Transcript 7.6 THE NATURAL BASE, E COMPOUND INTEREST  The compound interest formula is: nt  r  A  P1    n  Where A is the.

7.6 THE NATURAL BASE, E
COMPOUND INTEREST

The compound interest formula is:
nt
 r 
A  P1  
 n

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.
$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.

AS 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.
EXPONENTIAL 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) = ex.



Graph this on your calculator. You can find e
next to the 4 button. Push 2nd ln.
The domain of f(x) = ex is all real numbers, but
the range is y > 0.
GRAPHING WITHOUT USING THE GRAPH
KEY ON YOUR CALCULATOR.

Graph f(x) = ex – 3 by MAKING A TABLE OF
VALUES!!!!
THE NATURAL LOGARITHM
A logarithm whose base is e (so loge ) 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) = ex is?
SIMPLIFY (USING THE PROPERTIES OF
LOGS YOU ALREADY KNOW)

ln e0.15t

e3 ln (x + 1)

ln e2x + ln ex
CONTINUOUS COMPOUNDED INTEREST

The formula for continuously compounded
interest is A = Pert
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

TRY THIS:

What is the total amount for an investment of
$500 invested at 5.25% for 40 years, compounded
continuously?
HALF-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) = N0e-kt
PLUTONIUM

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?

ANOTHER 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.
SIMPLIFY:

ln e1

ln ex-y

ln e(-x/3)

eln 2x

e3lnx

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?
GRAPH:

f(x) = -e1-x