Calculus for AP Physics-C

Transcription

1 Calculus for AP Physics-C Adaped from: hps://preygoodphysics.wikispaces.com/pgp+calc+for+appc

2 ABLE OF CONENS DIFFEENIAL CALCULUS 3 SUM ULE 5 [EMINOLOGY] 6 SUM O DIFFEENCE ULE 7 PODUC ULE 8 QUOIEN ULE 9 HE CHAIN ULE INEGAL CALCULUS of 7

3 As you will see, Calculus is no really a subjec of AP Physics, i is a mahemaical ool ha we use o solve problems. However, many of you are now jus saring Calculus. For ha reason, his is a brief sudy o help you learn how o use he Calculus ools. While his maerial does involve some proofs (or parial proofs mos of he deails are lef o your mah eacher. Mos of he pages are devoed o differenial calculus, because essenially inegrals are ani-derivaives. You may no realize, bu we have already done Calculus in firs year Physics. Every ime you found he slope of a line you were doing Differenial Calculus. Every ime you found he area under a curve you were doing Inegral Calculus. he only differences were ha we usually used real daa poins whereas in his class (and Calculus we will use funcions fied o real daa o find he slopes (Differenial Calculus or he areas (Inegral Calculus. We found slopes of sraigh lines, which is easy o do from a graph. However, i is no so easy o find he slope of a curve because i is always changing. Calculus involves raes of change. I involves funcions ha are no linear and herefore requires a differen approach. Differenial Calculus Wha follows is a brief eplanaion of differenial calculus. Since we will be finding angen line slopes wihou using real daa poins, we need o find a way o sneak up on he slopes. angen lines o a curve ouch he curve locally a only one poin (he poin of angency. Since we need wo poins o compue a slope we will use he Secan Line slope mehod of finding approimae slopes and hen drive he secan line o a angen line by moving one end of he secan line oward he oher end. Suppose we have he following graph of a funcion : Using basic algebra, he slope of he line hrough he wo poins and is he slope of he secan line which is (recalling ha slope is rise over run (1 Now wha we really wan is he slope as approaches zero (moving he righ hand poin closer and closer o he lef hand poin and he secan line sars o look like he angen line ouching he funcion a only one poin near. (See hp:// for an animaion. Suppose, where is a consan. Wha is he slope of he curve a any poin? Well, le s use equaion (1 o find he slope of a paricular angen line. [ ] ( 3 of 7

4 hose of you who have aken Calculus know ha a large par of Calculus involves sneaky algebraic manipulaions, which ofen sar ou making hings messy, and hen proceed o reduce o somehing easy (relying on he old adage, someimes hings have o ge worse before hey can ge beer. here are HEE MAJO ICKS IN MAHEMAICS ha one can do o an epression ha DOES NO change he numeric value of he epression. We will use each of hese in he following derivaions: 1. Epand any epression ha can be epanded.. Add ZEO o he epression 3. Muliply he enire epression by ONE. We will begin wih mehod 1. Using Eq. ( as a saring poin: herefore, he slope of he secan line is. Now as approaches zero he slope becomes. Suppose we have. Wha is he slope of his funcion? Following he same procedure as before yields: hus he slope of he secan line is. Now as approaches zero boh he second and hird erms ge closer and closer o zero and he slope of he angen line becomes. Le s generalize his process and find he slope of? Well, firs we have o remember how o epand an iem like (remember he fibonacci riangle? Noice ha he firs erm is always and he second erm is always and he res of he erms have Using his paern, wha is? 4 of 7

