End Behavior and Increasing/Decreasing

Front

What is end behavior in a function?

Back

End behavior describes the behavior of a function as the input values (x) approach positive or negative infinity. It helps to understand how the function behaves at the extremes.

Front

Define relative maximum.

Back

A relative maximum is a point on the graph of a function where the function value is higher than the values of the function at nearby points.

Front

Define relative minimum.

Back

A relative minimum is a point on the graph of a function where the function value is lower than the values of the function at nearby points.

Front

What does it mean for a function to be increasing?

Back

A function is increasing on an interval if, for any two points in that interval, the function value at the second point is greater than the function value at the first point.

Front

What does it mean for a function to be decreasing?

Back

A function is decreasing on an interval if, for any two points in that interval, the function value at the second point is less than the function value at the first point.

Front

How do you identify intervals of increase and decrease from a graph?

Back

To identify intervals of increase and decrease, look for sections of the graph where the slope is positive (increasing) or negative (decreasing).

Front

What is a turning point in a function?

Back

A turning point is a point on the graph of a function where the function changes from increasing to decreasing or from decreasing to increasing.

Front

What is the significance of the first derivative in determining increasing/decreasing behavior?

Back

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

Front

What is the significance of the second derivative in determining concavity?

Back

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

Front

How do you find relative extrema using the first derivative test?

Back

To find relative extrema, set the first derivative equal to zero to find critical points, then test intervals around these points to determine if they are maxima or minima.