Scalars-3.1 OVERVIEW Derivatives 2018.11.24 Atheresmagic.com/PDF/Scalars-3.1.pdf · Scalars-3.1...
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Scalars-3.1 OVERVIEW Derivatives 2018.11.24 B .pages of 12 1 “… as any math student knows, the old problems you meet freshman year are some of the deepest you ever see” Ellenberg, Jordan (2014-05-29). How Not to Be Wrong: The Power of Mathematical Thinking (p. 435). Penguin Group US. Kindle Edition. On who gets to do mathematics “One of the most painful parts of teaching mathematics is seeing students damaged by the cult of the genius. The genius cult tells students it’s not worth doing mathematics unless you’re the best … We don’t treat any other subject that way! … The cult of the genius also tends to undervalue hard work. When I was starting out, I thought “hardworking” was a kind of veiled insult. … But the ability to work hard— to keep one’s whole attention and energy focused on a problem … is not a skill everybody has … and it’s impossible to do math without it. “ “I think we need more math majors who don’t become mathematicians. More math major doctors, more math major high school teachers, more math major CEOs, more math major senators. But we won’t get there until we dump the stereotype that math is only worthwhile for kid geniuses”. “ … I was pretty sure, when I went off to college, that the competitors I knew from Math Olympiad were the great mathematicians of my generation. It didn’t exactly turn out that way … most of the mathematicians I work with now weren’t ace mathletes [sic] at thirteen; they developed their abilities and talents on a different timescale. … Mathematics, mostly, is a communal enterprise, each advance the product of a huge network of minds working toward a common purpose”. Ellenberg, Jordan (2014-05-29). How Not to Be Wrong: The Power of Mathematical Thinking (pp. 412-415). Penguin Publishing Group. Kindle Edition. Page of 1 13 TomK Madison, WI
Transcript of Scalars-3.1 OVERVIEW Derivatives 2018.11.24 Atheresmagic.com/PDF/Scalars-3.1.pdf · Scalars-3.1...
Scalars-3.1OVERVIEWDerivatives2018.11.24B.pages ! of 12 1
“…asanymathstudentknows,theoldproblemsyoumeetfreshmanyeararesomeofthedeepestyoueversee”Ellenberg,Jordan(2014-05-29).HowNottoBeWrong:ThePowerofMathematicalThinking(p.435).PenguinGroupUS.KindleEdition.
Onwhogetstodomathematics
“Oneofthemostpainfulpartsofteachingmathematicsisseeingstudentsdamagedbythecultofthegenius.Thegeniusculttellsstudentsit’snotworthdoingmathematicsunlessyou’rethebest…Wedon’ttreatanyothersubjectthatway!…Thecultofthegeniusalsotendstoundervaluehardwork.WhenIwasstartingout,Ithought“hardworking”wasakindofveiledinsult.…Buttheabilitytoworkhard—tokeepone’swholeattentionandenergyfocusedonaproblem…isnotaskilleverybodyhas…andit’simpossibletodomathwithoutit.“
“Ithinkweneedmoremathmajorswhodon’tbecomemathematicians.Moremathmajordoctors,moremathmajorhighschoolteachers,moremathmajorCEOs,moremathmajorsenators.Butwewon’tgetthereuntilwedumpthestereotypethatmathisonlyworthwhileforkidgeniuses”. “…Iwasprettysure,whenIwentofftocollege,thatthecompetitorsIknewfromMathOlympiadwerethegreatmathematiciansofmygeneration.Itdidn’texactlyturnoutthatway…mostofthemathematiciansIworkwithnowweren’tacemathletes[sic]atthirteen;theydevelopedtheirabilitiesandtalentsonadifferenttimescale.…Mathematics,mostly,isacommunalenterprise,eachadvancetheproductofahugenetworkofmindsworkingtowardacommonpurpose”.
Ellenberg,Jordan(2014-05-29).HowNottoBeWrong:ThePowerofMathematicalThinking(pp.412-415).PenguinPublishingGroup.KindleEdition.
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Scalars-3.1OVERVIEWDerivatives2018.11.24B.pages ! of 12 2Derivatives.
INFORMALDEFINITION:Thederivativeruleof� isanexpressiondenotedby� whichevaluatesto
theinstantaneousrateofchange
ofthevalueof ateachvalue
of� .Surprise!It’salimit!Thusthevalueof� equalstheslopeofthegraphof� ateachvalueofthevariable� .� ispronounced“� primeof� ”.Theprocessofdeterminingaderivativeruleiscalleddifferentiation.
