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Math Help - local linear approximation

  1. #1
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    local linear approximation

    The local linear approximation of a function f will always be greater than or equal to the function's value if, for all x in an interval containing the point of tangency,

    (a) f ' less than 0
    (b) f ' greater than 0
    (c) f '' less than 0
    (d) f '' greater than 0
    (e) f ' = f '' = 0

    I need to be able to answer the question and explain why but I am not even sure what they mean by "local linear approximation"
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  2. #2
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    Quote Originally Posted by red_roses4 View Post
    The local linear approximation of a function f will always be greater than or equal to the function's value if, for all x in an interval containing the point of tangency,

    (a) f ' less than 0
    (b) f ' greater than 0
    (c) f '' less than 0
    (d) f '' greater than 0
    (e) f ' = f '' = 0

    I need to be able to answer the question and explain why but I am not even sure what they mean by "local linear approximation"
    a linear approximation is the approximation of a function value using a tangent line close to the location where one desires the approximation.

    here is a picture ...



    note that an approximation of f(a + \Delta x) taken from the line would be less than the actual value of f(a + \Delta x), because the tangent line lies below the curve.

    without looking at the picture, what property of the curve f(x) would tell you this?
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  3. #3
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    Quote Originally Posted by skeeter View Post
    a linear approximation is the approximation of a function value using a tangent line close to the location where one desires the approximation.

    here is a picture ...



    note that an approximation of f(a + \Delta x) taken from the line would be less than the actual value of f(a + \Delta x), because the tangent line lies below the curve.

    without looking at the picture, what property of the curve f(x) would tell you this?
    I am sorry. I am trying, but I really don't know what property you are talking about.
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  4. #4
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    think about the concavity of f(x)
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    if the f(a+delta x) is less than it should be, should the concavity always be positive?
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    ok, it took me a while, but I think i might be getting somewhere. Is the answer to the question C because, for the approximation, the blue line in skeeter's picture, to be above, or always greater than (except for at (a, f(a))) the actual line, the concavity must be negative?
    it would be greatly appreciated if someone could tell me if I am even headed in the right direction.
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  7. #7
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    Quote Originally Posted by red_roses4 View Post
    ok, it took me a while, but I think i might be getting somewhere. Is the answer to the question C because, for the approximation, the blue line in skeeter's picture, to be above, or always greater than (except for at (a, f(a))) the actual line, the concavity must be negative?
    it would be greatly appreciated if someone could tell me if I am even headed in the right direction.
    correct
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