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History of the Exponential Function:
Some background information about the number e. Early mathematical work tended to focus on logarithms, and although the natural log (base e) is fairly well-known now, the number we know as e managed to escape the attention of mathematicians for many years. Many early mathematical explorers and scientists danced around the discovery, coming very close to uncovering it only to move away from it by some other distracting aspect of their work. Consequently, e did not burst into the mathematical world by some significant understanding (like that of π), but rather slowly emerged onto the scene through random dabblings into the nature of things such as finance. Why is e important? Because y = e^x is the only equation known to man whose derivative is itself! To learn more about the history and other facts about exponential function click the button below! |
Exponential Functions in the Real WorldExponential functions can be used to explore finance, science and population growth in the real world. They can also measure things like radioactive decay. The following video will help go through some real world examples using the exponential functions.
Another example of a real world situation could be a problem like this one:
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Exponential Graph and Equation
The graph above shows the parent graph of an exponential function. The graph's equation is the exponential function f(x)= a^x. The properties of this function all depend on the value of a, which is any value greater than 0. If a is in between 0 and 1 the graph is decreasing from left to right. If the value of a is greater than 1, the graph increase from left to right (like the graph above). In general, the the graph never cross the x-axis and always passes through the points (0,1) and (1, a). The domain of this function is all real numbers. The range of the function is (0, positive infinity).
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Common Core StandardsThe North Carolina Common Core says that students must be able to:
Distinguish between situations that can be modeled with linear functions and with exponential functions
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Examples
Example 1
Solve for y when x=5.
y=1.2^x That means we need to plug-in x=5 and see what we get: y=1.2^5 y= (1.2) (1.2) (1.2) (1.2) (1.2) y= 2.48832 |
Example
Solve for x: 4 = 2^x
The problem says we have to multiply x number of two's together to get four. Well, everyone knows that 2*2=4, so the answer is two: 4=2^2 x=2 |