(This topic is also in Section 2.2 in Applied Calculus and Section 10.2 in Finite Mathematics and Applied Calculus)
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Brush up on your Algebra For all of the tutorials on nonlinear functions and models, you should be familiar with the algebra of exponents and radicals.
Exponential functions are frequently used to model the growth or depreciation of financial investments, population growth, radioactive decay, and phenomena where a quantity is allowed to undergo unrestrained growth.Exponential Function
An exponential function of the variable x is a function that can be written in the form
Examples |
In the following practice question, you are required to fill in some values of an exponential function and recognize its graph from the data. (See p. 95 of Applied Calculus or p. 573 of Finite Mathematics and Applied Calculus for a discussion of this example.)
Let f(x) = 3(2^{x}). Fill in the missing values in the following table, and then click on the correct graph.
In the following table, decide which of the rows represent exponential functions of x, which represent linear functions, and which represent neither.
We want to find an exponential equation of the form y = Ab^{x} for the curve passing through (1, 6) and (3, 8.64).
Suppose we invest $1 in the bank for 1 year at 100% interest, compounded m times per year. If m = 1 then 100% interest is added every year, then the compound interest formula tells us that, by the end of the year, you will have
| m |
dollars. |
What you might notice is, that as m gets larger and larger, the value of our investment seems to approach a fixed number:
Q What is the point of this number e?
A This number is extremely important to mathematicians and physicists. In calculus, e is the best behaved base for exponential functions. Besides its theoretical usefulness, e can be used to model continuous "exponential" growth:
Continuous Compounding: If $P is invested at an annual interest rate r compounded continuously, the accumulated amount after t years is
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