8 3 the number e
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2.7182818284590452353602874713526624977572470936999595749669676277240766303535 47594571382178525166427427466391932003059921817413596629043572900334295260595630 73813232862794349076323382988075319525101901157383418793070215408914993488416750

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8.3 The Number e

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8 3 the number e

2.7182818284590452353602874713526624977572470936999595749669676277240766303535

47594571382178525166427427466391932003059921817413596629043572900334295260595630

73813232862794349076323382988075319525101901157383418793070215408914993488416750

92447614606680822648001684774118537423454424371075390777449920695517027618386062

61331384583000752044933826560297606737113200709328709127443747047230696977209310

14169283681902551510865746377211125238978442505695369677078544996996794686445490

59879316368892300987931277361782154249992295763514822082698951936680331825288693

98496465105820939239829488793320362509443117301238197068416140397019837679320683

28237646480429531180232878250981945581530175671736133206981125099618188159304169

03515988885193458072738667385894228792284998920868058257492796104841984443634632

44968487560233624827041978623209002160990235304369941849146314093431738143640546

25315209618369088870701676839642437814059271456354906130310720851038375051011574

77041718986106873969655212671546889570350354021234078498193343210681701210056278

80235193033224745015853904730419957777093503660416997329725088687696640355570716

22684471625607988265178713419512466520103059212366771943252786753985589448969709

64097545918569563802363701621120477427228364896134225164450781824423529486363721

41740238893441247963574370263755294448337998016125492278509257782562092622648326

27793338656648162772516401910590049164499828931505660472580277863186415519565324

42586982946959308019152987211725563475463964479101459040905862984967912874068705

04895858671747985466775757320568128845920541334053922000113786300945560688166740

01698420558040336379537645203040243225661352783695117788386387443966253224985065

49958862342818997077332761717839280349465014345588970719425863987727547109629537

41521115136835062752602326484728703920764310059584116612054529703023647254929666

93811513732275364509888903136020572481765851180630364428123149655070475102544650

11727211555194866850800368532281831521960037356252794495158284188294787610852639

81395599006737648292244375287184624578036192981971399147564488262603903381441823

26251509748279877799643730899703888677822713836057729788241256119071766394650706

33045279546618550966661856647097113444740160704626215680717481877844371436988218

55967095910259686200235371858874856965220005031173439207321139080329363447972735

Standards 11, 14

8.3 The Number e

  • Goal 1: Use the number e as the base of exponential functions

Goal 2: Use the number e in real-life situations


Important numbers in history

Important numbers in History

  • 0: zero, null, nill. Invented about 825 by Al-Kwarizmi

  • π ≈ 3.141592654... The ratio of the circumference of a circle to the diameter.

  • ɸ ≈ 1.618...

  • 9,201,976: Swenson’s birthday


Euler s number

Euler’s Number

  • Natural base e

  • Discovered by Leonhard Euler (1707 - 1783)

  • e ≈ 2.718281828459...

Compound Interest

Compound Interest

Goal 1: Use the number e as the base of exponential functions


Where does e come from

Where does e come from?

asymptote is where?

Graph:

  • Goal 1: Use the number e as the base of exponential functions

2.718...


Natural base e

Natural Base e

  • Goal 1: Use the number e as the base of exponential functions


Review white boards

Review: White boards

Simplify the expression.

  • Goal 1: Use the number e as the base of exponential functions


Use e white boards

Use e: White boards

Simplify the expression.

  • Goal 1: Use the number e as the base of exponential functions


White boards

White boards

Use a calculator to evaluate.

  • Goal 1: Use the number e as the base of exponential functions

1

2


Describe graph of function

Describe graph of function

  • Goal 1: Use the number e as the base of exponential functions

Growth or decay?

y-intercept?

asymptote?


Graphing e functions

Graphing e Functions

  • Goal 1: Use the number e as the base of exponential functions


Using e

Using e

We learned that amount after compound interest can be calculated by:

as n →∞

But we know that

So...

for compounding continuously

Goal 2: Use the number e in real-life situations


Comparing compounding

Comparing Compounding

You deposit $1000 in an account that pays 8% annual interest. Find the balance after one year if the interest is compounded with the given frequency.

Daily

Continuously

Goal 2: Use the number e in real-life situations


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