EXAMPLES:Herearesomefunctionrulesandtheirderivativefunctionrules.
We’llshowhowwegotthoseexpressionsfor� later.
https://www.desmos.com/calculator/mixjf8gqvc
As� changesthevalueof� doesnotchange.� .Theslopeisconstant.
�
f (x) ′f (x)change in f (x)change in x
f (x)
x ′f (x)f (x) x ′f (x) f x
f (x) = 5 ʹf (x) = 0f (x) = x ʹf (x) = 1f (x) = 2x ʹf (x) = 2 ⋅1 = 2f (x) = x2 ʹf (x) = 2xf (x) = 4x3 ʹf (x) = 4 ⋅3x2
f (x) = x3 + x2 + x +10 ʹf (x) = 3x2 + 2x1 + x0 + 0= 3x2 + 2x + 1 + 0
′f (x)
f (x) = 5x f (x) ′f (x) = 0
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https://www.desmos.com/calculator/wyswbyq3p7As� changes,thevalueof� changesbytwiceasmuchso� .Thevalueof� doesnotdependon� .Theslopeisconstant.
�
https://www.desmos.com/calculator/0kszvcvv3bForagivenvalueof� thevalueof� Sothevalueof� dependsonthevalueof� .Theslopedependsupon� .
�
f (x) = 2x
x f (x) ′f (x) = 2′f (x) x
f (x) = x2 .
x ′f (x) = 2x ′f (x)x x
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Inadditiontopolynomialsandrationalfunctions� herearethemost
commontypesoffunctionsandtheirgraphs.Weintroducethemveryearlybecausetheyhelpusgiveniceexamplesofderivatives.Atthispointjustknowingthetheirnotation,theirgraphsandtheirderivativefunctionrulesisallweneed.WediscussthemindetailinScalars5.1,Scalars5.2andScalar5.3andshowhowtocomputevaluesforthemusingaddition,subtraction,multiplications,divisionandlimitsinScalars9.0andScalar9.1(TaylorSeries,atypeofin_initesum).
Movethereddotaroundandseehow� alwaysequalsthevalueoftheslopeof� .
�
https://www.desmos.com/calculator/4mhrg98kwv
�
https://www.desmos.com/calculator/q378arwffu
polynomialpolynomial
′f (x)f (x)
f (x) = sin x; ′f (x) = cos x; Taylor Series : sin x =−1( )k2k +1( )!k=0
∞
∑ x2k+1
f (x) = cos x ʹf (x) = − sin x
f (x) = cos x; ′f (x) = − sin x; Taylor Series cos x =−1( )k2k!
x2k=0
∞
∑
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�
TaylorSeries:� https://math.stackexchange.com/questions/770197/taylor-series-of-tan-xhttps://www.desmos.com/calculator/yrhkvsbhlw
�
https://www.desmos.com/calculator/9c9nk7dfo6
�
https://www.desmos.com/calculator/pum10pn60t
f (x) = tan x; ′f (x) =1
cos2 x= sec2 x;
tan x
f (x) = ln x; ′f (x) =1x; Taylor Series ln x =
−1( )k−1kk= 1
∞
∑ x −1( )k 0 < x ≤ 2
f (x) = ex ; ′f (x) = ex ; Taylor Series ex = 1k!k= 0
∞
∑ xk
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Wherethederivativerulescomefrom.
Part1:Howtoapproximatetherateofchange(slope)of� .Pleasemovethereddotaroundandseewhathappens.https://www.desmos.com/calculator/dlcvcaeqmp
� Firstpickpoints� .Theapproximaterateofchangeof
� .
As� approaches0theapproximationgetsbetter.Movethereddotaroundtowatchthishappen.
Part2:Theexactrateofchangeiscomputedwithalimitbecauseyoucan’tdivideby� when� .Wemakethefollowingde_inition(oneofthemostimportantonesinCalculus)fortheexactrateofchangeof� atanyvalue� .
DEFINITIONSofDerivativeandDifferentiable.Now� referstoanypoint.
wheneverthislimitexists.
