AP Calculus AB 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.
How can you justify that a function f(x) has a local maximum at a point x=c?
A local maximum at x=c can be justified by showing that the first derivative, f'(x), changes from positive to negative at that point.
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How can you justify that a function f(x) has a local maximum at a point x=c?
A local maximum at x=c can be justified by showing that the first derivative, f'(x), changes from positive to negative at that point.
What information does the first derivative, f'(x), provide about the behavior of the function f(x)?
The sign of the first derivative, f'(x), indicates whether the function f(x) is increasing (f' > 0) or decreasing (f' < 0) on an interval.
What is the relationship between the key features of the graph of f'(x) and the behavior of f(x)?
The y-values of f'(x) represent the slope of f(x), the x-intercepts of f'(x) are potential extrema for f(x), and the intervals where f'(x) is increasing/decreasing indicate the concavity of f(x).
What information does the second derivative, f''(x), provide about the behavior of the function f(x)?
The sign of the second derivative, f''(x), describes the concavity of f(x), indicating where the graph is concave up (f'' > 0) or concave down (f'' < 0).
Using analytical information, if you know that f'(5) = 0 and f''(5) > 0, what conclusion can you justify about f(x) at x=5?
Based on this information from the derivatives (the Second Derivative Test), you can justify the conclusion that the function f(x) has a local minimum at x=5.
If you are given the graph of f'(x), how do you determine where the original function f(x) is concave down?
The function f(x) is concave down on intervals where f''(x) is negative. This corresponds to the intervals where the graph of f'(x) is decreasing.
Given the graph of f'(x), how can you find the x-coordinates of the points of inflection for f(x)?
Points of inflection on f(x) occur where f''(x) changes sign. This corresponds to the x-coordinates where the graph of f'(x) changes from increasing to decreasing or vice-versa (i.e., at local extrema of f').
How can you justify that a function f(x) has a local minimum at a point x=c?
A local minimum at x=c can be justified by showing that the first derivative, f'(x), changes from negative to positive at that point.
What three forms of information from f' and f'' can be used to predict and explain the behavior of f?
Graphical, numerical, and analytical information from the first and second derivatives (f' and f'') can be used to predict and explain the behavior of the original function, f.
Given the graph of a function's derivative, f'(x), how do you determine the intervals where the original function, f(x), is increasing?
The function f(x) is increasing on intervals where its derivative, f'(x), is positive. This corresponds to the intervals where the graph of f'(x) is above the x-axis.