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Taylor Polynomials

  1. Mar 30, 2004 #1
    Let f be a function that has derivatives of all orders for all real numbers. Assume f(1)=3, f'(1)=-2, f"(1)=2, and f'''(1)=4

    a. Write the second-degree Taylor polynomial for f about x=1 and use it to approximate f(0.7).
    b. Write the third-degree Taylor polynomial for f about x=1 and use it to approximate f(1.2).
    c. Write the second-degree Taylor polynomial for f', the derivative of f, and x=1 and use it to approximate f'(1.2).

    a. T{2}(x) = f(1) + f'(1)(x-1) + f"(1)(x-1)^2/2!
    plug in .7 = 3.51
    b. T{3}(x) = f(1) + f'(1)(x-1) + f"(1)(x-1)^2/2! + f"'(1)(x-1)^3/3!
    plug in 1.2 = 2.64
    c. What do I do for c?
     
  2. jcsd
  3. Mar 30, 2004 #2

    Hurkyl

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    Well, what are the 0-th, 1-st, and 2-nd derivatives of f'?
     
  4. Mar 30, 2004 #3

    HallsofIvy

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    You know how to differentiate a polynomial don't you?

    The third order Taylor's polynomial for f is T{3}(x) = f(1) + f'(1)(x-1) + f"(1)(x-1)^2/2! + f"'(1)(x-1)^3/3!= 3- 2(x-1)+2(x-1)2+ (2/3)(x-1)3.

    The second order Taylor's polynomial for f' is the derivative of that:
    -2+ 4(x-1)+ 2(x-1)2. Now put x= 1.2.
     
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