5 Now, le s go back o our original problem and use his idea o find he slope of f( = k n Bu in he las power, he phrase is sill ONE or bigger since he original power. he slope of, since (3 Wha his means is ha for any simple power funcion, he slope is simply he original funcion imes he original power and hen he power is dropped by ONE. his is a derivaive. I is ha easy! his is known as he Power ule. Eample Suppose you are moving so ha your posiion is given by he funcion. hen your velociy (which is he slope of he posiion curve mus be. Your acceleraion (which is he slope of he velociy mus be. If you wish, you can even find he jerk (rae of change in acceleraion as he slope of acceleraion curve, which is. You can even find he snap (rae of change in jerk as he slope of jerk curve, which is, bu we digress. Noice ha las sep. his rule works for ANY power funcion wih any power (posiive, negaive or decimal. More Eamples his means ha he graphical physics we did las year can now be done very easily wih a simple formula. However (here is always a cach, isn here!?!, no all funcions are of he simple form. So, we need some ways o address oher funcions. he following describes how we will address more comple funcions. Sum ule Suppose we have a funcion ha is he sum of wo funcions? is he sandard way of saying he sum of wo funcions. An eample migh be,. 5 of 7

6 We now have o ask wha really means. Well, wha does mean? A rue mahemaician would say i is a se of numeric pairs where each unique firs number has EXACLY one second number. Or hey would say ha i is a way o mach up o ses so ha each member of he firs se (domain is conneced o eacly ONE member of he second se (range. For he purposes of Physics, we resrain he ses o be he EAL NUMBES and he connecion rule is ALWAYS an Algebraic Saemen (well a leas in mos of physics. hus means pu he number ino he algebraic rule called and spi ou is one and only value called. Above, in, we compue wo separae erms (independenly and hen sum he resuls. In he case above we could have facored firs o give bu why make i messy by urning a simple sum ino a muliplicaion!! When you have a sum of wo unknown funcions one MUS deermine each separaely AND HEN add. hus if one mus deermine and separaely and hen sum heir resuls. he slope equaion (1 becomes, [ ] [ ] or, rearranging (invoking ule 1 from page, we ge [ ] [ ] [ ] [ ] earranging furher, we ge [ ] [ ] [ ] [ ] he wo fracions on he righ are jus he slopes of funcion g( and h( added ogeher. hus, if a funcion is he sum of wo funcions hen is slope is jus he sum of he wo slopes. In our eample, from jus above. he slope is simply. [erminology] We are going o ge ired of consanly saying he slope of f( is so le s creae a shorhand way of saying he same hing. Say we have he funcion. 6 of 7

7 hen a, hus he slope is Of course, we are asking wha happens o hese raios when approaches zero. he asue will noice ha when approaches zero, so does and we end up wih he famous conundrum. he sneaky hing ha happens is ha boh op and boom go o zero ogeher bu he raio says he same as hey approach zero. We epress his approaching zero consan raio as simply he slope of he angen line. So we wrie he slope as (4 We also denoe he slope of he noaion and is spoken as prime of. hus we have for he slope of he ac of finding he slope is called differeniaion and he resul of differeniaion is called he derivaive. he ac of finding he slope of a line angen o a curve is called differeniaion. he resul, which is he slope of he angen line, is called he derivaive. If one wans o find a derivaive, hen one differeniaes a funcion according o he prescribed rules. epeaing he eamples from above using he new noaion: Wha is imporan o realize is ha all we are doing is finding he slopes of angen lines o curves. End of [erminology] Sum or Difference ule Wha happens if we have a funcion, which is he sum or difference of wo funcions? Doing he differeniaion we ge [ ] [ ] [ ] [ ] [ ] [ ] Noe ha he las wo erms are simply he derivaives of and. 7 of 7

8 hus he derivaive of he sum or difference of wo funcions is simply he sum or difference of he derivaives! herefore, we have he following: If, hen (5 If, hen (6 Warning: he above rule does NO work for producs, quoiens and composiions. We will now grind our way hrough hese oher operaions. Produc ule Suppose we have he funcion Jus in case you hink hese funcions don eis consider a rail coal car ha is having boh is speed and mass change as a funcion of ime. Momenum depends on mass and velociy, bu boh and are funcions of so he correc equaion for momenum is, which is he produc of wo funcions. We will see ha he ime derivaive of momenum is he echnical definiion of force. So, ha being said, how do we find he slope (derivaive of a produc of wo funcions? Well, here i is, bu wha o do wih i? Le s invoke ULE from page simply add zero o i in he following srange manner: As srange as ha seems, i is logically legal since adding zero does no change he epression on he righ side of he equaliy hus he equaliy is sill rue. he las wo erms are he same number ecep one is posiive and he oher is negaive. Now, rearranging he mess above we ge [ ] [ ] [ ] [ ] [ ] [ ] as hus, he derivaive of is (7 8 of 7

