivak Calculus 3 rd edition Publish or Perish 1994 http www amazon com gp product 09 14098896 Robert M Young Excursions in Calculus 1992 https www maa org ecomtpro Timssnet products TNT_products cfm CALCULUS PROBLEMS FOR A NEW CENTURY Editor Robert Fraga MAA publication ISBN 9780883850842 APPLICATIONS OF CALCULUS R Editor Philip Straffin Catalog Code NTE 29R ISBN 9780883850850 PROBLEMS FOR STUDENT INVESTIGATION Editors Michael B Jackson and John R Ramsay ISBN 9780883850869 Contact Key Curriculum Press www keypress com and order a copy of their Calculus Concepts and Applications along with the Instructor s Resource Book that has 100 explorations The topic of mathematical induction is presented in a way that anyone can understand There are projects and experiments including a pendulum rotating in an elliptical path for parametric function concepts a burette experiment where students verify that the solution of the hydrodynamic differential equation for fluid flow really is a quadratic function and lots more Paul Forester San Antonio 18 Free Software GeoGebra http www geogebra org cms GeoGebra is a dynamic mathematics software for education in secondary schools that joins geometry algebra and calculus On the one hand GeoGebra is a dynamic geometry2. www houstonact org Scroll to the bottom where they have a listing of all workshop materials organized by speaker and look under Nancy Stephenson and she has some good slope field stuff there Matching functions and derivative graphs http www univie ac at future media moe galerie diffl diffl html ablpuzzle http www southfield k12 mi us education components docmgr default php sectiondetailid 30542 amp catfilt er 1042 amp sc_id 1196026176 14 Jack Koenka has co written a textbook specifically with Calculus AB topics Applications of the derivative and integral have been enhanced Here is the website for the book http www trafford com 05 0775 There are two other resource videos about slope fields at AP Central http video collegeboard com apcentral ap03_04 hlarc_mcmullin_slopefields web pa_horizon_ui html http video collegeboard com apcentral ap03_04 hlarc_mcemullin winplot web pa_ horizon _ui html Math Videos on YouTube Abbott and Costello Math Ma and Pa Kettle Math http youtube com watch v 7W Mi5TUJDso http youtube com watch v MRiTzrISSfo New Math by Tom Lehler Quadratic Formula Song http youtube com watch v txSKDyvlG3Q http youtube com watch v H_7INT90DzI YouTube video on optimization http www youtube com watch v 4pzMFXprFQo Denise Fuji Mc Cleary Fairview High School Boulder CO Math Cartoons Math graduate student at U of Nebraska has a web site of mathematical cartoons ht
3. 2f x h fE Z x 2hy 2 xt h x eee ee _ Simplifying we have h h x 2h 2 x h x x 4hx 4h 2x 4xh 2h x _ k k g 2h 2 h So f x mf FEW re lim2 2 and the second derivative of x is2 That looks good b Use this limit definition to show that if f x a ax a x a x then f x 2a 6a x 1 pt OK Now let s try f x a a x a x a x We need expressions for f x 2h f x h and f x f x a t axtax a x f xth a a xth a x h a x h and f x 2h a a x 2h a x 2h a x 2h So substituting the above expressions into our f x 2h 2f x h f x h ay a x 2h a x 2h a x 2h 2 a a x h a x hy a x h ra tax ax ax E difference quotient gives the expanded expression Now multiply everything out and we have f x 2h 2f x h f x h ay a x 2aht a x 4a h 4a h a xX 6a hx 12a h x 8a h h 24 2a x 2a h 2a x 4a h 2a h 2a x 6a hx 6a h x 2a h h 2 3 A tax a x a x h Combining like terms this simplifies to 2a 6a x 6a h a auemaned e 2a h 6a h x 6a h k k lim 2a 6a x 6a h 2a 6a x ve ee f x 2h 2f x h f x so o Our limit definition seems to be working bes tt i EPDE c Generalize your limit definition to find the n derivative 7 f x x in terms of fonly 1 pt To generalize this re
4. Carolina AP biology or environmental sciences teachers AP calculus AP statistics or Honors Precalculus teachers Goal To develop curricular units emphasizing the power of mathematical modeling to address biological questions After the workshop teachers will conduct workshops in their local schools and districts and pairs of biology and math teachers will present their workshop projects jointly at annual meetings of the NCSTA and the NCCTM Leaders The workshop will be led by veteran teachers Dr Noreen Naiman and Dr Jack Bookman Dr Naiman has taught at the North Carolina School of Science and Mathematics for 15 years of teaching Dr Bookman is a member of the Duke Department of Mathematics with a special interest in math education Participants will receive e a 500 stipend e travel reimbursement for up to 200 e room and board on the Duke campus for 5 nights e support to present their workshop projects at the annual meetings of the NCSTA and the NCCTM e Credit for CEU s Application deadline is March 17 2008 Participants will be notified by April 11 Additional information and application forms for downloading may be obtained at the following website http howardhughes trinity duke edu outreach or by contacting Deborah Wahl at 919 684 6066 deborah wahl duke edu Don t Forget NCCTM in Fall 2008 Join us for the NCAAPMT session where we discuss the 2008 exam 24
5. figure out how to use the speeds they measure to approximate the distance they went and explain why it is different from the odometer reading Julia Guggenheimer Erica Wilson scanned and sorted multiple choice questions from four sources old AP exams Barron s George Best and D amp S into the following topics Precalculus Limits continuity differentiability definition of the derivative Rules of derivatives I power product quotient trig and motion Rules of derivatives II chain implicit inverse exponential log Graphing Applications of derivatives optimization and related rates Introduction to integrals average value FTC applications motion Riemann Rules of integration substitution parts partial fractions 9 Differential equations 10 Area volume and length of curve 11 L Hopitals and improper integrals 12 Series 13 Convergence of series 14 Parametric vector and polar BO TY US Using the 2nd edition of FDWK The website www ccsdk12 org mclemens has posted all assignment sheets and class notes for the year Some topics are supplemented they are listed on the assignment sheet as homework with no page number or problems listed Each row on the assignment sheet is one day This pacing leaves about 5 weeks of review before the AP exam as shown on the review assignment sheet Meg Clemens Canton NY 21 Discovery Activity Show kids integral of 1 t from 1 to x and talk about how it cannot be done
