AP Calculus BC Flashcards: Sketching Graphs of Functions and Their Derivatives
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Review key ideas with interactive flashcards. This set includes 10 cards to help you master important concepts.
What does the first derivative, f'(x), reveal about the behavior of the function f(x)?
The sign of f'(x) indicates whether f(x) is increasing (f' > 0) or decreasing (f' < 0). Critical points of f(x) occur where f'(x) is zero or undefined.
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What does the first derivative, f'(x), reveal about the behavior of the function f(x)?
The sign of f'(x) indicates whether f(x) is increasing (f' > 0) or decreasing (f' < 0). Critical points of f(x) occur where f'(x) is zero or undefined.
How can you justify that a function f(x) has a local maximum at a critical point x=c?
A local maximum can be justified by showing that f'(x) changes from positive to positive at x=c (First Derivative Test), or by showing that f'(c) = 0 and f''(c) < 0 (Second Derivative Test).
If you know that f'(5) = 0 and f''(5) > 0, what conclusion can you justify about the behavior of f(x) at x=5?
According to the Second Derivative Test, the function f(x) has a local minimum at x=5. This is because the function has a horizontal tangent and is concave up at that point.
What does the second derivative, f''(x), reveal about the behavior of the function f(x)?
The sign of f''(x) indicates the concavity of f(x). The function f(x) is concave up where f''(x) > 0 and concave down where f''(x) < 0.
If a function f(x) is increasing and concave down on an interval, what can you conclude about its derivatives?
Because the function is increasing, f'(x) must be positive. Because the function is concave down, f''(x) must be negative on that interval.
Given the graph of f'(x), how can you identify the x-coordinates of the inflection points of f(x)?
Inflection points of f(x) occur where its concavity changes, which is where f''(x) changes sign. This corresponds to the points where the graph of f'(x) changes from increasing to decreasing (or vice versa), which are the local extrema of f'(x).
What is the relationship between the graph of f'(x) and the key features of f(x)?
The x-intercepts of the f'(x) graph correspond to critical points of f(x). Intervals where the f'(x) graph is above the x-axis correspond to intervals where f(x) is increasing.
How is information from f'(x) and f''(x) used to predict and explain the behavior of f(x)?
By combining the sign analysis of f'(x) (for increasing/decreasing intervals) and f''(x) (for concavity), we can predict the overall shape and sketch a detailed graph of f(x).
How can numerical data from a table of f'(x) values be used to draw conclusions about f(x)?
By observing the sign of the f'(x) values in the table, one can determine the intervals where f(x) is increasing or decreasing. A change in sign indicates a local extremum.
What is a point of inflection?
A point of inflection is a point on the graph of a function where the concavity changes. This occurs where the second derivative, f''(x), changes sign.