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7.6 T HE N ATURAL B ASE, E

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C OMPOUND I NTEREST The compound interest formula is: Where A is the total amount, P is the principal (original amount), r is the annual interest rate, and n is the number of times the interest is compounded per year, and t is the time in years.

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$1 INVESTED AT 100% INTEREST COMPOUNDED N TIMES FOR ONE YEAR : As n gets very large, interest is continuously compounded (meaning you’re earning interest on the interest you’ve already earned, etc.) Play around with this function on your calculator for large values of n. (Graph it, substitute in values, try n = 1000, n = 10000, n = 100000.

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A S N GETS REALLY LARGE, WHAT HAPPENS ? It approaches a specific number. Look at the graph. There is a horizontal asymptote. Where is it located? This number is called e. e = 2.718281828459045…… e is an irrational number, just like pi.

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E XPONENTIAL FUNCTIONS WITH BASE E. Exponential functions with base e have the same properties as all other exponential functions we’ve studied. The function looks like f(x) = e x. Graph this on your calculator. You can find e next to the 4 button. Push 2 nd ln. The domain of f(x) = e x is all real numbers, but the range is y > 0.

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G RAPHING WITHOUT USING THE GRAPH KEY ON YOUR CALCULATOR. Graph f(x) = e x – 3 by MAKING A TABLE OF VALUES!!!!

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T HE NATURAL L OGARITHM A logarithm whose base is e (so log e ) is called the natural logarithm. It is “ln”. Again, natural logarithms have the same properties as all other logs, they just have this special name. So, that means that the inverse of f(x) = e x is?

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S IMPLIFY ( USING THE PROPERTIES OF LOGS YOU ALREADY KNOW ) ln e 0.15t e 3 ln (x + 1) ln e 2x + ln e x

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C ONTINUOUS C OMPOUNDED I NTEREST The formula for continuously compounded interest is A = Pe rt A is the total amount P is the principal (original amount) r is annual interest rate (make sure it’s changed to a decimal) t is time in years

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T RY THIS : What is the total amount for an investment of $500 invested at 5.25% for 40 years, compounded continuously?

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H ALF - LIFE Scientists are able to determine the age of a really old fossil or other substance by measuring a half-life. The half-life of a substance is the time it takes for half the substance to breakdown or convert to another substance during the process of decay. Natural decay equation: N(t) = N 0 e -kt

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P LUTONIUM Plutonium-239 (Plu-239) has a half-life of 24110 years. How long does it take for a 1 g sample of Plu-239 to decay to 0.1 g? What do we know? What do we need to find?

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A NOTHER ONE : Determine how long it will take for 650 mg of a sample of chromium-51, which has a half-life of about 28 days, to decay to 200 mg.

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S IMPLIFY : ln e 1 ln e x-y ln e (-x/3) e ln 2x e 3lnx

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Emma receives $7750 and invests it in an account that earns 4% interest compounded continuously. What is the total amount of her investment after 5 years?

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G RAPH : f(x) = -e 1-x

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