increasing, decreasing, and constant intervals

Front

What is an increasing interval in a function?

Back

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

Front

What is a decreasing interval in a function?

Back

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

Front

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

Back

A function is constant on an interval if its output (y-values) remains the same for all x-values in that interval.

Front

How can you determine if a function is increasing or decreasing on a given interval?

Back

You can determine if a function is increasing or decreasing by analyzing its derivative: if the derivative is positive, the function is increasing; if negative, it is decreasing.

Front

What is the significance of critical points in determining intervals of increase or decrease?

Back

Critical points are where the derivative is zero or undefined; they help identify potential intervals of increase or decrease.

Front

What does the notation (-∞, -2) U (0.5, ∞) represent?

Back

This notation represents the union of two intervals: the function is increasing on the interval from negative infinity to -2 and from 0.5 to positive infinity.

Front

How do you express an interval that includes its endpoint?

Back

An interval that includes its endpoint is expressed using square brackets, e.g., [a, b] means the interval includes both a and b.

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]).

Front

What is the role of the first derivative test in analyzing intervals of increase and decrease?

Back

The first derivative test involves checking the sign of the derivative before and after critical points to determine whether the function is increasing or decreasing.

Front

What does it mean if a function is always increasing?

Back

If a function is always increasing, it means that for any two points in its domain, the function's output at the higher input is always greater than or equal to the output at the lower input.