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Concavity quiz #1 Flashcards

Concavity quiz #1
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  • How can you determine the intervals where a function is concave up or concave down using its second derivative?
    Find the second derivative, locate where it’s zero or undefined, then check its sign on intervals to determine concavity.
  • What does it mean for a function to be concave up, and how is this related to its second derivative?
    A function is concave up if its graph curves upward like a smile, which corresponds to its second derivative being positive.
  • How can you visually distinguish between concave up and concave down on a graph?
    Concave up looks like a smile (curves upward), while concave down looks like a frown (curves downward).
  • What is an inflection point in terms of concavity and the second derivative?
    An inflection point is where a function changes concavity, occurring where the second derivative is zero or does not exist.
  • How do you determine intervals of concavity using the second derivative?
    Find where the second derivative is zero or undefined to identify potential inflection points, then test the sign of the second derivative in the intervals between these points.
  • If the second derivative of a function is negative on an interval, what can you say about the function's concavity there?
    The function is concave down on that interval.
  • When given the graph of a function’s second derivative, how do you determine where the original function is concave up or down?
    If the second derivative graph is above the x-axis (positive), the function is concave up; if below (negative), it is concave down.
  • How does the behavior of the first derivative relate to the concavity of the original function?
    If the first derivative is increasing, the function is concave up; if the first derivative is decreasing, the function is concave down.
  • What steps do you follow to find intervals of concavity for a function given only its formula?
    Find the second derivative, set it to zero or find where it does not exist, then test the sign of the second derivative in the resulting intervals.
  • Why do you use test values in intervals when determining concavity, and what do you look for?
    Test values help determine the sign of the second derivative in each interval; a positive sign means concave up, and a negative sign means concave down.