9 As an eample, consider he funcion wih and his is a silly eample since one can jus muliply he wo funcions and hen do he derivaive. If hen from wha we have already learned. However, using he new rule, we ge, which is he same. Warning: DO NO MULIPLY HE DEIVAIVES. If we muliply he derivaives in he previous eample, hen we would erroneously ge HIS IS WONG SO DON DO I. he ule for differeniaing funcions muliplied ogeher is: he FIS imes he derivaive of he SECOND plus he derivaive of he FIS imes he SECOND. Quoien ule Wha happens when you have one funcion divided by anoher? Now, o add (or subrac fracions, one needs a common denominaor. Bu, his common denominaor is in he numeraor of he larger fracion. hus I will call his number he common denumeraor o coin a phrase. Now, I will do he magic and add ZEO again. If you have o ask why I chose his paricular ZEO hen you have he makings of a mahemaician. he answer is: Afer monhs (years of rial and error Newon (Leibniz? found wha works!! 9 of 7

10 [ ] [ ] Please noe he facorizaion in he las line. ead i carefully. [ ] [ ] [ [ ] [ ] ] [ ] Wha his says is ha he derivaive of a raio of wo funcions is: he BOOM imes he derivaive of he OP minus he derivaive of he BOOM imes he OP, ALL divided by he BOOM squared. Here is a way of memorizing his echnique: Suppose (High over Hoe hen he derivaive is Ho deehi Hi deeho Over HoHo he Chain ule Suppose you have a funcion of a funcion. In oher words, compue a funcion hen use he resul of ha compuaion o compue a second funcion. I sounds uncommon bu i is very common. Here are some eamples: (he mus be compued firs hen he sin compued (Normally, one would compue he firs hen cube he resul hough i could be epanded Now, suppose we have he following: which simply means compue hen use he resul o compue he funcion. Wha is he derivaive of? Now, wha does one do wih his? Well he rick is o use ULE 3 from page and muliply by ONE. 10 of 7

11 dy g(h( + - g(h( h( + - h( = f ( = d h( + - h( g(h( + - g(h( h( + - h( = * h( + - h( g(h( + - g(h( = *h ( h( + - h( Now we have o ge sneaky and name dy d which also implies ha giving us g(h( + - g(h( = f ( = *h ( h( + - h( g(z + Dz - g(z = * h ( (z + Dz - z g(z + Dz - g(z = * h ( Dz = g (z* h ( = g (h(* h ( Wha he alphabe soup above means is ha one akes he derivaive of he ouer funcion and jus subsiues in he inner funcion, hen muliplies he whole hing by he derivaive of he inner funcion. Le s look a some eamples: Firs le s use he second eample from he sar of he Chain ule which was his is a combinaion of and or and herefore, Now le s show ha his is righ by doing he derivaive anoher way. he original funcion was when epanded ou. 11 of 7

12 Boh derivaives are he same. Muliplying a funcion by a consan: I have lef his iem o las simply because I forgo o do i earlier! Suppose we have he funcion where. Wha is is derivaive? y k dy k( f ( kf( f ( f ( f (, herfore y' k kf'( d Well, ha was easy. he derivaive of a consan imes a funcion is he consan imes he derivaive of he funcion. Before you can use Calculus you will need o do LOS of pracice aking derivaives of many funcions before you are ready o use Calculus as a ool in Physics and oher areas. eview of wha we have learned so far (yes here is more o learn. If hen. his is he Power ule and he ONLY funcion specific rule so far. If hen If hen If hen If hen If hen If hen Pracice will make you perfec EVENUALLY! 1 of 7

