First Derivative Test

Front

What is the First Derivative Test used for in calculus?

Back

The First Derivative Test is used to determine the relative extrema (maximum and minimum points) of a function by analyzing the sign of the first derivative.

Front

How do you find critical points of a function?

Back

Critical points are found by taking the derivative of the function, setting it equal to zero, and solving for the variable. Points where the derivative does not exist are also considered critical points.

Front

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

Back

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

Front

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

Back

If the first derivative changes from negative to positive at a critical point, it indicates that the function has a relative minimum at that point.

Front

What is the significance of a critical point where the first derivative is zero?

Back

A critical point where the first derivative is zero may indicate a relative maximum, relative minimum, or a point of inflection, and further analysis is needed to determine its nature.

Front

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

Back

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

Front

What is a relative maximum?

Back

A relative maximum is a point on the graph of a function where the function value is greater than the values of the function at nearby points.

Front

What is a relative minimum?

Back

A relative minimum is a point on the graph of a function where the function value is less than the values of the function at nearby points.

Front

How can you determine if a critical point is a relative extremum using the first derivative test?

Back

To determine if a critical point is a relative extremum, evaluate the sign of the first derivative before and after the critical point. If it changes from positive to negative, it's a maximum; if it changes from negative to positive, it's a minimum.

Front

What is the first derivative of a function?

Back

The first derivative of a function is the derivative that gives the rate of change of the function with respect to its variable.