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Access these online resources for additional instruction and practice with exponential functions.
definition of the exponential function | $f(x)={b}^{x}\text{,where}b0,b\ne 1$ |
definition of exponential growth | $f(x)=a{b}^{x},\text{where}a0,b0,b\ne 1$ |
compound interest formula | $\begin{array}{l}A(t)=P{\left(1+\frac{r}{n}\right)}^{nt},\text{where}\hfill \\ A(t)\text{istheaccountvalueattime}t\hfill \\ t\text{isthenumberofyears}\hfill \\ P\text{istheinitialinvestment,oftencalledtheprincipal}\hfill \\ r\text{istheannualpercentagerate(APR),ornominalrate}\hfill \\ n\text{isthenumberofcompoundingperiodsinoneyear}\hfill \end{array}$ |
continuous growth formula |
$A(t)=a{e}^{rt},\text{where}$
$t$ is the number of unit time periods of growth $a$ is the starting amount (in the continuous compounding formula a is replaced with P, the principal) $e$ is the mathematical constant, $\text{}e\approx 2.718282$ |
Explain why the values of an increasing exponential function will eventually overtake the values of an increasing linear function.
Linear functions have a constant rate of change. Exponential functions increase based on a percent of the original.
Given a formula for an exponential function, is it possible to determine whether the function grows or decays exponentially just by looking at the formula? Explain.
The Oxford Dictionary defines the word nominal as a value that is “stated or expressed but not necessarily corresponding exactly to the real value.” Oxford Dictionary. http://oxforddictionaries.com/us/definition/american_english/nomina. Develop a reasonable argument for why the term nominal rate is used to describe the annual percentage rate of an investment account that compounds interest.
When interest is compounded, the percentage of interest earned to principal ends up being greater than the annual percentage rate for the investment account. Thus, the annual percentage rate does not necessarily correspond to the real interest earned, which is the very definition of nominal .
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