课程目录
1.Introduction
1.1Howisthiscoursestructured
1.2Whyiscalculusgoingtobesomuchfun
2.FunctionsandLimits
2.1.1Howdowegetstartedwithcalculus
2.5.2Whatisthelimitofaquotient_
2.5.1Whatisthelimitofaproduct_
2.4.4Whatisthelimitof(x^2-1)_(x-1)_
2.4.3Whatisthelimitof(sinx)_x_
2.4.2Whatisthelimitofsin(1_x)_
2.4.1Morally,whatisthelimitofasum_
2.3.2Whatisthedomainofsquareroot_
2.3.1Whataresomereal-worldexamplesoffunctions_
2.2.3Howcanmorefunctionsbemade
2.2.2Whenaretwofunctionsthesame
2.2.1Whatisafunction
2.6.1Howfastdoesaballmove_
3.TheEndofLimits
3.1.1Whatelseistheretostudyaboutfunctionsandlimits_
3.5.2Whyisthelimitofx^2asxapproaches2equalto4_
3.5.1Whatistheofficialdefinitionoflimit_
3.4.3Howfastdoeswaterdripfromafaucet_
3.4.2Whatistheslopeofastaircase_
3.4.1Whatisthedifferencebetweenpotentialandactualinfinity_
3.3.4Whyisinfinitynotanumber_
3.3.3Whatisthelimitf(x)asxapproachesinfinity_
3.3.2Whatdoeslimf(x)=infinitymean_
3.3.1Whyisthereanxsothatf(x)=x_
3.2.4HowcanIapproximateroottwo_
3.2.3Whatistheintermediatevaluetheorem_
3.2.2Whatdoes_continuous_mean_
3.2.1Whatisaone-sidedlimit_
3.5.3Whyisthelimitof2xasxapproaches10equalto20_
4.TheBeginningofDerivatives
4.1Whatcomesnext_Derivatives_
4.5.3Whatisthederivativeofx^3+x^2_
4.5.2Whatisthederivativeofx^n_
4.5.1Whyisthederivativeofx^2equalto2x_
4.4.2Whatisthederivativeofaconstantmultipleoff(x)_
4.4.1Whyisadifferentiablefunctionnecessarilycontinuous_
4.3.2Whatinformationisrecordedinthesignofthederivative_
4.3.1Whyissqrt(9999)socloseto99.995_
4.2.4Howdoeswigglingxaffectf(x)_
4.2.3Whyistheabsolutevaluefunctionnotdifferentiable_
4.2.2Whatisatangentline_
4.2.1Whatisthedefinitionofderivative_
4.5.4Whyisthederivativeofasumthesumofderivatives_
5.TechniquesofDifferentiation
5.1.1HowdoIdifferentiateaproduct
5.2.2Morally,whyistheproductruletrue_
5.2.3Howdoesonejustifytheproductrule_
6.ChainRule
6.1.1Isthereanythingmoretolearnaboutderivatives_
6.6.3Howdoweprovethequotientrule_
6.6.2Howcanlogarithmshelptoprovetheproductrule_
6.6.1Howdowejustifythepowerrule_
6.5.1HowcanImultiplyquickly
6.4.3Whatislogarithmicdifferentiation_
6.4.2Whatisthederivativeoflog_
6.4.1Howdoesderivativeoftheinversefunctionrelatetothederivativeoftheoriginalfunction
6.3.2WhatisthefoliumofDescartes
6.3.1Whatisimplicitdifferentiation_
6.2.2Whatisthederivativeof(1+2x)^5andsqrt(x^2+0.0001)_
6.2.1Whatisthechainrule_
6.6.4BONUS_Howdoesoneprovethechainrule_
7.DerivativesofTranscendental(Trigonometric)Functions
7.1.1Whatarederivativesoftranscendentalfunctions_
7.5.3HowcanIapproximatesin1_
7.5.2Howcanwegetaformulaforsin(a+b)_
7.5.1Whydosineandcosineoscillate_
7.4.2Whatarethederivativesofinversetrigfunctions_
7.4.1Whatareinversetrigonometricfunctions_
7.3.4Whatarethederivativesoftheothertrigonometricfunctions_
7.3.3Whatisthederivativeofsin(x^2)_
7.3.2Whatisthederivativeoftanx_
7.3.1Whatisthederivativeofsineandcosine_
7.2.3Whyaretheretheseothertrigonometricfunctions_
7.2.2Whydoestrigonometrywork_
7.2.1Whataretranscendentalfunctions
7.6.1Howcanwemultiplynumberswithtrigonometry_
8.DerivativesintheRealWorld
8.1.1WhywouldIeverwanttotakederivatives_
8.4.4Howquicklydoesthewaterlevelriseinacone_
8.4.3Howquicklydoesabowlfillwithgreenwater_
8.4.2Howfastdoestheladderslidedownthebuilding_
8.4.1Howfastdoestheshadowmove_
8.3.2Whatdoesacarsoundlikeasitdrivespast_
8.3.1HowlonguntilthegraygoodestroysEarth_
8.2.3Whyshouldn'tIfallinlovewithl'H?pital_
8.2.2Howcanl'H?pitalhelpwithlimitsnotoftheform0_0_
8.2.1Howcanderivativeshelpustocomputelimits_
8.4.5Howquicklydoesaballoonfillwithair_
9.Optimization
9.1.1Whyisoptimizationpartofthiscourse_
9.5.2Howlargeofanobjectcanyoucarryaroundacorner_
9.5.1Wheredothreebubblesmeet_
9.4.2Howdoyoudesignthebestsoupcan_
9.4.1Howlargecanxybeifx+y=24_
9.3.4Howcanyoubuildthebestfenceforyoursheep_
9.3.3Whybotherconsideringpointswherethefunctionisnotdifferentiable_
9.3.2Wh
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