6. good mathematical notation and offer them an opportunity to problem solve using techniques of calculus The problems will attempt to follow the AP AB curriculum through the year We will make every effort to keep the problems challenging but within the capabilities of calculus students working together We encourage classes to participate as a team and submit solutions from the class This allows the students the opportunity to practice both writing and reading mathematics We want the students to have to work hard on the problems but if they do they should be able to get some good results Partial solutions will be accepted Some problems will involve theory some applications and modeling We hope that all of them will be different from the standard homework problems and require students to expand their capabilities We hope all of them will be interesting and fun and that students will look forward to working on them every two weeks The classroom teacher can act as an advisor but cannot contribute to the student s solutions The solutions from NC and NCAAPMT member classes will be scored and the teacher whose class scores the most points during the year will receive a plaque honoring their students efforts After two weeks a solution will be presented along with a new problem Particularly interesting student solutions will also be shared If you have any questions or concerns please contact Dan Teague at teague ncssm edu What follows are some
7. how to solve harder and proof based problems from participating in this contest Please share the website first round of problems for the coming year are up along with rules and other information http www usamts org Problems U_Problems php Susan Schwartz Wildstrom www wildstrom com As of June 29th 2007 Texas Instruments Inc will no longer be selling Derive However TI will continue to support Derive customers when possible As Texas Instruments moves forward our plan is to incorporate Derive features and compatibility into new TI products TI will continue to review our hardware and software product offerings and will develop products that meet the customer and market needs Customers that contact 1 800 TI Cares will receive confirmation of the above plans if asked and any support we may be able to provide Dave Slomer An amazing program called Jing which has a real potential for enhancing how to teach mathematics using an interactive whiteboard or even just a data projector and computer Jing is a free program that lets you record your screen actions which are then output as a flash program You can record the entire screen or a selected portion Once you stop recording you can save the flash file and or upload it for free to the web The web address is saved to your clipboard so it can be shared via email or a forum with students and or colleagues Here is the web address of a recent screen recording of me using the SmartView softw
8. l x _ t xy x 2 1 G1 x x E 5 whenever x lt 1 Now I x i 1 1 3 Evaluating at x 4 giveus G 3 gt So I x 1 4 144 2 9 4 16 4 256 14 4x 9x 16x 25xf G x c 1 4 4 9 4 16G 25 The coefficients are squares as before but they alternate in sign We know that 1 4x 9x 16x 25x pona whenever x lt 1 If we replace x with x we have t 2 3 4 1 x Betis 14 4 x 9 x 16 x 25 x whenever x lt 1 This is just I Cx 1 4x 9x 16x 2524 whenever x lt 1 So G x Z Evaluating at x 4 x x l 4 9 y 2 3 4 9 give us oO Gay sane This means that 1 4 9 4 16 4 25 4 sas d I t 8 4 274 64 4 412504 OK Now they are starting to get tricky We need to find the closed form for the series 1x 8x 27x 64x 125x The coefficients are cubes and the powers jump by two and they don t start with a constant l x 1 xy whenever x lt 1 Ifwe differentiate now we get 4 18x 48x That s not what we want If we xX 1 x Gea First let s deal with the powers We know that 1 4x 9x 16x 25x 4 multiply both sides by x we have x 4x 9x 16x 25x Now differentiate both sides of the equation and be careful with the quotient rule So _ x 4x41 e I x 14 8x 27x 64x 125x7 That gets the coefficie
9. them independently and tinker and try things out on their own If anyone else has used one and would like to contribute their thoughts that would be great Editor s Note The CAS version is just as exciting and the new software for your computer makes it much easier for the teacher to write activities for students There are already free downloads for teachers on the site Being able to type on the computer and see the calculator screen the students will see is a real plus Deb Britt College Board Resources Both the Fundamental Theorem of Calculus Special Focus materials and the Differential Equations Special Focus materials are now available as free downloads at the AP Calculus AB and BC Course Home Pages at AP Central They are the two items under For the Classroom under Teaching Resource Materials Please see also the new article Calculus at the Battle of Trafalgar under Resources for Teaching Differential Equations It s a classroom activity that helps both AB and BC students review solving differential equations and was contributed by Dan Teague of the North Carolina School of Science and Mathematics 16 By the Numbers David Bressoud Macalester College Saint Paul MN bressoud macalester edu http www macalester edu bressoud 276 004 number of AP Calculus exams in 2007 380 000 number of students who took AP Calculus in 2006 07 College Board estimate 500 000 number of students who took calculus in high scho
10. The North Carolina Association Of Advanced Placement Mathematics Teachers Newsletter a a a a a a a a a 7 Volume 16 Winter 2008 Issue No 1 Board Members Web Address Deborah Britt Executive Secretary www ncaapmt org calculus Jeff Lucia Treasurer Membership Chair Stephen Davis Web Master Rhea Caldwell Western Region Gloria Dupree President Elect Western Region Ray Jernigan President Eastern Region Emogene Kernodle Central Region Patricia Morris Immediate Past President Central Region Martha Ray Central Region David Royster College Representative Western Region Dan Teague Central Region Sue Wall Eastern Region Notes from the President s Desk I hope you re having a great year You job is always busy trying to find ways to integrate a variety of topics in problem solving contexts differentiate tasks to meet the needs of students encouraging students to maximize their potential finding and plugging the points of discontinuity and more Summing it up you have a difficult sometimes pushing you to your limit but very important role as a teacher As I conclude my term as president I d like to say thank you for serving in the teaching profession and thank you for teaching and developing mathematicians of the future Please invite a friend to join NCAAPMT and please let us know how we can better support you in the classroom Ray Jernigan Advanced Placeme
11. To find a generating function like this one we assume one exists and try to find its structure In this case we say that G x a a x a x a x a x a x and we know that the coefficients are defined by the Fibonacci recursion a 0 a 1 and a 2 3 5 8 and other times 0 1 1 2 3 5 8 In this case we want the first term to be 0 4 a sometimes it is convenient to start with 1 1 n l So if a 4 a we multiply both sides of this equation by x and evaluate for n 0 1 2 3 n 0 1 2 3 a x a x 4a n 0 a a 4 n a x a x a x n 2 n 2 a 2 0 2 2 n 3 a a aG n 4 aeaa rae n 5 A a x a 2 Now add up the equations in the second column We have a a X a x a X tax a a x ax tax a Hax tax a x We have three series that are each a modification of our proposed generating function G x The left side of the equation is G x with the first two terms gone and some x s divided out We can write G x a a x a Xx 2 3 4 5 a A X A X asx Ax a x ea Make sure you believe that statement before proceeding The right hand side has two variations of G x The first is G x a a a x a x a x a x si ON xX while the second is just a ta x a x a x G x So a X 0 x a x tax a a x a x a a Hax a x a
