Positive/Negative and Increasing/Decreasing Intervals

Front

What are increasing intervals in a function?

Back

Increasing intervals are the ranges of x-values where the function's output (y-values) is rising as x increases.

Front

What are decreasing intervals in a function?

Back

Decreasing intervals are the ranges of x-values where the function's output (y-values) is falling as x increases.

Front

How do you determine the increasing and decreasing intervals of a function?

Back

To determine increasing and decreasing intervals, find the derivative of the function. If the derivative is positive, the function is increasing; if negative, it is decreasing.

Front

What is a positive interval in a function?

Back

A positive interval is a range of x-values where the function's output (y-values) is greater than zero.

Front

What is a negative interval in a function?

Back

A negative interval is a range of x-values where the function's output (y-values) is less than zero.

Front

How can you identify the positive and negative intervals of a function?

Back

To identify positive and negative intervals, analyze the function's graph or evaluate the function at various x-values to see where it is above or below the x-axis.

Front

What does it mean for a function to have a local maximum?

Back

A local maximum is a point where the function's value is higher than the values of the function at nearby points.

Front

What does it mean for a function to have a local minimum?

Back

A local minimum is a point where the function's value is lower than the values of the function at nearby points.

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 potential local maxima, minima, and points of inflection, which are essential for determining increasing and decreasing intervals.

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 critical point.