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Linear estimate

  1. Jun 3, 2007 #1
    Hey guys I was just taking a look at a sample exam and I came across this, with no recollection of ever learning it.

    I don't really want anyone to solve a problem for me per se, I just want an explanation for what is being asked.

    So anyway, I am asked to find the linear estimate for a function f(x) for all small values of x (close to 0).

    I'm not quite sure what a linear estimate is. Is it some sort of application of the linearization formula?

    L(x) = F(a) = F'(a)(x-a)

    Help would be appreciated. Thanks.
     
  2. jcsd
  3. Jun 3, 2007 #2

    cepheid

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    Yeah, exactly. Except that you mistyped your formula. It should be:

    L(x) = F(a) + F'(a)(x - a)

    In this problem, a = 0

    This is just a Taylor series expansion (about 0) in which you only keep the terms up to the first order term (that's called a first order Taylor expansion for short).

    Think about it...some well known functions do look approximately linear for values of x close to zero. Example:

    f(x) = sin(x)

    You can see that this is the case if you take a first order Taylor expansion, but you can also just look at a plot of sin(x) and see that it sure looks that way. As a result, for small values of x:

    sin(x) ~ x
     
  4. Jun 3, 2007 #3

    HallsofIvy

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    Or, equivalently, use the tangent line approximation.
     
  5. Jun 4, 2007 #4

    Gib Z

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    Basically you could think of this approximation as replacing a curve with its tangent. That is quite accurate close to the point of tangency, because at that point it has the same value and gradient, sort of like heading in a similar direction to the curve and therefore still somewhat accurate near the point of tangency.
     
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