Applied: Circular Motion

A. The Natural Exponential Function (e^x) Definition: The natural exponential function, denoted as e^x, is a mathematical function where 'e' is Euler's number (an irrational constant approximately 2.71828) and 'x' is the exponent. It is formally defined by the infinite series: e^x = 1 + x/1! + x²/2! + x³/3! + ... Explanation: This function describes continuous growth or decay. Unlike other exponential functions like 2^x or 10^x, e^x has a unique property where its rate of change (derivative) at any point is equal to its value at that point. This makes it fundamental in calculus and various scientific applications. Its domain is all real numbers (ℝ), and its range is all positive real numbers (0, ∞). Think of it as: Imagine a bank account where interest is compounded continuously, not just annually or monthly. The growth of your money in such an account would be modeled by an exponential function with base 'e'.