Increasing and 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

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 increasing and decreasing intervals from a graph?

Back

To determine increasing and decreasing intervals from a graph, observe the slope of the function: if the slope is positive, the function is increasing; if the slope is negative, the function is decreasing.

Front

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

Back

If a function is constant on an interval, it means that the output (y-values) does not change as the input (x-values) changes within that interval.

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 locations where the function changes from increasing to decreasing or vice versa.

Front

How can you find the increasing intervals algebraically?

Back

To find increasing intervals algebraically, take the derivative of the function, set it greater than zero, and solve for x.

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.

Front

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

Back

The second derivative indicates the concavity of the function; if the second derivative is positive, the function is concave up, and if negative, concave down.

Front

What is a local maximum?

Back

A local maximum is a point where the function value is higher than all nearby points, indicating a peak in the graph.

Front

What is a local minimum?

Back

A local minimum is a point where the function value is lower than all nearby points, indicating a trough in the graph.