To determine intervals of increase and decrease, analyze the function's derivative for positive or negative values, respectively. A function's graph with varying slopes indicates increasing or decreasing growth rates over specific intervals.
To determine intervals of increase and decrease in a mathematical function, you must first analyze the function's derivative. The derivative of a function provides information about the slope of the tangent line at any given point on the function's graph. An increasing interval on a function's graph is characterized by a positive slope, where the derivative is greater than zero, indicating that the function is going uphill in that interval. Conversely, a decreasing interval is where the slope is negative, with the derivative being less than zero, showing that the function is going downhill.
Let's consider a line graph depicting a function's growth rate over time. The slope of the line segments will reveal whether the growth rate is increasing, decreasing, or constant (zero growth). For instance, suppose we have a graph with section A featuring a negative slope that levels off to a zero slope, section B with a positive slope that decreases until the curve levels off, and section C with a positive slope that increases until it hits a consistent upward slope. Section A demonstrates a decreasing interval, section B an increasing interval with a decreasing growth rate, and section C an increasing interval with a consistent growth rate.
So, intervals of increase are where the derivative is positive, and intervals of decrease are where it's negative. By plotting the derivative or examining the slope for linearity, consistency, or patterns, we can identify such intervals.