Increasing Functions in Calculus: Definition, Examples and Derivative Techniques

Function is increasing

First derivative greater than zero

A function is said to be increasing if its output values increase as the input values increase. More formally, if for any two inputs x1 and x2, where x1 < x2, we have f(x1) < f(x2), then the function is said to be increasing over the interval [x1, x2]. For example, consider the function y = 2x. As x increases, y increases as well. For instance, if we take x = 1, then y = 2*1 = 2. If we take x = 2, then y = 2*2 = 4. Therefore, the function y = 2x is increasing. Another example of an increasing function is the exponential function y = e^x. As x increases, y increases at a faster and faster rate. In the context of calculus, we can use the derivative of the function to determine if it's increasing or decreasing. If the derivative is positive, then the function is increasing. If the derivative is negative, then the function is decreasing.

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