e^x
The expression e^x represents the exponential function, where e is a mathematical constant approximately equal to 2
The expression e^x represents the exponential function, where e is a mathematical constant approximately equal to 2.71828, and x is the exponent.
When we raise e to the power of x (e^x), we are essentially evaluating the exponential function at the value x. The result is equal to the value of e multiplied by itself x times.
For example, if we have e^2, we would calculate it as e * e = 2.71828 * 2.71828 = 7.38906. Similarly, e^3 would be e * e * e = 2.71828 * 2.71828 * 2.71828 = 20.08554.
The exponential function e^x has several important properties. Firstly, e^0 equals 1, regardless of the value of e. This is because any number raised to the power of 0 is always 1.
Secondly, e^x increases rapidly as x becomes larger or approaches positive infinity. For example, e^1 is approximately 2.71828, but e^10 is approximately 22026.46579. This showcases the exponential growth of e^x.
On the other hand, as x approaches negative infinity, e^x approaches 0. For instance, e^-1 is approximately 0.36788, whereas e^-10 is approximately 0.0000453. This demonstrates the exponential decay of e^x.
Furthermore, the exponential function e^x is used in various mathematical and scientific applications. It is particularly important in calculus, where it serves as the base function for differentiation and integration.
In conclusion, e^x represents the exponential function, where e is a constant and x is the exponent. It results in a value equal to e multiplied by itself x times and exhibits rapid growth as x increases and exponential decay as x decreases.
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