End-of-Year AP Calculus Projects

Transcription

1 End-f-Year AP Calculus Prjects Cngratulatins! Yu ve cnquered the AP Calculus BC Exam! Fr the next three weeks, yu will be respnsible fr cmpleting three prjects. The first tw prjects will cunt as test grades fr furth quarter. Yur final prject will cunt as yur exam grade. Please nte that each prject has an expected due date as well as a rubric. Please pay special attentin t the rubric as it will explain hw each prject will be graded. Prject 1 Can Can!: An Investigatin* Due Friday, May 19 th at the beginning f class Yu will prbably never walk int a grcery stre and say, h I need t use calculus t get thrugh this shpping trip! That s still ging t be true. But that isn t t say there isn t calculus hidden behind everyday bjects. Fr this grup prblem set, yu may wrk in grups f up t three t analyze cans. Guiding Questin: Hw vlume ptimized are the cans in the stre? Each can in the stre is made f a certain amunt f metal. 1 Culd yu melt that metal dwn, refrge it int a different sized cylinder which hlds even mre vlume? In ther wrds, des that metal enclse the mst vlume it culd? Yur prcess: Yu re ging t find 3 different sized cans (e.g. sda can, sup cans, red bull cans, Arizna iced tea cans, tuna fish cans, etc.) and calculate the amunt f material used t make these cans (the surface area will suffice we aren t ging t take int accunt the thickness f the cans). Fr each can, yu need t calculate what the maximum amunt f vlume the metal culd hld in thery. Then yu re ging t calculate hw vlume ptimized each can is, by calculating vlume f can best pssible vlume f can This decimal can be cnverted t a percent, which represents hw clse the actual can is t being the highest vlume can. Yur prduct: Yu will need t make a visually arresting, clrful pster (a) with phtgraphs f each f yur cans, (b) with the height, radius, surface area, and vlume f each can labeled, (c) with a clear explanatin f hw yu algebraically calculated the maximum pssible vlume fr yur cans, and (d) with a calculatin f hw vlume ptimized each can is. Yu will want t (fr each can) prduce a graph f the can s vlume versus the can s radius, and mark the pint n the graph with the maximum pssible vlume, and mark the pint n the graph which represents yur actual can. Advice 1. Please d all yur wrk in centimeters, rund t the nearest tenth, and keep in all calculatins until the end. (Use units!) 1 We ll use surface area t talk abut hw much metal is used t build the can. The assumptin is that all cans are made f metal with the same thickness, which I knw is nt true. It s a simplifying assumptin.

2 2. When measuring the radius f each can, yu need t be as accurate as pssible. Yu can get the mst accurate radius if yu measure the circumference f the can (wrap a piece f paper arund the can and mark the circumference, and then measure the amunt yu ve marked ff) and then calculate the radius using C 2 r. 3. When graphing, yu shuld use (an nline graphing calculatr). Type the equatin in n the left hand windw, and click n settings t set the x-min/x-max, y-min/y-max. T save the graph, click n save/share (yu may need t create an accunt t d this). 4. Yu may nt want t write ut the calculatin fr each can n yur pster. Instead, after wrking ut the prblem fr a cuple cans, can yu cme up with a way t shw hw yu d get the answer fr a can with any surface area? Putting that n the pster, with sme explanatin f what yu re ding in each stp, will be useful. Checklist/Rubric: This prject will be graded as a percentage f 83 pts as distributed belw. 1. A clever title (mre clever than the title f this wrksheet) /2 2. A phtgraph f each can /6 3. Can 1: /20 a. Height, radius, surface area, and vlume f each can (in cm, t the nearest tenth) /4 b. Derivatin fr hw t calculate the maximum pssible vlume fr each can i. Derivatin must be neatly written; each step must be clearly explained. /10 c. A Calculatin f hw vlume ptimized each can is (the percent) /2 d. A graph with the Vlume vs. Radius graphed /4 i. Graph must have a prper Xmin, Xmax, Ymin, Ymax and axes must be labeled. ii. Graph has a clearly marked pint which represents the mst vlume ptimized iii. the graph has a clearly marked pint which represents the actual can 4. Can 2: /20 a. Height, radius, surface area, and vlume f each can (in cm, t the nearest tenth) /4 b. Derivatin fr hw t calculate the maximum pssible vlume fr each can i. Derivatin must be neatly written; each step must be clearly explained. /10 c. A Calculatin f hw vlume ptimized each can is (the percent) /2 d. A graph with the Vlume vs. Radius graphed /4 i. Graph must have a prper Xmin, Xmax, Ymin, Ymax and axes must be labeled. ii. Graph has a clearly marked pint which represents the mst vlume ptimized iii. the graph has a clearly marked pint which represents the actual can 5. Can 3: /20 a. Height, radius, surface area, and vlume f each can (in cm, t the nearest tenth) /4 b. Derivatin fr hw t calculate the maximum pssible vlume fr each can i. Derivatin must be neatly written; each step must be clearly explained. /10 c. A Calculatin f hw vlume ptimized each can is (the percent) /2 d. A graph with the Vlume vs. Radius graphed /4 i. Graph must have a prper Xmin, Xmax, Ymin, Ymax and axes must be labeled. ii. Graph has a clearly marked pint which represents the mst vlume ptimized iii. the graph has a clearly marked pint which represents the actual can 6. The pster is neatly put tgether, rganized in a lgical, cherent way /5 7. The pster is free frm calculatin/algebraic/calculus errrs. /10 Edited frm

