Graphs of functions and their derivatives

Front

What does it mean when a graph is increasing?

Back

When a graph is increasing, the function's values rise as the input values increase, and its derivative is positive.

Front

What does it mean when a graph is decreasing?

Back

When a graph is decreasing, the function's values fall as the input values increase, and its derivative is negative.

Front

What is the relationship between a function and its derivative when the function is increasing?

Back

If a function is increasing, its derivative is positive.

Front

What is the relationship between a function and its derivative when the function is decreasing?

Back

If a function is decreasing, its derivative is negative.

Front

What does a zero derivative indicate about a function?

Back

A zero derivative indicates that the function has a horizontal tangent line at that point, which could be a local maximum, minimum, or a point of inflection.

Front

If f'(x) > 0, what can be said about f(x)?

Back

If f'(x) > 0, then f(x) is increasing at that point.

Front

If f'(x) < 0, what can be said about f(x)?

Back

If f'(x) < 0, then f(x) is decreasing at that point.

Front

What is the significance of a positive derivative on an interval?

Back

A positive derivative on an interval indicates that the function is increasing throughout that interval.

Front

What is the significance of a negative derivative on an interval?

Back

A negative derivative on an interval indicates that the function is decreasing throughout that interval.

Front

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

Back

A function has a local maximum at a point where it changes from increasing to decreasing, and its derivative is zero or undefined at that point.