12. an to have my students begin using it in the freshman year Some of the things that I liked about it are but not limited to the following 15 1 Pretty print finally my students will be able to see if what they entered into their calculator is what they intended 2 The higher resolution screen is great 3 The ability to work on a computer exactly how you would on the hand held This is very nice for teachers who want to create files for their students but would rather use a real key board and mouse to work much more quickly 4 The ability to really see things graphically numerically algebraically and geometrically in some cases on the same screen 5 While there is a learning curve the user interface makes some things much easier such as stat plots and regressions No more 2nd statplot turn plot on define the variables set the window oh yeah the dim mismatch error if you need to do another problem and forget to turn off the plot Or linreg L1 L2 vars gt y vars gt function gt y 1 etc just to get a graph N spire does most of it automatically 6 This may be hard to explain but you are able to trace a function and not only put in an x to see what the y value will be but you can put in the y value and see what x value will give you that result 7 The geometry program is nearly as easy to use as GSP or Cabri both on the calculator and on the computer software 8 Able to save files like on a computer with a My Docume
13. anks Or maybe it would be a sequel it s all so derivative after all Together the AP exams would be Seven Days in May 1964 Burt Lancaster and Kirk Douglas 2 If AP Calculus was a food it would be Fig Newtons although integration by shells does make me lust for bundt cakes 3 If AP Calculus was starring in a movie it would be played by actor actress ZERO Mostel would be the choice for CRITICAL acclaim Elizabeth TAYLOR would star in the T V SERIES 4 If AP Calculus was a color it would be TANgent of course And if it were a sports team Maybe the Houston Eulers http education ti com educationportal activityexchange Activity do ald 7734 I stumbled upon this link today and was quite impressed I just got back from visiting NASA Marshall Space Flight Center and learning more about the Ares I and Ares V Orion is the manned upper portion on Ares I Ares looks to be substantially more cost effective and thousands of times safer than the shuttle 20 Want a weird function On a TI 83 or TI 84 1 Graph yl x x ZDecimal 2 Change MODE to atbi and graph it again 3 Change function to yl abs x x and graph WOW My new favorite graph John Hanna Has anyone else seen or used this activity Also TI s activity exchange already has some great TI Nspire activities Sean Bird http cs3 covenantchristian org bird Calculus htm Has everyone seen the new AP national report http www collegebo
14. ant s principal investigator The continuing development of MathPlayer is an outgrowth of Design Science s commitment to creating software tools needed by the education scientific and publishing communities to make math accessible We believe math can and should be made accessible and we are working with the accessibility community to make this goal a reality said Steve Noble Director of Accessibility Policy for Design Science MathPlayer can be downloaded free from the MathPlayer area of the Design Science website www dessci com mathplayer Anyone publishing web pages that include MathML can use the company s Download MathPlayer button on their web pages linking their readers to the free software Design Science was founded in 1986 and headquartered in Long Beach California Design Science develops software used by educators scientists and publishing professionals including MathType Equation Editor in Microsoft Office WebEQ MathFlow MathPlayer and TeXaide to communicate on the web and in print For more information please visit www dessci com 19 Ideas from Calculus Teachers Successive Differences This is a somewhat low tech approach without real world data Are you familiar with the method of successive differences If you create a polynomial function and take successive differences of the value of the function for evenly spaced values of the independent variable the number of differences it takes to arrive at a column of
15. ard com prod_downloads about news_info ap 2007 2007_ap report nation pdf Karl M Bunday Minnetonka MN http Aearninfreedom org Try GeoGebra This is a Java Computer Algebra System that can do what WinPlot does and more http www geogebra org cms GeoGebra is the way to go Its capabilities are amazing Check out the site http mathplotter lawrenceville org mathplotter mathPage index htm Geogebra also permits the sorts of constructions that Sketchpad GSP does Mark Howell has some TI programs on his site You can download those programs from http www gonzaga org teachers mhowell index htm There s documentation included and versions for the TI 89 as well Some of the resources available on the site include detailed syllabi student notes projects and other AP Exam prep materials You will also find resources for AP Statistics Some of you may have visited the site in the past http mshipke com to download various teaching resources This site has now been relocated to http mrskrummel com Michelle Krummel A good project for the week you start derivatives if you have drivers in your school Pair up students and have them go out on 10 minute rides As soon as the students get in the car have them set the trip odometer to zero The non driving student then reads off the speed every minute for the next 10 minutes of the trip At the end of the ten minutes they note how far the odometer says they have gone Their goal is to
16. are to solve a quadratic equation graphically http screencast com t 618IpF4sJDW What a great way to teach 22 the students how to use a graphics calculator And the students will have access to a permanent record of how to do this And you can literally do this is just a few minutes You can even do it in front of a live audience i e my class and save it on the spot to the web for later viewing I also embedded the flash file into an ActivStudio flipchart with plenty of space on the WB page for annotations So now I can stop the movie to discuss what we have seen replay a section if necessary and annotate either alongside or over the flash file I have been using Jing for less than a day but can think of a myriad or even a gigiad of ways I might use it Jing can be found at http www jingproject com Rex For teachers teaching AP for the first time the site has a wealth of materials for AP calculus all are in PDF formats The materials were written by me an experienced AP teacher for many years There are also free materials for AP statistics and within a few weeks a good number of precalculus topics will appear as well A comprehensive list of websites that enhance teaching math is also onthe site The website is www mastermathmentor com Stu Schwartz Pictures from AP reading are at http arapaho nsuok edu math ap calculus html Smothers and Debbie De Maria made NCTM presentation in San Antonio entitled Hands on Cal
