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Math Help - Linear Approximation Question

  1. #1
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    Linear Approximation Question

    Local Linear approximation
    Locar Linearization
    Tangent Line Approximation


    Do they all mean the same?
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  2. #2
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    Re: Linear Approximation Question

    Hey Mathinik.

    The Tangent Line Approximation usually refers to a one-dimensional approximation (or a particular approximation with respect to a partial derivative) or with respect to a directional derivative, but not in a multi-dimensional sense.

    The local linearization in a multi-dimensional senses uses what is called the Jacobian and this produces a linear approximation for all variables given all the partial derivatives and thus is not just a line but a plane in n-dimensions. This is usually an approximation (unless the object is linear itself which is a special case).

    The approximation is the same as the local linearization.

    You also have to consider what the word local means: local usually refers to have an approximation that is based around a particular point or region in the same way that you can expand a function with a known taylor series around a particular point rather than around 0. If you know derivative information around a certain point then you can use a linear approximation (keeping only the first order terms) around that point than having to use higher order terms and this is what is done frequently.
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