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8.3 The Number e

2.7182818284590452353602874713526624977572470936999595749669676277240766303535 47594571382178525166427427466391932003059921817413596629043572900334295260595630 73813232862794349076323382988075319525101901157383418793070215408914993488416750

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8.3 The Number e

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  1. 2.7182818284590452353602874713526624977572470936999595749669676277240766303535 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

  2. 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

  3. 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

  4. Where does e come from? asymptote is where? Graph: • Goal 1: Use the number e as the base of exponential functions 2.718...

  5. Natural Base e • Goal 1: Use the number e as the base of exponential functions

  6. Review: White boards Simplify the expression. • Goal 1: Use the number e as the base of exponential functions

  7. Use e: White boards Simplify the expression. • Goal 1: Use the number e as the base of exponential functions

  8. White boards Use a calculator to evaluate. • Goal 1: Use the number e as the base of exponential functions 1 2

  9. Describe graph of function • Goal 1: Use the number e as the base of exponential functions Growth or decay? y-intercept? asymptote?

  10. Graphing e Functions • Goal 1: Use the number e as the base of exponential functions

  11. 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

  12. 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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