AP Calculus BC · Unit 5: Analytical Applications of Differentiation
5.6 Determining Concavity of Functions over Their Domains
In short
You must determine the concavity of a function and locate points of inflection using its first and second derivatives. The focus is on understanding that a function is concave up (or down) on an open interval if its first derivative is increasing (or decreasing).
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What else do you need to know about Determining Concavity of Functions over Their Domains?
You must analytically determine these behaviors and provide formal conclusions rather than relying on superficial visual descriptions from a calculator graph.
Can you do these?
- I can determine if the graph of a function is concave up or down on an open interval by checking if its derivative is increasing or decreasing.
- I can use the second derivative of a function to find intervals of upward or downward concavity.
How is Determining Concavity of Functions over Their Domains tested on the exam?
On the AP Exam, sound reasoning must be accompanied by clear communication. When justifying concavity or points of inflection, you must use extreme precision in their language. You are expected to refer to functions by their specific names (e.g., f, f', and f'') rather than using ambiguous pronouns like 'it' or 'the function', which conflate different functions and are penalized. When given a graph of the derivative f', it is safer and easier to make arguments about f based directly on the graph of the derivative (e.g., 'f is concave up on a < x < b because the graph of f' is increasing on a < x < b').
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Determining Concavity of Functions over Their Domains: common questions
What do you need to know about Determining Concavity of Functions over Their Domains for AP Calculus BC?
You must determine the concavity of a function and locate points of inflection using its first and second derivatives. The focus is on understanding that a function is concave up (or down) on an open interval if its first derivative is increasing (or decreasing).
How is Determining Concavity of Functions over Their Domains tested on the AP Calculus BC exam?
On the AP Exam, sound reasoning must be accompanied by clear communication. When justifying concavity or points of inflection, you must use extreme precision in their language. You are expected to refer to functions by their specific names (e.g., f, f', and f'') rather than using ambiguous pronouns like 'it' or 'the function', which conflate different functions and are penalized.
How do I know I have mastered Determining Concavity of Functions over Their Domains?
Check you can do each of these: determine if the graph of a function is concave up or down on an open interval by checking if its derivative is increasing or decreasing; use the second derivative of a function to find intervals of upward or downward concavity.
Where does Determining Concavity of Functions over Their Domains fit in AP Calculus BC?
Determining Concavity of Functions over Their Domains is topic 5.6 in Unit 5: Analytical Applications of Differentiation of AP Calculus BC.
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