17. culus and Webbed Feats Their packet of info is located at this link http www feps edu LakeBraddockSS high school math hands_document pdf Why is it that the derivative of the volume for a sphere gives us the surface area of a sphere but that the derivative of the volume of a cube does not gives us the surface area of a cube Oh but it does First you need the radius of the cube Let the cube be centered at the origin so that it intersects the x axis at r and r Then the edge of the cube is 2r and the volume is 8r 3 The derivative of the volume is 24r 2 6 4r 2 the surface area There is a variation on this idea in Teaching AP Calculus second edition page 139 www LinMcMullin net Conferences Radical Math Conference Brooklyn NY April 4 6 2008 http radicalmath org conference Program Theme Creating Balance in an Unjust World Radical Math encourages educators to incorporate issues of social and economic justice into their teaching and curriculum work Visit the site for information about the 2008 conference and to order the 2007 DVD NCTM 2008 Annual Meeting and Exposition Salt Lake City Utah April 9 12 2008 http www nctm org Research Presession on April 7 9 2008 Theme for Conference is Becoming Certain about Uncertainty CAS Computer Algebra Systems Conference New Trier High School Northfield Illinois Saturday June 28 Sunday June 29 2008 dhall elmhurst205 org Computer algebra systems CAS have the potent
18. ean to solve a differential equation 7 What is Newton s Law of Cooling Describe in words and give the basic equation 8 Perform the Newton s Law of Cooling Experiment described on the back of this page Describe your experiment What substance did you use Did it start in the oven or the freezer How often did you measure and record temperature Insert all required items into your essay 9 Make a conclusion type of statement that ties together all of the above info on integrals Grading Rubric Explained classic problem of calculus Described two approximation methods for finding areas Gave function interval number of subintervals Showed graph with rectangles trapezoids drawn under it Showed approximate area Gave definition of integral Stated what must be true of a function for it to be integrable Integrated the function from 2 Explained all steps adequately Compared results and explained how to get better approximation Stated what a differential equation is Explained what it means to solve a differential equation Explained what Newton s Law of Cooling is and gave equation Described experiment Picture of yourself performing experiment Data table Steps used to solve for C and k and final model equation Graph of data and model Statement regarding accuracy Conclusion statement AMANU NOTE There is an opportunity to earn 5 extra credit points with this essay it is described below 12 Experimen
19. en 1 7225 y 2 4406 and the height of the brick is 152 4406 feet If the distance between buildings is 100 feet then L 100 g 32 B 150 and h 30 Then t 2 5344 y 10 5505 and the height of the brick is 139 4495 feet 10 Calculus Lab Volumes of Solids of Revolution Doris Rowe West Rowan High School Salisbury NC This is a group activity I ve chosen and numbered six containers mainly bottles each with a different shape from the others Each group will draw a number to determine which container they will use for the activity Materials Needed Container provided by teacher Ruler tape measure or calipers for measuring length in centimeters Graphing calculator and Maple software Measuring cups or graduated beakers for measuring volume in mL Water Camera Coordinate grid with centimeter increments Directions Rubric 5 5 10 20 10 10 5 25 5 5 1 2 Read all the directions before beginning lab Also determine what you think the objective of this lab is Measure and record the various radii of your container as well as the overall length use centimeters Depending on your container you may want to measure several sub lengths Based on your measurements and your knowledge of functions determine a piecewise function that could be rotated around the x axis to create the shape or a close approximation of your container Also make the bottom of your container coincide w
20. er against grid Integrals used to calculate the volume and their total Measured volume of container Comparison of volumes percent error and statement regarding accuracy Five additional points will be awarded to the group with the smallest percent error 11 AP Calculus Essay Integration amp Differential Equations Doris Rowe West Rowan High School Salisbury NC You SHOULD work with a partner on this essay Type your responses to the following prompts For the examples use either the TI connect MS Equation Editor or cut and paste from web sites or other resources If you use a resource other than your textbook be sure to cite it 1 In your own words explain the classic problem of calculus that integrals solve 2 Describe the two approximation methods for finding areas Give an example of a function choose one that would require more than a simple power rule when antidifferentiating see 5 an interval and a number of subintervals Show a graph of the function on the interval with the rectangles or trapezoids drawn under it Show the approximate area determined by that method 3 Give a definition of an integral 4 What must be true of a function for it to be integrable 5 Integrate the function you gave in 2 on the interval you chose Show all steps and explain each step Compare this result with your approximation What would have made your answers even closer 6 What is a differential equation What does it m
21. ial to revolutionize mathematics education at the secondary level They do for algebra amp calculus what calculators do for arithmetic simplifying expressions solving equations factoring taking derivatives and much more CAS grants teachers new freedom Come explore the future of mathematics education Discover how secondary and middle school teachers are using CAS in their own classrooms Get classroom tested lesson ideas developed for CAS enhanced classroom environments Learn what other countries are doing with CAS Interact with prominent CAS pioneers from the USA and beyond For more information or questions contact Dan Hall at dhall elmhurst205 org Or Ilene Hamilton at ihamilton district125 k12 il us 23 International Congress on Mathematical Education 11 Monterey Mexico July 6 13 2008 http www icmel 1 org mx icme1 1 Between 3000 and 4000 professionals from 100 countries in the mathematics education area including researchers educators and teachers will gather The International Congress on Mathematical Education ICME aims to 1 Show what is happening in mathematics education worldwide in terms of research as well as teaching practices 2 Inform about the problems of mathematics education around the world 3 Learn and benefit from recent advances in mathematics as a discipline Summer Science and Math Workshop July 13 18 2008 INTEGRATING THE QUANTITATIVE AND LIFE SCIENCES ELIGIBLE TEACHERS North
