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Math Help - Numerical Methods Taylor Polynomial

  1. #1
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    Numerical Methods Taylor Polynomial

    Hey all, this one has me stumped,

    Q. The nth Taylor Polynomial for a function f at xo is sometimes referred to as the polynomial of a degree at most n that "best" approximates f near xo.

    Find the quadratic polynomial that best approximates a function f near xo = 1 if the tangent line at xo = 1 has equation y = 4x - 1, and if f''(1) = 6.

    I'd appreciate anyone taking a look at this
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by Locke333 View Post
    Hey all, this one has me stumped,

    Q. The nth Taylor Polynomial for a function f at xo is sometimes referred to as the polynomial of a degree at most n that "best" approximates f near xo.

    Find the quadratic polynomial that best approximates a function f near xo = 1 if the tangent line at xo = 1 has equation y = 4x - 1, and if f''(1) = 6.

    I'd appreciate anyone taking a look at this
    Observe that from this information, we can also conclude that f\!\left(1\right)=3 and f^{\prime}\!\left(1\right)=4.

    Thus, we have the taylor polynomial f\!\left(x\right)\approx f\!\left(x_0\right)+f^{\prime}\!\left(x_0\right)\l  eft(x-x_0\right)+\tfrac{1}{2}f^{\prime\prime}\!\left(x_0  \right)\left(x-x_0\right)^2 =3+4(x-1)+3(x-1)^2

    I leave it for you to simplify that expression.

    Can you take it from here?
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  3. #3
    MHF Contributor Calculus26's Avatar
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    the 2d Taylor polynomial is:

    f(x0) + f ' (x0)(x-x0) + [f "(x0)/2](x-x0)^2

    You already have the first 2 terms from the tangentline we have

    -1 + 4x +[ f " (1)/2](x-1)^2

    = -1 + 4x + 3(x-1)^2

    = 3x^2 - 2x + 2
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  4. #4
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    Thanks to the both of you, your posts were a great help
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