Characteristics of Graphs

Front

What is a decreasing function?

Back

A function is decreasing on an interval if, for any two points x1 and x2 in that interval, if x1 < x2 then f(x1) > f(x2).

Front

What is an increasing function?

Back

A function is increasing on an interval if, for any two points x1 and x2 in that interval, if x1 < x2 then f(x1) < f(x2).

Front

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

Back

To determine intervals of increase and decrease, observe the slope of the graph: if the graph rises as you move from left to right, it is increasing; if it falls, it is decreasing.

Front

What is a local maximum?

Back

A local 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

What is a local minimum?

Back

A local 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 the term 'critical point' refer to?

Back

A critical point is a point on the graph of a function where the derivative is zero or undefined, indicating potential local maxima or minima.

Front

What is the significance of the first derivative test?

Back

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

Front

What is the second derivative test?

Back

The second derivative test uses the second derivative of a function to determine the concavity of the function at a critical point, helping to classify it as a local maximum or minimum.

Front

What is the difference between absolute and local extrema?

Back

Absolute extrema are the highest or lowest values of a function over its entire domain, while local extrema are the highest or lowest values within a specific interval.

Front

How do you find the maximum value of a function on a closed interval?

Back

To find the maximum value on a closed interval, evaluate the function at the endpoints and at any critical points within the interval, then compare these values.