Applications of the First Derivative

This set of flashcards covers key concepts related to the applications of the First Derivative, including determining intervals of increase and decrease, critical numbers, and the First Derivative Test.

What happens to the function f(x) if f'(x) > 0 in an interval (a, b)?

The function f is increasing on (a, b).

What happens to the function f(x) if f'(x) < 0 in an interval (a, b)?

The function f is decreasing on (a, b).

What happens to the function f(x) if f'(x) = 0 in an interval (a, b)?

The function f is constant on (a, b).

What is a critical number of a function f?

A critical number is any number x in the domain of f such that f'(x) = 0 or f'(x) does not exist.

What is the procedure for finding relative extrema using the First Derivative Test?

1. Determine the critical numbers of f. 2. Determine the sign of f'(x) to the left and right of each critical number.

What can you conclude if f'(x) changes sign from positive to negative at a critical number c?

The function f has a relative maximum at x = c.

What can you conclude if f'(x) changes sign from negative to positive at a critical number c?

The function f has a relative minimum at x = c.

What does it mean if f'(x) does not change sign at a critical number c?

The function f has neither a relative maximum nor a relative minimum at x = c.

In the context of the First Derivative, what do you determine in order to find intervals where a function is increasing or decreasing?

Find all values of x for which f'(x) = 0.