1.6 Class Notes - Feb 8 Lecture Compound interest Consider...

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Feb. 8 Lecture 2.5 and 4.4. The exponential function e x and its derivative. Picture from Wikipedia Compound interest Consider 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 twice 1 1.5 (1.5)(1.5) = 2.25 once = 2, twice = 2.25 If the payment is made 4 times 1 1.25 (1.25) 2 (1.25) 3 (1.25) 4 = 2.44 Monthly payments (1 + 1/12 ) 12 = 2.613 Daily payments (1 + 1/365 ) 365 = 2.7145 Daily payments (1 + 1/365 ) 365 = 2.7145 Hourly payments (1 + 1/8,760 ) 8760 = 2.718127 Every minute (1 + 1/525,600 ) 525,600 = 2.71827924 Every second (1 + 1/31,536,000) 31,536,000 = 2.71828178 Limit e = 2.7 1828 1828 45 90 45 … The derivative of e x ( ) 0 Rule 8 Algebra 4. 1 1 li . ' m 1 x x x y x y x h x h x h h a a a e e e e h h e e e h + + = = = = A differential equation The chain rule says ( )' '( ) ' If ( ) then '( ) Rule 8B. ( ) solves '( ) ( ) ( )' ' g g g g rt f g f g g f g e f g e e h t e h e g t r h t = = = = = =
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