Derivative Graphs

Front

What does the graph of a function's derivative, y=f'(x), indicate about the original function y=f(x)?

Back

The graph of the derivative indicates where the original function is increasing or decreasing. If f'(x) > 0, then f(x) is increasing; if f'(x) < 0, then f(x) is decreasing.

Front

What is the relationship between the slope of a function and its derivative?

Back

The derivative of a function at a point gives the slope of the tangent line to the graph of the function at that point.

Front

What does it mean if the derivative of a function changes from positive to negative?

Back

If the derivative changes from positive to negative, it indicates that the original function has a relative maximum at that point.

Front

What is the significance of a zero derivative at a point?

Back

A zero derivative at a point indicates a potential relative extremum (maximum or minimum) or a point of inflection.

Front

How can you determine if a function is concave up or concave down using its derivative?

Back

If the derivative of a function is increasing, the function is concave up. If the derivative is decreasing, the function is concave down.

Front

What does it mean for a function to be decreasing in terms of its derivative?

Back

A function is decreasing when its derivative is negative (f'(x) < 0).

Front

What is the relationship between the second derivative and the concavity of a function?

Back

The second derivative indicates concavity: if f''(x) > 0, the function is concave up; if f''(x) < 0, the function is concave down.

Front

What is a relative minimum in terms of the first derivative test?

Back

A relative minimum occurs at a point where the derivative changes from negative to positive.

Front

What is a relative maximum in terms of the first derivative test?

Back

A relative maximum occurs at a point where the derivative changes from positive to negative.

Front

What does the graph of a function look like if its derivative is always positive?

Back

If the derivative is always positive, the graph of the function is always increasing.