1.8 Increasing and Decreasing 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) increases 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) decreases as x increases.

Front

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

Back

You can determine this by analyzing the slope of the function on that interval. If the slope is positive, the function is increasing; if negative, it is decreasing.

Front

What does it mean if a function is constant on an interval?

Back

It means that the function's output does not change as x varies within that interval.

Front

What is the significance of critical points in determining increasing and decreasing intervals?

Back

Critical points are where the derivative is zero or undefined. They help identify where the function changes from increasing to decreasing or vice versa.

Front

If a function is increasing on the interval (2, 4), what can be said about its derivative on that interval?

Back

The derivative of the function is positive on the interval (2, 4).

Front

If a function is decreasing on the interval (0, 3), what can be said about its behavior at the endpoints?

Back

At the left endpoint (0), the function is higher than at the right endpoint (3), indicating a decrease.

Front

What is the relationship between the first derivative and increasing/decreasing intervals?

Back

The first derivative indicates the slope of the function. If the first derivative is positive, the function is increasing; if negative, it is decreasing.

Front

How do you find the increasing intervals of a polynomial function?

Back

To find increasing intervals, take the derivative of the polynomial, set it to zero to find critical points, and test intervals around those points.

Front

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

Back

It means that the output (y-value) of the function is greater than zero for all x-values in that interval.