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AP Calculus BC Flashcards: Connecting a Function, Its First Derivative, and Its Second Derivative

Written by AP Content Team, Verified for 2026 AP Exams, Last updated: May 2026

Review key ideas with interactive flashcards. This set includes 10 cards to help you master important concepts.

What does the sign of the first derivative, f'(x), indicate about the function f(x)?
The sign of f'(x) indicates whether the function f(x) is increasing (f'(x) > 0) or decreasing (f'(x) < 0) on an interval.
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What does the sign of the first derivative, f'(x), indicate about the function f(x)?
The sign of f'(x) indicates whether the function f(x) is increasing (f'(x) > 0) or decreasing (f'(x) < 0) on an interval.
How are the critical points of a function f(x) related to its first derivative, f'(x)?
The critical points of f(x) occur at x-values where its derivative, f'(x), is equal to zero or is undefined.
If the graph of f'(x) is above the x-axis, what can you conclude about the behavior of the graph of f(x)?
If f'(x) is above the x-axis, then f'(x) > 0, which means the function f(x) is increasing.
How do you justify that a function f(x) has a local minimum at x=c using its first derivative?
A local minimum at x=c can be justified by showing that f'(c) = 0 or is undefined, and that f'(x) changes sign from negative to positive at x=c.
To justify that a function f(x) is concave up on an interval, what must you show about one of its derivatives?
To justify that f(x) is concave up, you must show that its second derivative, f''(x), is positive on that interval.
What does the sign of the second derivative, f''(x), indicate about the function f(x)?
The sign of f''(x) indicates the concavity of the function f(x). If f''(x) > 0, f(x) is concave up; if f''(x) < 0, f(x) is concave down.
What feature on the graph of the first derivative, f'(x), corresponds to a point of inflection on the graph of f(x)?
A local extremum (maximum or minimum) on the graph of f'(x) corresponds to a point of inflection on the graph of f(x), as this is where f''(x) changes sign.
If f(x) has a local maximum at x=c, what can be concluded about the values of f'(c) and f''(c)?
If f(x) has a local maximum at x=c, then f'(c) = 0 (or is undefined) and f''(c) must be less than zero (indicating it is concave down).
What is the relationship between the slope of f'(x) and the function f(x)?
The slope of the first derivative, f'(x), is the second derivative, f''(x). Therefore, the slope of f'(x) determines the concavity of f(x).
How is a point of inflection on the graph of f(x) related to its second derivative, f''(x)?
A point of inflection on the graph of f(x) is a point where the concavity changes. This corresponds to a sign change in the second derivative, f''(x).