Derivative Graphs

Front

What is a local minimum in the context of derivative graphs?

Back

A local minimum occurs at a point where the derivative changes from negative to positive, indicating that the function has a lowest value in a small neighborhood around that point.

Front

What does it mean when the derivative f'(x) is equal to zero?

Back

When f'(x) = 0, it indicates a stationary point, which could be a local maximum, local minimum, or a point of inflection.

Front

How can you determine if a function is increasing or decreasing using its derivative?

Back

If f'(x) > 0, the function f(x) is increasing. If f'(x) < 0, the function f(x) is decreasing.

Front

What is the significance of the sign chart for f'(x)?

Back

The sign chart for f'(x) helps to determine the intervals where the function f(x) is increasing or decreasing, as well as identifying local maxima and minima.

Front

What does a positive derivative indicate about the graph of a function?

Back

A positive derivative indicates that the graph of the function is rising or increasing at that point.

Front

What does a negative derivative indicate about the graph of a function?

Back

A negative derivative indicates that the graph of the function is falling or decreasing at that point.

Front

What is a point of inflection?

Back

A point of inflection is where the concavity of the function changes, which occurs when the second derivative f''(x) changes sign.

Front

How can you find local maxima and minima using the first derivative test?

Back

To find local maxima and minima, identify critical points where f'(x) = 0 or is undefined, then test the sign of f'(x) around these points.

Front

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

Back

The graph of the derivative f'(x) provides information about the slope of the function f(x), indicating where it is increasing, decreasing, and where it has local extrema.

Front

What does it mean if f'(x) changes from positive to negative at a point?

Back

If f'(x) changes from positive to negative at a point, it indicates that the function f(x) has a local maximum at that point.