22. identical differences gives the degree of the function Brad Huff Fresno CA Ten Day Syllabus for Precalculus DAY 1 Teach them that a b c is a c b c DAY 2 Teach them that a b c is NOT a b a c DAY 3 Teach them that x In x is NOT 1 In DAY 4 Teach them that you can t solve sin kx 1 by saying x 1 sin k DAY 5 Remind them that a b c is NOT a b a c DAY 6 Show them a movie of a student sitting in a field writing a b 2 a2 b 2 and then getting HIT BY A TRAIN DAY 7 Remind them that a b c is NOT a b a c DAY 8 Teach them that if the domain of the a function f is the reals the graph of y f x is NOT a blank pair of axes that perhaps they should adjust the window DAY 9 Teach them that x y z is NOT x y x z DAY 10 Group work Bring a trout to class Have them solve sin kx 1 If they get x 1 sin k hit them with the trout Make it a big trout The Exam About 80 is calculator inactive There are maybe 8 9 MC questions out of 45 for which a calculator is required not optional and perhaps 9 points on the 54 point free response Lin Mc Mullin Writing Assignment 1 If AP Calculus AB was a movie it would be Vertical Limit 2000 Chris O Donnell or perhaps A Private Function 1984 Michael Palin and Maggie Smith or The Sum of All Fears 2002 Ben Affleck BC Calculus would also include clips from The Polar Express 2004 Tom H
23. ies 12 Inyers Functions lt ivusiccassessasesssnndswarsypioan seseeansaseneinseaadadwasdaderedidaan 14 Slopefield Resources sseessoessoossssesssesssesssocesoosssosssesssocssoosssosssossssesssosssosso ossos 14 Math Videos amp Cartoons seesessoosessssssessssssesssesesssssssesseesoeosoosso 15 FIENSPI E cei eean ne EE T EAE Enara 15 College Board Resources sseseeseesoososssssesssessesesccesosossosssssseeeese 16 By the Numbers js xosadscsceasadiacsdccrneasirciadaassadainaddeamumiaaaloeseueanss 17 AP Course Audit cosine craiaansorwecscncaasdenacogswaianeris sueann Er en ans 17 RESOUTCES edeeecesssrsssccnsssssessd ed osdans eT eene assoi E e i aa 18 Ideas from Teacher Sas csvessnstiassisasscatensaillvaediicessssneectenactessenctioun 20 CONIERENCES srest saser EE E ENEEK SES 23 Calculus Challenge The North Carolina Association of Advanced Placement Mathematics Teachers website can be found at http www ncaapmt org calculus is hosting a series of challenge problems for calculus classes this year The Calculus Challenge is open to any calculus class in NC and to the classes of NCAAPMT members from outside NC Of course the problems will be available to everyone and any class is welcome to work on the problems if they find them interesting New problems will be presented every two weeks These will be somewhat non standard problems that will allow students to work together and share ideas and approaches practice writing solutions with
24. ig functions y x2 etc then a restriction of the range is often used to make the inverse a function I believe calculators graph inverses by plotting the point y x for each x y that belongs to the function Incidentally since you cannot trace the graph produced by the Draw Inverse operation you may want to do the following which allows you to trace and also reinforces the definition by using it to draw the inverse In Parametric graphing mode enter the function as X1t t Y1t f t where f x is your function Then for the inverse enter X2t YIt t Y2t X1t t Graph in Zoom Square of course to better see the symmetry Trace and then use the up and down arrows to jump between corresponding points on the function and inverse watch the coordinates on the bottom of the screen as well Lin McMullin Slopefield Resources John Hanna s jehanna optonline net TI Nspire website for information on slopefields and other items is www johnhanna us http www linmemullin net Winplot html Wwww mastermathmentor com In the free calculus download sections Thomas Santowski has a couple of worksheets which involve having the students plot slopes and draw the slope field http apcentral collegeboard com apc public repository slopefieldhandoutapne_11870 pdf http apcentral collegeboard com apc public homepage 11871 html The FDWK text has a whole section on slope fields with which you could create your own worksheet if you wish
25. ith the y axis Graph this piecewise function in your calculator or with Maple see me for help Use Maple to graph the solid formed by revolving your piecewise function around the x axis see me for help Using your piecewise function and your knowledge of calculus compute the volume of your container in cubic centimeters Measure the capacity volume of your container in mL Compare your calculated volume to your measured volume recall from science class that 1 cubic centimeter 1 milliliter Compute your percent error Make a statement regarding the accuracy of your calculation That is make a judgment on how good or accurate you think your calculated volume was and explain the difference between it and the measured volume Take a picture of your container in front of the coordinate grid marked by centimeters I m providing one for the class to use Be sure to orient your container so that its axial center coincides with the x axis and its bottom coincides with the y axis Type your lab report including everything listed in the rubric below With the exception of the description of your measurements use compete sentences throughout your report Objective of lab Name and description of the shape of your container ex wine bottle wide base narrow top Recorded measurements neatly presented in a table with written description Piecewise function Graph of piecewise function Graph of solid of revolution Picture of contain
26. n ae is known as the closed form of the series Using the closed form we know that x 1 G G 1 3 1 6 2 a First show that if 1 x x x whenever x lt 1 then 1 2x 3x7 4x 5 I x 1 x 2 3 whenever x lt 1 Find 1 2 4 3 4 4 4 1 pt We know that Lach bat txt pe whenever x lt 1 so by differentiating both sides of this equation we have 1 2x 3x 4x 5 whenever x lt 1 To determine the sum of the infinite 1 series 1 2 1 3 4 4 5 4 we just evaluate the closed form at x The closed form for a series is known as the generating function for that series so we typically denote it as G x If Gaja tha CMS Ss HOBO HO Oe 1x ss 4 b Find a closed form for x 2x 3x 4x 5x Modify your expression to find the sum of 14 4 2 9 4 16 4 25 4 We know that Dacha bat tat pe and 42x43 AP te How do we find a x l x closed form for x 2x 3x 4x 5x If we multiply our second series by x x 0 we have x X 2x7 43x 44x74 4 05 5 whenever x lt l I x How do we use this to sum 1 4 9 4 16 4 25 4 We need to find the closed form for 14 4x 9x 16x 25x and evaluate at x h Notice that 1 4x 9x 16x 25x is the derivative of x 2x 3x 4x 5x so its closed form should be the derivative of I xy x 2x 3x 4x ie es whenever x lt 1 then