13 Ouside of polynomials, here are as many specialy (non-polynomial funcions as here are grains of sand in he universe. We will concern ourselves wih a very useful (and small subse of hese funcions. hey are: Le s find he derivaive of. dy d = sin( + -sin( Now wha o do? he rick is o conver he lef erm in he numeraor using a rig ideniy ha we all know and remember wih fondness. If you are no in ha group ha remembers such suff, hen ge a mah book ou and look up rig ideniies. I will use he rig. ideniy:. dy sin( + -sin( sin(cos( + sin(cos( - sin( = = d Now, slighly rearrange and hen do a bi of facoring. dy d sin(cos( + sin(cos( - sin( = = sin( [ cos( -1] = + sin(cos( = sin(* sin( [ cos( -1] + sin(cos( [ cos( -1] + cos(* sin( Now i is ime o bypass he Limi mehod you will learn (or have already seen in Calculus and jus do some arihmeic. (Noe o suden: his is a live Ecel Char jus double click on i if you are reading his file in MS Word. 13 of 7

14 Δ Δ cos(δ cos(δ-1 (cos(δ-1/δ sin(δ sin(δ/δ Degrees adian he firs column sars ou wih common angles in degrees unil 1 o and hen jus sars halving he previous angle. he second column is he degree convered ino radians. I probably is worh noing (maybe even making a big deal ha a radian is NO a uni. I is simply a fancy way of saying ONE he raio of he arc lengh o he radius, hus is consrucion unis cancel. In fac, we will see eamples laer (paricularly in Simple Harmonic Moion and oaion where he word radian is purposely ignored and in fac deleed. By he way, 1 radian = 180 o /π = abou o. All he oher columns do rig. wih he radian value since Ecel does rig in radians. cos( 1 he fifh column is he raio which approaches ZEO as 0. sin( he sevenh column is he raio which approaches ONE as 0. Small digression before reurning o Calculus: his las fac above is very useful when we need o simplify a hard o compue problem. Having a raio approach ONE simply implies ha he numeraor and denominaor become he SAME number as he denominaor approaches zero. hus if one has a process ha uses he sine of small angles, one can ofen jus use he angle iself insead of he sine of he angle. Look back a he able. A 30 o, he sin(30 0 and 30 0 in radians are wihin 5% of each oher. A 15 o he wo numbers are wihin 1.1%. A 5 o, he numbers are wihin 0.1%. his means ha if one is measuring o wihin 5%, he angle (in radians can be used insead of he sine(angle up o 30 o. Errors less han 1% allow he subsiuion below 5 o. We will use his Small Angle Conversion when we sudy Simple Harmonic Moion and when we look a Inerference (assuming we have he ime. End small digression. On wih Calculus. 14 of 7

15 he raios above imply he following: dy d sin(cos( + sin(cos( - sin( = = sin( [ cos( -1] = + sin(cos( cos( -1 = sin(* = sin(* ZEO + cos(*one = cos( sin( [ cos( -1] + sin(cos( [ ] + cos(* sin( hus he derivaive of is. Now ha is a nice and easy fac! Previously, i was shown ha if, hen. his is he Chain ule and is ofen applied o he derivaives of rig funcions like sine and cosine. he argumens of rig funcions are ofen funcions hemselves, so he derivaive of a rig funcion is he derivaive of he funcion muliplied by he derivaive of he argumen. Now le s find he derivaive of. dy d cos( + - cos( = = [ cos( -1] = cos(* We used he rig Ideniy cos(cos( - sin(sin( - cos( - sin(* sin( hose wo raios should look very familiar from above so le s jus use hem. = cos( [ cos( -1] - sin(sin( dy d [ ] cos( -1 = cos(* - sin(* sin( = cos(* ZEO - sin(*one = -sin( hus, he derivaive of is. 15 of 7

16 Noe ha he slope numbers of he sine curve are he same numbers as he cosine curve which implies ha If hen. Noe also ha he slope numbers of cosine are he negaive of he sine numbers implying ha If hen. Wihou furher ado, le s find he derivaive of he angen funcion. We don really need o know i for AP Physics bu you now have ALL he ools you need o find i. y = an( = sin(, herefore he division rule cos( y = dy boom *op slope - op*boom slope cos(cos( - sin((-sin( = = d boom cos( [ ] + [ sin( ] 1 = [ ] [ cos( ] = [ sec( ] = cos( cos( [ ] his is jus an eample of how we can handle wha look like hard problems by breaking hem down ino known chunks and applying he rules. Now for your firs major EAL PHYISCS SUFF using Calculus! 16 of 7