.
f (x) = x2 at x = x1
x1 and x2 = x1 + h
f (x) at x1 isf (x1 + h)− f (x1)
h= change in f (x)
change in xh
h h = 0f (x) x
x
ʹf (x) =def
limh→0
f (x+ h) − f (x)h
ʹf (x) is called the derivative rule (function) of f (x)
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IMPORTANTByde_initionyoumustalsogetthesameanswer(comingfromtherightortheleft).IfthisdoesnothappenthenbydeIinitionthederivative� doesnotexist..If� existsthenwesaythat� isdifferentiableat� ..Ifthederivativeexistsforallvaluesintheopeninterval� thenwesaythat� isdifferentiableinthatopeninterval..If� doesnotexistthenwesaythat� isnotdifferentiableat� .InAppendixLweshowhowtousethisdeIinitiontocomputethederivativesforvariousfunctions.
AnalternatewaytodeIinethederivative.Itissometimesmoreconvenienttousethefollowing(equivalent)formula.
DEFINITION:AlternatebutEquivalentDeIinitionofDerivativeandDifferentiable
�
Examplesofusingthetwode_initionstocomputeaderivativerule:
�
′f (x)′f (x) f x
a < x < bf
′f (x) f x
′f (x) = limt→x
f (t)− f (x)t − x
Let f (x) = x2
′f (x) =def
limh→ 0
f (x + h)− f (x)h
= limh→ 0
x + h( )2 − x2h
= limh→ 0
x2 + 2x h + h 2 − x2
h= limh→ 0
2x + h( ) = 2x + 0 = 2x.
′f (x) = limt→ x
f (t)− f (x)t − x
= limx→ c
t2 − x2
t − x= limt→ x
t − x( ) t + x( )t − x( )
= limt→ x
t + x( ) = 2x
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Examplesofwhenthederivativedoesn’texist.
Frequently,whenagraphlooksunusual,� isnotde_inedattheunusualplaces.Forexample� isnotde_inedatthexvaluesfortheverticalorangelinesbelow.It
isunde_inedfor�
�
� isnotde_ined(� isnotdifferentiable)at0belowbecauseyougetdifferentanswerswhenyouapproach0fromtheleftandtheright.However,� iscontinuousat0.RecheckthedeIinitionsandmakesurethatyouunderstandwhythisistrue.
https://www.desmos.com/calculator/_j5s3aheov
�
Atthispointifyouaretakingaformalcalculuscourseorjustwanttodomoreadvanceddifferentiation,readaboutthechainrulein3.1ADV-0.
′f (x)′f (x)
x =2n+1( )π2
, n = 0, ±1,±2, ...
For f (x) = x =def x : x ≥ 0
−x : x < 0⎧⎨⎩
⎫⎬⎭
′f (x) ff
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Aninterestingquestion:
�
�
Youmustforgetaboutanyintuitivenotionsofdifferentiabilityandappealdirectlytothede_initionofderivative.Thequestionthusbecomesdoes
� .
Adddetails.
Somemorenotation.
�
You’llseealotoffactorialnotation� intheveryimportantTaylorSeriesinScalars9.1andit’sthepatternabovethatgeneratesthosefactorials.
Is f (x) =x2 sin 1
x⎛⎝⎜
⎞⎠⎟: x ≠ 0
0 : x = 0
⎧
⎨⎪
⎩⎪
⎫
⎬⎪
⎭⎪differentiable at x = 0?
′f (0) = limh→ 0
f (x + h)− f (x)0
exists
Examples
f (x) x4
′f (x) is called the1st derivative of f (x) 4x3
′′f (x) =def
′f (x)( )′ is called the 2nd derivative of f (x) 4 ⋅3x2
′′′f (x) =def
′′f (x)( )′ is called the 3rd derivative of f (x) 4 ⋅3⋅2x1 = 4!x1
etc.
!( )
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Tableofthebasicrulesofdifferentiation
u and v are function namesDerivative rulesNotation Results Name
1. (c)ʹ = 0 Constant Rule
2. (xn ʹ) = nxn−1 Power Rule n ≠ 0 n is any real number.
