AP Calculus Extreme Values and Critical Numbers

Front

What is a critical number in calculus?

Back

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

Front

How do you find critical numbers of a function?

Back

To find critical numbers, take the derivative of the function, set it equal to zero, and solve for x. Also, identify points where the derivative does not exist.

Front

What is the significance of critical numbers?

Back

Critical numbers help identify potential local maxima and minima of a function, which are important for understanding the function's behavior.

Front

What is an absolute minimum?

Back

An absolute minimum is the lowest point over the entire domain of a function.

Front

What is a relative maximum?

Back

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

Front

How do you determine if a critical number is a maximum or minimum?

Back

Use the first derivative test: if the derivative changes from positive to negative at the critical number, it's a maximum; if it changes from negative to positive, it's a minimum.

Front

What is the first derivative test?

Back

The first derivative test involves analyzing the sign of the derivative before and after a critical number to determine if it is a local maximum or minimum.

Front

What is the second derivative test?

Back

The second derivative test uses the second derivative of a function: if it's positive at a critical number, it's a local minimum; if negative, it's a local maximum.

Front

How do you find the absolute extrema on a closed interval?

Back

Evaluate the function at critical numbers and at the endpoints of the interval, then compare these values to find the absolute maximum and minimum.

Front

What is the role of endpoints in finding extrema?

Back

Endpoints are crucial in finding absolute extrema on closed intervals, as they can be the highest or lowest points of the function.