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Math Help - graphing w/out function

  1. #1
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    graphing w/out function

    I have an exam tomorrow and I've been going through old practice exams to prepare. Until the most recent, they've all had questions about finding various aspects of a graph given a function. Now I'm faced with a question asking me to sketch a graph and come up with a function given a few facts about the function.

    I have no idea how to tackle this and I can't find any examples online or in my book.

    Here's the problem:

    Sketch the graph of a function y = f (x) satisfying f (0) = 1,
    f ′ (0) = −3, f ′′ (0) = 6 and f ′′′ (0) = −6 in the vicinity of x = 0. Then find an example of a function f (x) satisfying these conditions. (Hint: Use the ‘shifting coefficients’ ( x^n /n! ) form of the polynomial and
    the power rule, or else just use standard power rule antidifferentation.) Identify its critical points, local maxima and minima, inflection points, and intervals of increase, decrease, and upward and downward concavity.


    Another question asks for the same facts for a function where the following is true:

    f is odd,
    f has exactly one local minimum at (2, 4)
    the intervals on which f is concave up are [-3, -1] and [0,1]

    If anyone can help me see how figuring this out is even possible and where to start, I would be very grateful.
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  2. #2
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    Lexington, MA (USA)
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    Hello, littlejodo!

    I'll do part of the first one . . .


    Sketch the graph of a function y = f(x)

    satisfying: . \begin{Bmatrix}f(0)&=&1 \\ <br />
f '(0) &=& \text{-}3 \\ f''(0) &=& 6 \\ f'''(0) &=& \text{-}6 \end{Bmatrix} .in the vicinity of x = 0.
    Examine what each statement tells us . . .

    f(0) = 1
    The graph has the point (0, 1)

    f'(0) = \text{-}3
    The slope is negative; the function is decreasing: \searrow

    f''(0) = +6
    The graph is concave up: \cup

    f'''(0) = \text{-}6
    The concavity is decreasing (it is "flattening out").


    So, near x = 0, the graph might look like this:
    Code:
              |
          *   |
              |
           *  |
            * |
             1*
              |   *
        - - - + - - - - * - - - - - -
              |                   *
              |
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