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This is one of those magical numbers like
.
We see it in continuous
compounding - exponential growth - entities that grow like
.
Sometimes called the ``base of natural logarithms'',
.
Look at a
graph.
of e to the power t,
et = exp(t).
Notes
- Defined for all values of t, even negative ones.
- Crosses that y-axis at y = 1, why?
- This relationship really grows fast, hence the term
exponential growth.
- As t heads to negative infinity, et goes to 0.
- As t heads to positive infinity, et goes to infinity.
Why don't we call this a power function?
Because the object we are changing, t, is in the exponent,
not the base.
The more general form of the function:
Here
- C is a multiplicative constant (how much money at the start).
- e is the magical number.
- r is a fixed constant (the interest rate).
- t is what we are changing (how long the investment is held for).
Example.
- C = 100.
- r = 0.05 is a fixed constant (the interest rate).
- t is time in years.
Rules for exponential functions (same as power functions):
-
.
-
Exponential decay.
A useful modeling function: the logistic:
This is in fact the formula for the
graph.
Next: 5.
Up: Business
Previous: 3.3
Richard Waterman
1999-05-06