Natural exponential function

Definition

The natural exponential function is defined as $e^x$, where $e$ is Euler's number, approximately equal to 2.71828. It is a fundamental function in calculus with unique properties related to growth and decay.

5 Must Know Facts For Your Next Test

1. The derivative of the natural exponential function $e^x$ is itself, $\frac{d}{dx} e^x = e^x$.
2. The integral of the natural exponential function $e^x$ is itself plus a constant, $\int e^x \, dx = e^x + C$.
3. The natural exponential function has a horizontal asymptote at $y = 0$, meaning it never touches the x-axis but gets infinitely close.
4. Exponential growth and decay problems often use the natural exponential function due to its continuous growth rate property.
5. $e^{a+b} = e^a \cdot e^b$: The addition of exponents property for the natural exponential function.

Review Questions

• What is the derivative of $e^x$?
• How do you integrate the natural exponential function?
• State one real-world application where the natural exponential function is used.

Related terms

Euler's Number: A mathematical constant approximately equal to 2.71828, which serves as the base for natural logarithms.

Natural Logarithm: The inverse function of the natural exponential function, denoted as $\ln(x)$, and satisfies $\ln(e^x) = x$.

Exponential Growth: A process that increases quantity over time at a rate proportional to its current value, often modeled by functions of the form $e^{kt}$.