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Math Help - Finding f'(x) by looking at a graph.

  1. #1
    Member Marconis's Avatar
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    Finding f'(x) by looking at a graph.

    Reviewing for my final, I came across a question on an old final that is rather unfamiliar. The question has a graph of a function f(x), and it asks you to find the limits at certain values. The last two questions are f'(2) and f"(3). Strangely, we have never learned in class how to find the derivative by looking at the graph of a function, and my textbook barely touches upon it. I know what the definition of a derivative is, but when viewing it this way I am finding it difficult.

    How do I go about this? It is probably the most basic part of derivatives, but we never learned it! Thanks!
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  2. #2
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    Post the graph.
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  3. #3
    Member Marconis's Avatar
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    Ok, here's the graph. Just thinking about it, I'd say that f'(2) is 0, since the tangent line has a slope of 0 at that point. Is that correct?
    Now, what about the second derivative?

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    looks like an inflection point at x = 3 ... what does that tell you?
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    Now is this the f graph or f' graph?

    If it is the f graph, I would think f'(2)=0.
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  6. #6
    Member Marconis's Avatar
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    It's the f graph.

    I thought an inflection point is where concavity changes...it's changing at x=3?
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  7. #7
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    Quote Originally Posted by Marconis View Post
    It's the f graph.

    I thought an inflection point is where concavity changes...it's changing at x=3?
    yes ... what does that tell you about f''(3) ?
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    What does the 2nd derivative test tell you about the values of inflection points?
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    Member Marconis's Avatar
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    Quote Originally Posted by dwsmith View Post
    What does the 2nd derivative test tell you about the values of inflection points?
    Uh, where they are concave up or concave down?

    I don't know :-(.
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  10. #10
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    if f(x) is concave up, f''(x) > 0

    if f(x) is concave down, f''(x) < 0

    so ... what would f''(x) equal if it changes concavity at a differentiable point on the curve?
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  11. #11
    Member Marconis's Avatar
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    I'm lost. Sorry.
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  12. #12
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    f''(3) = 0
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  13. #13
    Member Marconis's Avatar
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    Why?
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  14. #14
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    Example

    \displaystyle f(x)=x^3+x^2+x+1\rightarrow f'(x)=3x^2+2x+1 \ \mbox{and} \ f''(x)=6x+2

    When set these equation equal to zero, we obtain points where the concavity changes (first derivative) and inflections change (2nd derivative).
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  15. #15
    Member Marconis's Avatar
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    Right, I forgot about that.

    So everytime it asks f"(x), it will always =0?
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