27. nt is a registered trademark of The College Board which was not involved in the production of and does not endorse this publication Notes from the Secretary s Desk It has been a fast year My dissertation is coming along All my interviews are complete transcribed and coded The writing is going well at times and then slow at times But it is exciting to see that there really is an end in sight I am job hunting for Fall of 2008 so all the paperwork is a pain at times and it keeps me busy If anyone knows of a position that allows me to still live in Mars Hill NC please contact me I have one AP workshop to do in March and that should complete the year I will be attending the AP reading this summer so our question writers for the summer newsletter will know me personally Please remember to renew your membership to receive the two newsletters You can send your 5 00 check payable to NCA PMT to Jeff Lucia 718 Lansdowne Road Charlotte NC 28270 His email address is jeff lucia providenceday org We want to serve you If you have items of interest please pass them on to me Deb Britt Mars Hill NC dgb531 aol com Table of Contents Page Numbers Calc lus CHAMCN G6 sicscdwansnsstiscacansdavesncavensavecieawciiwecdinanel Guxctexey 2 Calcul s Lap ci ciwsaveeersaccnsiaccsvnecaweanancevecnakarasndusannceesscauanneuieaun 11 Calculus ESSAY scisaccsccccocsessnenconssiscsnmssedencondevcavendddnnnuncvensnenandsanerdsecstannaknccenddeec
28. nts folder 26MB of memory undo and redo copy paste etc even with the same shortcuts ctrl C for copy as on a computer The undo was used A LOT There were many more but I don t want to give away all the surprises Basically it does every thing that an 84 does but does it better There were some things that we didn t like but the TI tech guys that were there said that they would try to address those in the next software upgrade which is now available for download at www ti com One big example that I can think of was the lack of a calc menu on the graphing screen They decided to put in something with the trace function where if you traced to a max an M would appear and all you had to do was look at the ordered pair to see the value or when you got to a zero a z would appear with the ordered pair This was a nice idea but sometimes it didn t work and we couldn t figure out why We insisted as a group that a calc menu is needed It is still a product in development but what is already there is pretty easy to use and the upgrades that are coming will even make it better In short it is a really good product which is likely to only be slightly more expensive than the 84 silver and yet there is an interchangeable keypad so that the calculator becomes an 84 if needed Two calculators in one in a way The teaching materials were very well put together the teachers were able to for the most part work through
29. nts right x 4x41 x _ x8 4x 1 To Now for the powers If 1 8x 27x 64x 125x whenever x lt 1 then by replacing x with x we have 1 8x 27x 64x 125x whenever x lt 1 Weare close to what we want 6 ei 4 4 fe 2 Now multiply both sides of the equation by x So Ix 8x 27x 64x ore 1 x 2 x 4x7 x whenever x lt 1 Since lt 1 we are OK So G x _ _ and 3 1 2 y 4 G _ 531 7 GY 2048 531 The sum is 1G 8G 27GY 646 1256 oO 16 QEY HOHA HAG apo What is the generating function for 1x x 4 x G x 4 x We need to think backwards on this one We must notice that 1 x x x xt x is the derivative of 1 aimed Ixt 4 x E E E So jas must be the derivative of the generating function we x 1 are after What function has L as its derivative Well In 1 x I don t he x x forget the chain rule So does 1x G Jx G Jx 4 x G J In 1 x whenever x lt 1 Notice that the values match at x 0 So G x n 1 x and G 4 In 3 The sum is approximated by 0 405465 Super Challenge 0 1 1 4 2 3 4 5 4 8 4 13 4 the coefficients are the Fibonacci numbers No one is supposed to have gotten this one I just wanted you to see how these kinds of things are done
30. ol includes AP IB dual enrollment amp other courses with calculus in title in 2006 07 very rough estimate based on US Dept of Ed figures 125 000 number of students who are entitled to and take college credit for calculus studied in high school based on AP score CB estimate 20 000 25 000 number of other students who use IB or dual enrollment to obtain college credit for calculus studied in high school personal guess based on total number studying IB and dual enrollment 500 000 number of students who take Calculus I in college 2 year and 4 year each year estimate based on Conference Board of the Mathematical Sciences data includes Business Calculus CB estimates that about 2 3 of those who study calculus in HS and do not receive college credit for it will retake Calculus I in college so about 750 000 students each year take calculus for the first time Breakdown of bachelor degrees US Department of Education data from 2004 the most recent year for which full data is available 78 200 Engineering and engineering technology 61 500 Biological sciences 59 500 Computer and Information sciences 18 000 Physical sciences 13 300 Mathematics and Statistics 230 500 Total out of 1 400 000 bachelor degrees awarded in US in 2004 There were also 307 000 Business majors This does not include Mathematics Education or any of the other majors such as Economics or Architecture that typically require Calculus AP C
31. om en company press releases 070917 htm It s called MathPlayer 2 1 from the same folks who make Math Type Here is their press release Long Beach CA September 17 2007 Design Science Releases MathPlayer 2 1 The new version supports Internet Explorer 7 and adds accessibility features Design Science Inc today announced the release of MathPlayer 2 1 adding support for Microsoft s Internet Explorer 7 and new features to help students with vision and learning disabilities This new version reinforces the company s commitment to furthering math accessibility technology for web pages and braille MathPlayer s math to speech technology has been enhanced to synchronously highlight the mathematics as it is spoken Synchronized highlighting can be a learning aid for all students and has been shown to be particularly effective for people with learning disabilities such as dyslexia MathPlayer 2 1 also supports DotsPlus a braille math format for use with Tiger braille printers and seamlessly integrates with leading screen reader software such as JAWS Window Eyes HAL Read amp Write and BrowseAloud MathPlayer 2 1 contains the initial results of our NSF SBIR grant There are many more exciting features coming in the next release both for people with vision impairments and for the general user We are committed to making MathPlayer the best viewer for math and MathML for everyone said Dr Neil Soiffer Senior Scientist at Design Science and the gr