3 Prject 2 Calculus Recipe Prject Due Friday, May 26 th Each student r pair f students will find a recipe with at least 10 numerical values in the recipe (ingredients amunt, cking temperature, baking time, serving size, etc.). Yu must re-write all numerical values in the recipe using calculus prblems that represent the crrect amunts in the recipe. Fr example: Old Recipe New Recipe 2 eggs lim 16x 2 3x x 8x 2 5 eggs yields 8 dzen yields 3x dx dzen All students must include each f the fllwing types f prblems nce: 1. Limit evaluatin 2. Definite Integral 3. Average Value 4. Slpe f a Tangent Line 5. Vlume f a Slid (by crss-sectins OR revlutin) Additinally, yu must chse three frm the list belw: Other ptins inlcude: Relative Minimum/Maximum Value Abslute Minimum/Maximum Value Inflectin Pint Area bunded by curves Related Rates Plar area Apprximatin using Euler s Methd Values crrespnding t Lgistic Mdels (carrying capacity, ppulatin at greatest grwth rate, etc.) Errr Bund 6. Speed f a parametric particle 7. Arc Length (in any crdinate system) 8. Cnverging Imprper Integral Value 9. Radius f Cnvergence fr an Infinite Pwer Series 10. Cnverging Infinite Series Value (gemetric series, pwer series, etc.)

4 ADDITIONAL CRITERIA: Yu must include a cpy f the riginal recipe, separate frm yur typed calculus versin f the recipe. Yur grup must als hand in an answer sheet that shws FULL SOLUTIONS fr every value. Yur wrk des nt need t be typed, but it must be neat and rganized. Yu are nt required t actually make the recipe but it definitely sweetens the deal by 5 pints! This prject is wrth 50 pints. The pints will be divided amng the individual prblems yu create and graded fr accuracy. If yu have a recipe that has the minimum 10 prblems, each will be wrth 5 pints.

5 Prject 3 Mathematics Interest Prject All papers due Tuesday, May 30 th Presentatins fr exam exempt senirs will begin n Tuesday, May 30 th Presentatins fr nn-exempt students will be during the teacher-made exam perid. Final Prject Mathematics Interest Prject Research a tpic that interests yu that pertains t any mathematical cncept. (It des nt necessarily have t be a high-level math cncept smetimes the simplest cnnectins are the mst interesting!). Write a synpsis f yur interest tpic t be submitted. (31 pints see attached rubric) Present yur interest prject t the class, including audi/visual aids. (19 pints see attached rubric) Pssible Interest Prject Tpics Fractals Tessellatins (and MC Escher) Mbius strip The gemetry f Tetris Rubik s Cube Lgic Prblems (Lewis Carrll) Sudku Card cunting Casin games Origami Optical illusins Card tricks Fibnacci Sequence/The Glden Rati (math in nature) Cryptgraphy and cdes Bard games Vting methds Graph thery Alternative numbering systems/the histry f numbers and numbering systems Abacus/slide rule (calculating befre the calculatr) The histry f Calculus (Newtn and Liebniz) Any ther tpic apprved by Ms. Anthny

6 Interest Tpic Rubric Name Synpsis (33 pints): Frmatting Typed 12 pint Times New Rman fnt Duble-spaced Standard 1 margins 2 3 pages Surces must have 4 credible surces listed n a Wrks Cited page, and cite each surce at least nce in yur paper Citatins prperly included in MLA frmat in bdy f paper Mathematical Relevancy Knwledge f Tpic Ability t Explain Tpic Ability t Explain Math Mechanics/Grammar n persnal prnuns (I, me, yu, us, we, etc.) must include an intrductin and cnclusin standard requirements fr en essay 1 pint deductin per spelling errr 1 pint deductin per punctuatin/capitalizatin errr Subttal: / 33 Presentatin (19 pints): Eye Cntact Speaking Clearly Enthusiasm Cmprehensibility Audi/Visual Aid Subttal: / 19 TOTAL / 52