17 Suppose we have an objec moving in a circle of radius wih a period. he speed of he objec is simply he circumference divided by he period, hus we have. From las year, we also know he cenripeal acceleraion is: [ ]. We did no derive his epression las year. You now know he calculus necessary o do so. Le s see If he objec is moving around he circle wih a velociy hen where is i a any inermediae ime? Leing (Basic physics we ge where is lengh along he circumference. he cenral angle shown above as is hen simply (in radians. Z A ime, he objec has moved from o. Now, we acually know by doing rigonomery! cos( cos( or cos( epeaing for y sin( we ge y sin( sin( or Boh and are funcions of. We can now acually compue how fas he objec is raveling along each ais by differeniaing each funcion (Slope of posiion is velociy. Noe he use of he Chain ule below!! cos( or v d d sin( sin( 17 of 7

18 18 of 7 I may be hard o see bu I have invoked HEE (3 of he differeniaion rules: 1. he las rule Consan imes a funcion. he derivaive of he Sine funcion 3. he Chain ule epeaing for we ge Noe ha and are boh componens of he acual slan or angenial velociy as he objec moves along he circle. his means we can use he Pyhagorean heorem o calculae he acual velociy. Period Circumference V V V y 1 ( cos ( sin cos( sin( So, Calculus has solved a problem for which we already knew he answer. So wha? Bu, wai, here is more. We have he - and y- velociy equaions. We can compue he acceleraions along each ais doing eacly he same hing. sin( 4 sin( '( cos( 4 cos( '( d dv a v and d dv a v y y y Now using he Pyhagorean heorem again, we find he ne (Cenripeal Acceleraion as follows: 4 (1 4 ( cos ( sin 4 sin( 4 cos( 4 a a a y c cos( cos( sin( d dy v or y y

19 Noe ha his is he same equaion as on p. 13. he only difference is ha we derived i from basic physics and Calculus wihou he usual hand waving we used in Physics 1. Jus one of MANY eamples you will see during he year of he power of Calculus. Now we have o address inerlocking funcions ha appear from ime o ime in his course. hey are he eponenial funcions given by: y = a and y = e and heir INVESE funcions given by: y = log a ( and y = ln( We will derive he derivaives using he same compuer approimaion as we did for sine and cosine. We will do firs and hen eend i o and. y = a, herefore dy d = a+ - a = a a - a a Now, jus wha is ha srange funcion spreadshee. As you can see, he srange lile fracion a = a a -1 1? Using Ecel once again le s play wih i (his is also a live a= a= 100 Δ a Δ a Δ -1 aio Δ a Δ a Δ -1 aio Ln(= Ln(100= a= 10 a= Δ a Δ a Δ -1 aio Δ a Δ a Δ -1 aio Ln(10= Ln(143.96= approaches as 0. We sugges you open he spreadshee and play wih he values of o see wha happens. hus he derivaive of is simply for any posiive. 19 of 7

20 As you migh guess, he above ables are NO proofs, hey are demonsraions. I leave he hard job of proving all his o he Calculus eachers For Base 10 we have y = 10 hus dy d = 10 ln(10 = *10 Also, for y = e dy d = e ln(e = 1*e = e (ecall ha ln(e = 1 his lile derivaive fully eplains why he same as he original funcion! is so common in upper mah and science courses. he derivaive is Las, bu no leas, we will deermine he derivaive of. Given y = ln( y = dy d = ln( + - ln( emember your logarihm manipulaion rules! Given y = ln( y = dy ln( + - ln( = = 1 d = 1 é ù ln 1+ ë ê û ú = 1 Now, wha does one do wih he epression é + ù ln ë ê û ú = 1 é ù ln 1+ ë ê û ú é ù ln 1+ ë ê û ú = 1 é ù ln 1+ ë ê û ú 1? Well, once again we go o Ecel and jus le i play wih he numbers and see wha happens. BW: Mess 1 Noe ha in every case above, he epression which is beer known as e. hus we have 1 regardless of he value of, always becomes of 7