32. ourse Audit New information on the AP Course Audit is now posted on AP Central http apcentral collegeboard com apc public courses teachers_corner 46361 html It contains the AP Course Ledger a comprehensive and public registry of all courses by school authorized to use the AP label on student transcripts Information is also provided on how to renew in March authorization for 2008 2009 John F Mahoney Moderator AP Calculus EDG 17 Resources One good resource is the following paper the text of which can be found on the internet at the URL below Dan Kennedy Things I have learned at the AP reading College Mathematics Journal 30 5 November 1999 346 355 http tinyurl com ebvdq see middle of web page Below are some books Dave Renfro thinks any serious calculus teacher should consider having on hand Apostol et al A Century of Calculus Part 1 1894 1968 1992 https www maa org ecomtpro Timssnet products TNT_products cfm Apostol et al A Century of Calculus Part 2 1969 1991 1992 http www amazon com gp product 0883852063 James Callahan et al Calculus in Context The Five College Calculus Project 1995 http www amazon com gp product 0716726300 Richard Courant and Fritz John Introduction to Calculus and Analysis Volume 1 1965 1999 http www amazon com gp product 354065058X E Hairer and G Wanner Analysis by its History 1996 http www amazon com gp product 0387945512
33. ra Credit opportunity If your experimental substance is something that the class can eat bring it in to earn 5 extra points 13 Inverse Functions Y1 X Sabs X 1 or any other fctn in question Y2 solve Y1 Z X Z 2 to get the inverse of Y1 ZOOM 4 to have a square window Note that the graphical CALC features work Try 2nd TRACE 6 to be tracing on Y2 the inverse of Y1 and then ENTER to get the slope of Y2 at the abscissa you had Try option 7 to be dazzled with the shading On the Home screen nDeriv Y2 X X 2 gives the slope to be 2 fnInt Y1 X X 0 2 gives 1 333333 TI 82 83 and 84 support this Steve Just a bit about the TI drawinv command I don t use it when demonstrating on my overhead display calculator because it is easier for me to just turn the display screen over on its back and give it a quarter turn This is equivalent to reflecting the graph over y x It really reinforces the idea of the reflection in students minds There s no possibility that the calculator is doing some kind of magic to get the picture of the inverse Of course students can t do this trick with their own calculators Dave When does a function have an inverse that is also a function Answer When it is one to one Obviously some functions have inverses that are not function e g y x 2 becomes x y 2 not a function You need to make a distinction between the two possibilities If the inverse is not a function e g the inverse tr
34. samples of our work so far Selected Problems from the NCAAPMT Calculus Challenge Problem 3 h In class you have learned the definition of derivative We know that f x lim fe F h whenever the limit exists The derivative of a function is another function so we can find its derivative as well The derivative of the derivative is known as the second derivative and is denoted f x a Find a limit definition for f x similar to f x lim and using only values of f f x h f x h do not use f in your limit statement We know that r 4 noai LO h gt 0 and that f EG To make it a little easier to read define function g to be the derivative of f so g x f x g xth g x f x h f x h h We want g x g x lim and IX 7 S th h f h h So g x lim g xt h lim im ott Sth imf Ct C a y n0 h ie h So f x lim lim h gt 0 h h gt 0 h We know that the limit of a sum is the sum of the limits so we have Pae a OOE h gt 0 h gt 0 h and eta FG 2h 2F th F h gt 0 40 h Also lim limQ x is just lim Q x The composed limit statements are redundant So we have oa f x 2h 2f x h f x f x inl 7 Before we try our definition on a messy cubic let s try it on something simple like f x x before we go too far down the wrong track If f x x then S xt 2h
35. source code for the applets In addition to the calculus applets mentioned previously there is a free graphing program WCMGrapher targeted at making 2D graphs for precalculus and calculus that can be copied pasted into word processing programs for handouts quizzes tests etc and presentation programs for projected presentations PowerPoint based games like Jeopardy etc WCMGrapher runs as a standalone application on Windows Mac Linux and uses Java 5 or later It is free and you can download it from http sourceforge net projects webcompmath Click the big green download button then select the latest version of WCMGrapherInstaller xxxx jar where xxxx is the version number Double click the file that downloads to run the installer which will install the software and a user s manual You can also view an extract of the users manual at http webcompmath sourceforge net html WCMGrapher 20Users 20Guide htm which has an overview and more detailed installation instructions Tom Downey A place that you can get some really cool stuff is www houstonact org Go to the bottom of the page where you will find materials archived by speaker and you can download really cool stuff Dixie Ross USA Math Talent Search is a free contest run by the Art of Problem Solving for students in all levels of mathematics USAMTS is a four round five problem per round contest with interesting problems of a non calculus type Students can learn a lot about
36. sult let s consider the third derivative We need to replace function g in h 2h 2 t h pp EHAEO in ze EG pattern try to explain why the pattern exists or how it is created and see if we can see a pattern Ifyou see a Replacing function g in lim gwth el ih G h h gives us k k h gt 0 h Ereren AEren lim which simplifies to fam Ot et 2h 43 f x th fC h gt 0 he Notice how the terms in the numerator combined The like terms in each expression have the same sign and add together apei Aei lim k K 30 h AGRE et AG ii h For the 4 derivative we replace function g in lim P h 0 f 3h 3f 2h 3f eth FO P This gives f x 4h 3f x 3h 4 3f x 2h f xth f x 3h 3f x 2h 3f x h f x lim he he which simplifies to jml CTA AAS et 3h Of 2h 4F et h t FO h gt 0 h Again look how the terms in the numerator combine S x 4h 3f x 3h 3f x 2h x h f x 5h 3f x4 2h 3 f x4 h r x lim i i If this looks familiar you are right The coefficients combine in the same added way as the elements in Pascal s triangle These coefficients are known as binomial coefficients denoted In general we see that n APG Sith y h gt 0 Problem 4 Adding with Derivatives Aies 1 We know that the infinite geometric series 1 x x x x Er whenever x lt 1 j di The expressio
37. system You can do constructions with points vectors segments lines conic sections as well as functions and change them dynamically afterwards On the other hand equations and coordinates can be entered directly Thus GeoGebra has the ability to deal with variables for numbers vectors and points finds derivatives and integrals of functions and offers commands like Root or Extremum These two views are characteristic of GeoGebra an expression in the algebra window corresponds to an object in the geometry window and vice versa Graph http www padowan dk graph WinPlot from Peanut Software http math exeter edu rparris winplot html Peanut Software http math exeter edu rparris winplot html EquationPlotter http mathplotter lawrenceville org mathplotter index htm Resources for Disabled Calculus Students John Burrell wrote One of my Calculus AB students is blind and I am looking for any suggestions on teaching methodologies to provide him with the best possibility to learn Tony responded This is the site that has the best products for the blind math student VEO self voicing tactile audio system NEW Emprint SpotDot Embosser http www viewplus com This site is the location of some of the sites that I used when I was teaching my second blind student http www psja k12 tx us dvillarreal NorthSite MiniPortals Mathematics htm 381 Free For students with visual and learning disabilities http www dessci c
