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anonymous

  • 5 years ago

WHY do we use the second derivative to find the inflection point? Couldn't quite grasp the concept.

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  1. shadowfiend
    • 5 years ago
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    The first derivative will tell you where the slope of the function is 0. This is a prerequisite for an inflection point.

  2. shadowfiend
    • 5 years ago
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    The other requirement is that the function change `directions' from one side of the point to the other.

  3. shadowfiend
    • 5 years ago
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    In order for that to happen, the slope must go from being negative to being positive or positive to negative. The second derivative can tell us whether or not that is happening.

  4. shadowfiend
    • 5 years ago
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    Sorry, that's completely wrong.

  5. shadowfiend
    • 5 years ago
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    Heh.

  6. shadowfiend
    • 5 years ago
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    Started explaining the wrong concept.

  7. shadowfiend
    • 5 years ago
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    Let's start from scratch! An inflection point is where a function changes from being concave upwards to being concave downwards.

  8. shadowfiend
    • 5 years ago
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    Functions that are concave upwards have a slope that is increasing (either becoming more positive or less negative), and those that are concave downwards have a slope that is decreasing (either becoming less positive or more negative).

  9. anonymous
    • 5 years ago
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    Remember the inflection point tells us where the concavity of the underlying function changes from concave up to concave down.. So tracing this through the first derivative....an inflection point (on the original function) becomes a max or min on the first derivative function and therefore a zero on the second derivative function. That is why it is easiest to identify the inflection points from the second derivative....they are the zeros!

  10. shadowfiend
    • 5 years ago
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    To convince yourself of this, you can do a quick sketch of a regular parabola, which is concave up, and sketch tangent lines at the edges and in the middle of the curvatures. You should be able to see that the tangent lines go from a very negative slope to a mid-negative slope to a zero slope to a mid-positive slope to a very positive slope. You can do the same with a parabola flipped over the y axis to convince yourself of how concave down implies a tangent slope that is decreasing. Then yes, apply descartes's information :)

  11. anonymous
    • 5 years ago
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    thank you both! But I guess I am confused with what the second derivative IS.

  12. anonymous
    • 5 years ago
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    I know that the first derivative is the expression for the slope of the tangent line; therefore, when it is zero, and there is a change in sign, we can find the extrema. But why is it when the second derivative=0, we find the inflection point? descartes sort of lost me on tracing through the first derivative...

  13. shadowfiend
    • 5 years ago
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    So, the first derivative describes the slope of the function at any given point.

  14. shadowfiend
    • 5 years ago
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    Basically it's a visualization (or description) of how the slope changes.

  15. anonymous
    • 5 years ago
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    yes.

  16. shadowfiend
    • 5 years ago
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    Similarly, the second derivative describes the slope of the *first* derivative at any given point.

  17. anonymous
    • 5 years ago
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    i depende on how the slope is

  18. anonymous
    • 5 years ago
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    thanks. I am still sort of muddy. Descartes said, "an inflection point (on the original function) becomes a max or min on the first derivative function and therefore a zero on the second derivative function. "

  19. anonymous
    • 5 years ago
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    why is that?

  20. shadowfiend
    • 5 years ago
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    In terms of the first function, it describes how fast and in what direction the function's slope is changing.

  21. shadowfiend
    • 5 years ago
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    So, as I was saying earlier, the inflection point is when the function changes from concave up to down or vice versa.

  22. shadowfiend
    • 5 years ago
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    This means that the *slope* of the function changes from *increasing* to *decreasing* or the other way around.

  23. shadowfiend
    • 5 years ago
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    This means that the slope of the first *derivative* changes from *positive* to *negative* or vice versa.

  24. shadowfiend
    • 5 years ago
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    Which means that the *value* of the *second* derivative changes from positive to negative or vice versa.

  25. shadowfiend
    • 5 years ago
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    And between positive and negative is zero :)

  26. anonymous
    • 5 years ago
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    that can make a diffrence in the question itself so the slop might be *increseing* or *decreseung* ethir way

  27. shadowfiend
    • 5 years ago
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    Does that make a little more sense?

  28. anonymous
    • 5 years ago
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    'slope* of the function changes from *increasing* to *decreasing* or the other way around"

  29. anonymous
    • 5 years ago
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    do you mean the original function?

  30. anonymous
    • 5 years ago
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    isn't that the extrema though?

  31. anonymous
    • 5 years ago
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    wait. I think I get it...

  32. shadowfiend
    • 5 years ago
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    I mean the original function, yes. And the extrema are where the slope of the original function changes from positive to negative, not where it changes from actually increasing to decreasing.

  33. anonymous
    • 5 years ago
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    :( what is the difference? I'm sorry.

  34. shadowfiend
    • 5 years ago
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    Increasing is a description of the *rate* at which something is changing. So if the slope of the original function is increasing, its slope is getting more positive *faster*. Instead of just getting more positive.

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