3.5 Intervals of increase and decrease

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-values) rises 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-values) falls as x increases.

Front

How do you determine where a function is increasing or decreasing?

Back

You can determine this by analyzing the first derivative of the function. If the derivative is positive, the function is increasing; if negative, it is decreasing.

Front

What does it mean if a function has a critical point?

Back

A critical point occurs where the derivative is zero or undefined, indicating potential locations for local maxima, minima, or points of inflection.

Front

What is the significance of the first derivative test?

Back

The first derivative test helps identify whether a critical point is a local maximum, local minimum, or neither by checking the sign of the derivative before and after the point.

Front

Define the term '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

Define the term '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.

Front

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

Back

The second derivative can indicate concavity and help confirm whether a critical point is a maximum or minimum.

Front

How can you express an interval in interval notation?

Back

An interval in interval notation is expressed using parentheses or brackets to indicate whether endpoints are included (closed) or excluded (open). For example, (a, b) or [a, b].

Front

What does the notation (-∞, a) represent?

Back

The notation (-∞, a) represents all real numbers less than 'a', extending indefinitely to the left on the number line.