ÿWPC« f§±V1l¥“ƒ™jáOÖ;`©¢›—ÿ%êÀ,}ßzŠÂA57õrÚg.*s<”JˆÈÇ ›ì3 \Q<{@ÙyçIjé@rÿ‹]O}":`«G|‘…¹­$a‘ïÛ‹¨i­übú¯ÉNyèwu?à”N‘>’!£|§ªPŽxoß7¤P>N#¾Å—muþpV½¿/5OÐå/>‰Iƒa‚Ì%1yöâ= ž¡‘‡pfé9ùõÓ‚Tb k@Áºæmÿ•1G“ÄnKßpž¹yïS™¼Ï±r`Ùy)á•åÝq꜆¦j@E8 ±‚±G>3¨H| g?î:<Ù ÁEû¡­‹‘,yÁIŽÕvÜ>Kük¨2$’´Z}dMųX‰5a›uíáS _}`s 7å“á$ZqA5×2ªDàDJ(v±umVœy÷ØûÀâ.à]áÖ£ˆ$G(%"»C.[Gû\¼ÝWÒE O8Ä  H>ë_‘ÆÊT.TÒ Kˆ¸ÌN’±ŠŠE; üݲ¢WÛ¸eæzhS že¬•ÑÅé[DÜâ  ~\ß#†ãmø™0¦ÞqÏõA})+ÖtĦۋo]BØ ¯n‰jEÞ U<¨ %äên­ú 04§Ûw@Û4U>/U>m<pðø9`($Prestige 10cpiX3|x»«<pðø9`($Prestige 10cpiXÂxpðø@Zhõ X@( ¤T$¡¡Ó  Ó ,U-D.U/0U12U34U5éQ`abcde<@hÐ9Z Š&Courier Regular<@hÐ9Z Š&Courier RegularÝ ƒ¤T!ÝÓ  ÓÝ  Ý€ÌÌÓ  Óñ»ñÔ‡ X(]XXXÔñ»ñUsing€Physics€to€Teach€Calculus€in€an€Integrated€CourseÐ €Ð ÐÓ*ÓÓÓÌÓ»ÓÓ  Ó€Turner€HoganÌDepartment€of€Mathematics€and€Computer€ScienceÌTexas€Woman's€UniversityÌP.O.Box€425886ÌDenton,€TX€76204„5886ÌD_Hogan@twu.eduÌÓÜÓÌÌÓ  ÓÓÓ€€€ñ»ñÔ‡ X(]XXXÔñ»ñ€€In€calculus€the€derivative€measures€the€rate€of€change€and€theÏintegral€computes€the€limit€of€the€sum€of€incremental€parts€so€thatÏthere€are€many€applications€in€physics.€€For€example,€theÏderivative€is€velocity,€the€rate€of€change€of€distance€with€respectÏto€time,€and€the€integral€is€work,€the€limit€of€the€sum€of€theÏproduct€of€force€and€increments€of€distance.€€To€individuals€whoÏhave€a€thorough€understanding€of€the€definitions€of€these€twoÏfundametal€concepts€of€calculus,€the€above€connections€are€clear.€ÏTo€students€struggling€with€these€definitions,€the€connections€areÏoften€clouded.ÌÌ€€€€€In€an€integrated€calculus/physics€course€such€as€that€taughtÏat€Texas€Woman's€University€as€a€part€of€the€Foundation€CoalitionÏEngineering€Education€program,€there€is€an€opportunity€to€useÏapplications€of€calculus€in€physics€to€reinforce€(or€perhaps€toÏreteach)€€calculus€concepts.ÌÌ€€€€€Access€to€reform€calculus€materials€such€as€the€CalculusÏConsortium€at€Harvard€text€support€these€connections€through€theÏrepresentation€of€functions€as€data€sets€and€graphs€as€well€as€inÏanalytical€form.€€This€gives€students€an€opportunity€to€experienceÐ -à(0 Ðderivatives€and€anti„derivatives€as€something€other€than€equationsÏgenerated€from€given€equations.€€The€derivative€of€a€functionÏhaving€a€graphical€representation€is€easily€interpreted€as€theÏlimit€of€the€slope€of€the€tangent€and€hence€the€slope€of€the€curve.ÏThe€derivative€of€a€function€having€a€numerical€representation€letsÏstudents€see€how€the€average€rate€of€change€becomes€theÏinstantaneous€rate€of€change€as€limits€are€evaluated.€€The€integralÏof€a€function€pictured€graphically€motivates€the€relationshipÏbetween€area€under€a€curve€and€the€value€of€the€integral.€€InÏnumerical€form€the€Riemann€sums€strongly€motivate€the€definition€ofÏthe€integral.€€Since€differentiation€and€anti„differentiation€areÏintroduced€almosttogether,€easy€differential€equations€found€earlyÏin€physics€can€be€ð ðsolvedðð€in€an€intuitive€way.Ð pÀ ÐÌ€€€€€An€example€of€a€physics€concept€that€can€be€used€either€as€aÏteacher€presented€topic€or€as€a€team€assignment€to€be€done€in€classÏis€the€derivation€of€the€formula€for€the€moment€of€inertia€of€someÏsolid€of€uniform€density€about€an€appropriate€axis.€€The€teacherÏcan€select€a€solid,€choose€a€subdivision€of€n€equal€parts,Ïdetermine€the€moment€of€inertia€of€each€part,€sum€the€partialÏmoments€and€somehow€take€a€limit€to€complete€the€derivation.€€SinceÏthe€emphasis€is€on€the€concept€of€integration€this€process€isÏuseful,€although€somewhat€tedious.€€Throughout€the€computations€oneÏquestion€should€be€repeatedly€asked€of€the€class.€€"Is€there€anÏeasier€way€to€do€this?"€€Answers€might€include:€€"Use€DERIVE€(orÏsome€other€computer€algebra€system€or€a€graphing€calculator)€toÏwork€out€the€summations€and/or€take€the€limit."€€"That's€anÐ P- (2 Ðintegral.€€Use€the€Fundamental€Theorem€to€evaluate€it."€€"That's€anÏintegral.€Use€DERIVE€to€evaluate€it."€€Since€the€desired€calculusÏconcept€is€the€definition€of€the€integral,€either€of€the€last€twoÏanswers€is€best.ÌÌ€€€€€The€best€use€of€an€example€of€this€sort€is€to€give€each€ofÏyour€classroom€teams€a€different€solid€to€work€with€and€hope€thatÏat€least€one€of€them€will€recognize€the€definition€of€a€definiteÏintegral.ÌÌ€€€€The€end€result€of€activities€such€as€this€one€should€be€aÏbetter€understanding€of€what€an€integral€is€and€improvement€in€theÏproblem€solving€skills€of€the€class€members.ñ»ñÔ#†XU¸XX X(]n#Ôñ»ññ»ñÔ#†XU¸XX X(];#Ôñ»ñ