Using the First Derivative Test in Calculus

Front

What does the First Derivative Test help determine?

Back

The First Derivative Test helps determine the intervals where a function is increasing or decreasing, as well as identifying local extrema.

Front

If f'(x) > 0 on an interval, what can we conclude about f(x)?

Back

If f'(x) > 0 on an interval, then f(x) is increasing on that interval.

Front

If f'(x) < 0 on an interval, what can we conclude about f(x)?

Back

If f'(x) < 0 on an interval, then f(x) is decreasing on that interval.

Front

What is a critical point?

Back

A critical point is a point in the domain of a function where the derivative is either zero or undefined.

Front

True or False: All critical points are local extrema.

Back

False. Not all critical points are local extrema; some may be points of inflection.

Front

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

Back

A zero derivative at a critical point indicates a potential local maximum or minimum.

Front

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

Back

If f'(x) changes from positive to negative at a critical point, then f(x) has a local maximum at that point.

Front

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

Back

If f'(x) changes from negative to positive at a critical point, then f(x) has a local minimum at that point.

Front

What is the relationship between the first derivative and the slope of the function?

Back

The first derivative of a function represents the slope of the tangent line to the function at any given point.

Front

How do you find intervals of increase for a function using its derivative?

Back

To find intervals of increase, identify where the first derivative f'(x) is greater than zero.