Graph of y = e^x
Interactive graph of y = e^x (natural exponential function). Explore exponential growth, its unique derivative property, and asymptotic behavior.
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Understanding the Function
The natural exponential function y = e^x, where e is approximately 2.71828, is one of the most important functions in mathematics. It grows faster than any polynomial as x increases and approaches zero (but never reaches it) as x decreases. The y-intercept is at (0, 1) since e^0 = 1.
What makes e^x unique is that it is its own derivative: d/dx(e^x) = e^x. This self-replicating property makes it the natural choice for modeling continuous growth and decay processes, including population growth, radioactive decay, compound interest, and heat dissipation.
Key properties: domain is all real numbers, range is (0, infinity), horizontal asymptote at y = 0, y-intercept at (0, 1), always positive, always increasing, always concave up. The inverse function is the natural logarithm ln(x).