21 = 1 = 100 Δ Δ/ Mess Δ Δ/ Mess = 10 = 0.1 Δ Δ/ Mess Δ Δ/ Mess Given y = ln( y = dy ln( + - ln( = = 1 d = 1 é ù ln 1+ ë ê û ú = 1 é + ù ln ë ê û ú = 1 é ù ln 1+ ë ê û ú é ù ln 1+ ë ê û ú = 1 é ù ln 1+ ë ê û ú = 1 ln(e = 1 hus he derivaive of is simply. Again, his is a very simple resul. his maerial represens all of he derivaive informaion you need for he AP-C Physics course. However, you jus knew here was a however! I is only half of he calculus we will use. 1 of 7

22 Inegral Calculus here is a srong connecion beween finding slopes of curves and finding he area under he curve. Wihou doing he enire fancy mah done in a EAL Calculus class le s jus lis wha we learned las year. 1. he slope of a posiion-ime curve is velociy.. he slope of a velociy-ime curve is acceleraion. 3. he area under a velociy-ime curve is change in posiion. 4. he area under an acceleraion-ime curve is he change in velociy. Suppose we have he equaion which jus says ha he velociy is he slope of posiion graph. We can rewrie he equaion is In he graph, he gray shaded area is approimaely a rapezoid wih an average heigh of beer known as he average velociy. If he inerval is very small he Average Velociy will simply be he average of he end poins. he area beneah ha segmen of he curve will ell us how far we raveled during he inerval. he area is simply which is also from above. hus he OAL change in is simply he sum of he lile changes in each lile area of widh. Bu, he change in can also be compued by simply subracing he iniial posiion from he final posiion. Wha we have is he sum of all he s from o will give us he OAL. Now, suppose we have more han jus daa. We have an acual funcion ha represens he velociy curve. Suppose. We already know ha his funcion is a slope of a posiion funcion (las year. We also know how o find slopes. hus, we need o UNSLOPE. We need o find a funcion ha has as is slope,. of 7

23 Eamples Suppose. Wha funcion when differeniaed gives? I doesn ake oo much hough o realize i is. (Jus differeniae he second equaion. hus we know ha will ge he job done. If, he. However, here are acually infiniely many such equaions as we will see. Suppose, hen, also. In fac he 3 could be any consan and i would sill work. Wha we do is simply add a C (consan o he UNSLOPE o ge he general equaion. So, if hen. Now, going back up, we know ha he oal area under he curve beween and is he oal for he curve bu we also know ha. his allows us o deermine he area under he curve by simply compuing and and hen subracing hem. In our case above, which means ha and. oal (. Wha we have done is simply find he ANIDEIVAIVE of he funcion and plug in he endpoins and subrac. he consan C is called he Consan of Inegraion and depends on knowing he values of he variables in he Inegral a some poin. Bu we are also adding up infiniely many minue area slices under he curve which is called inegraion. We are really finding a sum, which is why he symbol for inegraion is (see he s shape?. If, hen, and. his is he noaion for an indefinie inegral. INEGAL is a fancy word for ani-derivaive which is a fancy word for UNSLOPE which is an ugly word meaning undo he derivaive you did o make he funcion we are now rying o undo (EAD HA SEVEAL IMES FAS. hus, a curve s definie inegral is he difference in area under he curve beween wo poins on he -ais. his is known as he Fundamenal heorem of Calculus. Eamples 1. Find he area under beween and. (i.e. find he inegral of he Inegral (ani-derivaive = unslope is where. Compuing. If velociy is given by, how far do I ravel beween and? (i.e. find he inegral of he inegral is. he area is 3. Wha is he area under a sine curve from o? 3 of 7

