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Feb. 8 Lecture2.5 and 4.4.The exponentialfunction exand its derivative.Picture from WikipediaCompound interestConsider a savings account that pays 100% interest per year.If the payment is made only once then $1 becomes $2 at the end of the year.If the payment is made twice1 →1.5 →(1.5)(1.5) = 2.25once = 2, twice = 2.25If the payment is made 4 times1 →1.25 →(1.25)2 →(1.25)3 →(1.25)4= 2.44Monthly payments(1 + 1/12 )12= 2.613Daily payments(1 + 1/365 )365= 2.7145Daily payments(1 + 1/365 )365= 2.7145Hourly payments(1 + 1/8,760 )8760= 2.718127Every minute(1 + 1/525,600 )525,600= 2.71827924Every second(1 + 1/31,536,000)31,536,000= 2.71828178Limit e = 2.7 1828 1828 45 90 45 …The derivative of ex()0Rule 8Algebra 4. 11li.'m1xxxyxyxhxhxhhaaaeeeehheeeh++→⋅=−−=⋅=−=A differential equationThe chain rule says ()''()'If ()then '()Rule 8B. ( )solves '( )( )()''ggggrtf gfggf gefgeeh tehe gtr h t=⋅===⋅==
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