Increasing, Decreasing, Constant Intervals

Front

What is an increasing interval in a function?

Back

An increasing interval is a range of x-values where the function's output (y-value) rises as x increases.

Front

What is a decreasing interval in a function?

Back

A decreasing interval is a range of x-values where the function's output (y-value) falls as x increases.

Front

What does it mean for a function to be constant over an interval?

Back

A function is constant over an interval if its output (y-value) does not change as x varies within that interval.

Front

How can you determine if a function is increasing, decreasing, or constant from its graph?

Back

You can determine this by observing the slope of the graph: a positive slope indicates increasing, a negative slope indicates decreasing, and a horizontal line indicates constant.

Front

What is the significance of critical points in analyzing intervals of a function?

Back

Critical points are where the function's derivative is zero or undefined, indicating potential changes in increasing or decreasing behavior.

Front

How do you find the intervals of increase and decrease for a polynomial function?

Back

To find intervals of increase and decrease, calculate the derivative of the function, set it to zero to find critical points, and test intervals around those points.

Front

What is the first derivative test?

Back

The first derivative test is a method used to determine whether a critical point is a local maximum, local minimum, or neither by analyzing the sign of the derivative before and after the point.

Front

What does it mean if a function is never decreasing?

Back

If a function is never decreasing, it means that the function is either increasing or constant for all x-values in its domain.

Front

How can you express an interval in interval notation?

Back

An interval in interval notation is expressed using parentheses and brackets, e.g., (a, b) for open intervals and [a, b] for closed intervals.

Front

What is the difference between open and closed intervals?

Back

Open intervals do not include their endpoints (e.g., (a, b)), while closed intervals do include their endpoints (e.g., [a, b]).