AP Calculus AB Flashcards: Connecting a Function, Its First Derivative, and Its Second Derivative
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Review key ideas with interactive flashcards. This set includes 10 cards to help you master important concepts.
If you know f'(c) = 0 and f''(c) > 0, what can you conclude about the function f(x) at x=c?
According to the Second Derivative Test, the function f(x) has a relative minimum at x=c.
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If you know f'(c) = 0 and f''(c) > 0, what can you conclude about the function f(x) at x=c?
According to the Second Derivative Test, the function f(x) has a relative minimum at x=c.
What feature on the graph of f(x) corresponds to a relative extremum (max or min) on the graph of f'(x)?
A relative extremum on the graph of f'(x) corresponds to a point of inflection on the graph of f(x). This is where the concavity of f(x) changes.
How are the critical points of a function f(x) related to the graph of its derivative, f'(x)?
Critical points of f(x) occur where f'(x) is zero or undefined. These correspond to the x-intercepts or discontinuities on the graph of f'(x).
What does the sign of the first derivative, f'(x), indicate about the original function, f(x)?
If f'(x) is positive on an interval, f(x) is increasing on that interval. If f'(x) is negative, f(x) is decreasing.
Given the graph of f'(x), how would you determine the intervals where f(x) is decreasing?
You would identify the intervals on the x-axis where the graph of f'(x) is below the x-axis. On these intervals, f'(x) < 0, so f(x) is decreasing.
What is the relationship between the intervals where f'(x) is increasing and the graph of f(x)?
If the graph of f'(x) is increasing on an interval, its derivative, f''(x), is positive. This means the graph of the original function, f(x), is concave up on that interval.
How is a point of inflection on the graph of f(x) related to its second derivative, f''(x)?
A point of inflection on f(x) is where the concavity changes. This corresponds to a point where f''(x) changes sign (often where f''(x) = 0).
How do you justify that a function f(x) is concave up on an interval?
You can justify that f(x) is concave up on an interval by showing that its second derivative, f''(x), is positive over that entire interval.
What does the sign of the second derivative, f''(x), indicate about the original function, f(x)?
If f''(x) is positive on an interval, f(x) is concave up on that interval. If f''(x) is negative, f(x) is concave down.
How do you justify that a function f(x) has a relative minimum at x=c using the First Derivative Test?
A relative minimum at x=c can be justified by showing that f'(c)=0 or is undefined, and that f'(x) changes from negative to positive at x=c.