3. ( c ⋅u(x) ʹ) = c⋅ ʹu (x) Constant Multiple Rule4. ( u(x) + v(x) ʹ) = ʹu (x) + ʹv (x) Sum Rule5. ( u(x) ⋅ v(x) ʹ) = u ʹ(x) ⋅ v(x) + u(x) ⋅ v ʹ(x) Product Rule
6.u(x)v(x)
⎛⎝⎜
⎞⎠⎟ʹ
=u ʹ(x) ⋅ v(x) − u(x) ⋅v ʹ(x)
v(x)2 Quotient Rule
The trig functions.7. (sin x)ʹ = cos x8. (cosx)ʹ = − sin
9. (tan x)ʹ =1
cos2 x= sec2 x
10. (sin−1 x) ʹ =1
1− x2−1< x <1, 1≤ y
11. (cos−1 x)ʹ = −1
1− x21 −1< x <1, y ≥1
12. (tan−1 x)ʹ =1
x2 +1The logarithmic and exponential functions:
13. (ln x)ʹ =1x
0 < x
14. (ex )ʹ = ex
15. (ax)ʹ = ax lna
16. (loga x)ʹ =1
xlna
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Anothernotationforderivatives:Thereisanothernotationthatisusedtorepresentaderivative.Supposethatwehaveafunctionrule� andit’sderivativerule� .
Ifweintroduceanewvariable� andspecifythat� (whichmeansthatvariable� takesonvaluesgeneratedbytherule� ;itdoesnotmeanthat� because� isavariableand� isnotationrepresentingafunction
rule).Then� isanothersymbolforthederivativerule� .Butitis
interestingthatnooneeverwrites� .Whyisthat?
Somepeoplealsousethenotation:�
The� ”differential”notationisusefulasfollows:
1.Whenperformingamechanicalprocess calledseparationofvariableswhich1
isusedforsolvingcertainkindsofdifferentialequations.
2.Whendeterminingreversederivatives(antiderivatives)akaevaluatinganinde_initeintegral.Inthatcase� issometimesmanipulatedasanindependentalgebraicobjectinamechanicalprocesscalled� substitution
Inthesetwocaseswemanipulatethenotationinwaysthatdonotdirectlyfollowfromthede_initions.JustbeawarethattherearetheoremsinAdvancedCalculusthatsaythatyoucandothosemanipulations.
Asidefromthosetwosituationswedon’tusethe“� differentialnotation”inthese
notes.
f (x) ′f (x)
y y = f (x)y f (x)
y equals f (x) y f (x)dydx
⎛⎝⎜
⎞⎠⎟
′f (x)
dydx
⎛⎝⎜
⎞⎠⎟= limh→ 0
f (x + h) − f (x)h
d 2 ydx2
=def
′′f (x)
dydx
⎛⎝⎜
⎞⎠⎟or just
dydx
dxu −
dydx
ApostolVol.1p.3461
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Asde_inedabovethesymbols arenotde_inedwhenusedalone.However,thereisacontextinwhichthesymbol� doeshaveawellde_inedmeaningalthoughwedon’tuseitunlesswehaveto.DEFINITIONS:Differentials :
2
Givenafunctionnamed� anditsderivative(function)named� wedeIinetwovariables asfollows.https://www.desmos.com/calculator/r5twyoxv6n(Clickanddragtherightpoint)
�
�
Weprefer totheothernotationsunlessweareusingthe
mechanicalproceduresdescribedabove.Youcanvalidlyadd,subtract,multiplyand
dividethedifferentials justlikeanyotheralgebraicobjects.
dx and dydx
f ′fdf and dx
f and ʹf are the names of functions See 2.1 Functions and Their Notationf (x) and ʹf (x) are the corresponding rules
′f (x) and ddxf (x)
df and dx
BasedonThomas’Calculus,11th.Ed.,2008,PearsonEducation,Addison-Wesley.pp.225-2262
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(Wedon’tusetheword,inIinitesimal,inthesenotes).Youmayseeitinothertexts.
“ForaperiodoftimeencompassingNewton’sworkinglife,thedisciplineofanalysiswasasubjectofcontroversyinthemathematicalcommunity.Althoughanalytictechniquesprovidedsolutionstothelong-standingproblems,…theproofsofthesesolutionswerenotknowntobereducibletothesyntheticrulesofEuclideangeometry.Instead,analystswereoftenforcedtoinvokein_initesimal,or“in_initelysmall”quantitiestojustifytheiralgebraicmanipulations.SomeofNewton’smathematicalcontemporaries,suchasIsaacBarrow,werehighlyskepticalofsuchtechniques,whichhadnocleargeometricinterpretation.AlthoughinhisearlyworkNewtonalsousedin_initesimalsinhisderivationswithoutjustifyingthem,helaterdevelopedsomethingakintothemodernde_initionoflimitsinordertojustifyhiswork“.
NewtonwasastudentofIsaacBarrow.
https://en.wikipedia.org/wiki/Isaac_Barrow
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