38. t Portion of Essay Newton s Law of Cooling Materials needed e Thermometer preferably a meat thermometer if using the oven e Watch e A substance such as a cake or a large piece of meat nothing too small into which you can stick the thermometer e Camera Directions 1 Start with your substance in an oven or a freezer or some place that is significantly different in temperature from your house 2 Choose a room in your house whose temperature does not fluctuate easily Measure and record the temperature of that room 3 Have someone take a picture of you doing this step Measure and record the temperature of your substance when first removed from oven freezer or other location Move the substance to the room you chose in step 2 4 Continue to measure and record the temperature of your substance three more times for a total of four data points be sure to record the times at which you take the temperatures 5 Use the first two time temperature data points to solve for C and k in Newton s Law of Cooling 6 With your graphing calculator plot the points and your model together Use the data matrix editor to enter the points Enter your equation in the y screen 7 In your essay insert a picture of yourselves conducting the experiment your data table your equation and the steps you took to solve for C and k and your graph 8 Make a statement about the accuracy and prediction value of your equation Ext
39. tp brownsharpie courtneygibbons org Calculus Tutorial Videos We have been creating calculus tutorial videos on a wide range of AB BC and college level calculus They are all free and can be viewed at www midnighttutor com As the name implies we are trying to create a resource for the student who is lost on a problem set and has no other resource available to him her The project is a direct outcropping of my own experience teaching AB calculus two years ago Streaming video turns out to be a wonderful medium for this kind of instruction since the video can be paused rewound etc Please make this resource known to your students Also if you have any problems which you would like for us to add to the collection please feel free to email them to solve midnighttutor com Tom the Midnighttutor The New TI Nspire Dan Dudley Verde Valley School Sedona AZ I don t know how many have had the opportunity to attend a workshop on the N spire but I was lucky enough to attend one for the last 3 days Not only was I able to spend 8 hours a day for 3 days working with it but I was able to own one at the end of the conference I was very impressed with the product There obviously was a lot of thought into what teachers can use in a classroom to have children learn things in different ways I was using the non CAS version which since I will be an IB only school starting next year and this is the most likely version that IB will allow I pl
40. value for y indicates a position below eye level We see that tan 0 so 0 Tan KI n d d ac Also note that F 2 g so gt c Since the initial velocity is t 0 the brick is dropped not thrown we have a gt Further if t gt then y h t gt If 0 Tan Gi then ae 5 Simplify this expression dt q af L 2 la and substitute y gt and y h 1grt So a 2 a 5 E dt L y L ty y 1 L Finally a dy z ket z We want to maximize zhet 7 Now we have a classic optimization problem Differentiate using the quotient rule and set the derivative equal to zero G h gt The denominator is always positive so we don t need to worry about a critical value where the derivative does not exist From the numerator factor out Lg and multiply everything out so 2 4 0 cefe h hgt E tagh gt 3 r Combine terms to arrive at the 4 degree equation 0 r h het We can solve this equation for t which is all we need to find y h t gt Using the quadratic formula we have pa 2ht V4h 3P 3 Since must be positive we use the sum rather than the difference solution amp 2n V4n 30 2h V4 30 Finally the height of the brick So p h Lgt h 4 2h_ V4h 30 6 at this pointis B y ar As a test case consider L 50 g 32 B 150 and h 30 Th
41. with power rule Okay so what does it look like Hand out graph of 1 t on a very fine grid 100 squares per square unit and assign each kid different x values and have them approximate area by counting squares Talk about why some will report a negative area since their x value is to the left of 1 our defined starting point Once all kids have their areas collect the data x value vs area and plot it Lo and behold it s the In curve Dixie Ross http www calculusapplets com is a website that provides interactive Java applets for teaching and learning single variable calculus The applets use graphs and tables to illustrate concepts in calculus and allow the user to dynamically change the functions involved or the point on the graph that is of interest These pages can be explored by students learning calculus or used by teachers while teaching calculus and are geared to the topics covered in AB and BC Advanced Placement Calculus Each applet comes in a version shown on a web page suitable for viewing in a browser window and also as a separate resizable window with larger fonts and line widths suitable for projection on a screen Note that this set of pages is not a complete text for calculus but is intended to supplement a standard textbook or web site Use of the web site and the software is free and should work on Windows Macintosh Linux systems that have Java 5 or later installed You can also download the entire web site and download the
42. x can be rewritten as G x a ax _ G x a 2 X x G x Solve for G x AyX Ay A X So G x a a x xG x a x x G x and G x We also know that x x 1 a 0 a 1 so x G 3 G x _ Finally we have G 4 _ _ _ x axe Finally we have O a 3 x a Ai 3 The question of for what values G x anra actually represents the infinite series x x O lx lx 2x 3x 5x 8x 13x is an important one that we cannot answer with the x O lx lx 2x 3x7 5x 8x 13x7 _ for x lt mathematics that you presently know It turns out that V5 1 5 x x 1 2 To convince yourself that G x ae actually generates the Fibonacci series just do the long x x division and see how the terms are created Problem 7 A student is standing on top of a building B feet tall At a point h feet above her eye on the top of a building L feet away a brick is dropped Assume air resistance is negligible so that the acceleration due to gravity is g Ifthe student rotates her head to keep her eye on the brick as she watches it fall sec determine the height of the brick at the point in the brick s descent when the student s head is moving most quickly First we draw a diagram The figure at right represents the situation We let the variable y represent the position of the brick measured from horizontal So a negative
Download Pdf Manuals
Related Search