Positive/Negative and Increasing/Decreasing 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) increases as x increases.

Front

How do you determine the increasing intervals of a function?

Back

To determine the increasing intervals, find where the derivative of the function is positive.

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) decreases as x increases.

Front

How do you determine the decreasing intervals of a function?

Back

To determine the decreasing intervals, find where the derivative of the function is negative.

Front

What does it mean for a function to have a positive interval?

Back

A positive interval refers to the range of x-values for which the function's output (y-values) is greater than zero.

Front

How can you find the positive intervals of a function?

Back

To find the positive intervals, solve the inequality f(x) > 0.

Front

What is the significance of critical points in determining 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

What is the relationship between the first derivative and the behavior of a function?

Back

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

Front

What is the notation for representing intervals?

Back

Intervals are represented using parentheses for open intervals (not including endpoints) and brackets for closed intervals (including endpoints).

Front

What does the interval notation (4, 8) represent?

Back

The interval (4, 8) represents all real numbers x such that 4 < x < 8.