24 Find he inegral of. he slope of herefore he inegral mus be. Plug in he endpoins and. hus he area under a half hump from o mus be! 4. Suppose a force ha is applied o a car is a funcion of he car s posiion. How much work mus be done o move he car from o if he force funcion is? (Newons his could easily be accomplished by a srong spring. In fac, we did his las year when we sudied springs. he work done is simply he area under a Force v. Disance curve. Since we need an area, jus find he inegral.. Do you remember he formula for spring energy? [ ] Plug in he endpoins and we ge. 5. Suppose we have he following informaion: and a. Wha equaion represens he objec s acceleraion and wha equaion represens he objec s posiion? he acceleraion is simply he derivaive of he velociy, hus, and we are done. he posiion is he inegral, which is ; however, we know ha when. Using ha informaion, we solve for C, and ge, or. he posiion equaion is. 6. Suppose we cause a car o accelerae consanly a and require ha, and a. Wha will be he car s velociy and posiion a some laer ime? he relaionship beween acceleraion and velociy is he same as he relaionship beween velociy and posiion. hus, o find velociy we simply inegrae he acceleraion. Given ha, he inegral MUS be. Bu, a,. herefore,, or. herefore. his is he same equaion we learned las year. o find posiion, we jus inegrae velociy ( which gives (. (Noe: his is a differen han he one in he velociy equaion above. herefore ( or. he final funcion is (. his also is he same as he equaion from las year. 7. Demonsrae ha he ani-derivaive is he same as he area under he curve. Suppose an objec moves in such a way ha is velociy is given by. How far would i ravel beween and? Using Calculus, we simply find he inegral (ani-derivaive, unslope and plug in he -ais end poins and subrac,. Compuing, we ge, 4 of 7

25 (81 (9 6 C ( (1 (1 (1 C C0.5 4 C d ( (3 (3 (3 C - ( (1 (1 (1 C his is so simple (on he surface! Now, I am going o use a special mahemaical mehod o find he acual areas by adding up a whole bunch of small rapezoids under he curve. Suppose we have he funcion. Wha is he change in posiion beween and? Using wha we have jus seen, we inegrae he funcion, plug in he end poins and subrac. (. he change in from o is simply ( ( or ( ( ( Now, le s acually compue he area by adding up lile rapezoids. he graph above shows he velociy curve,, and I have added some sample rapezoids beween and. Assume here are such rapezoids from o. If he gaps are all he same widh, hen each is wide. he average heigh of each rapezoid is approimaely he heigh or he righ edge of he rapezoid. (We could jus as easily use he lef edge or he average heigh. he righ edge of rapezoid #1 ends a (. hus he heigh is simply ( (. he area of he rapezoid is ( (. 5 of 7

26 he righ edge of rapezoid # ends a (. hus he heigh is ( (. he area is ( (. he righ edge of rapezoid #3 ends a (. hus he heigh is ( (. he area is ( (. And so on he righ edge of he rapezoid ends a ( (dah!. he heigh is ( (. he area is ( (. Now he oal area (approimaed is ( ( ( ( ( ( ( (. Now, using ULE 1 from page, epand his epression. ( ( ( ( ( ( ( ( or ( ( ( ( ( ( ( ( ( or ( (. Since here are recangles, adding up he same number imes gives. In our case he same number is hus (, giving ( Now is here a nify epression,?! Suppose Also, When added,. herefore (! hus or ( ( ( ( (. 6 of 7

27 If we le hen and we have, which is he same as wha we obained previously, using Calculus. his should convince you ha we don really wan o calculae areas his way. However, i is acually raher common when he funcion o be inegraed is oo hard o inegrae symbolically. BW: In his paricular case we could have simply calculaed he area since he OIGINAL area is iself a simple rapezoid.. Wha you AE seeing is ha Calculus is ready-made o make Physics easier (if and only if, one can use funcional equaions insead of discree daa poins as we did las year. his means ha here is going o be a srong reliance on fiing funcions o daa in his class. Of course we also did i las year wih linear regression. We will coninue wih linear regression and may add polynomial and log-log curve fis as he year progresses. Simply saed, if a funcion represens a derivaive, hen boh he area under ha derivaive and he funcion ha produced he derivaive (he Inegral are obained by essenially he same process. Wha you are NO seeing is he HAD fac ha he ac of Inegraion is far more difficul han he ac of differeniaion. Differeniaion is augh in a week. Inegraion akes muliple courses and you have only begun. 7 of 7