Approximating functions with Taylor polynomials and error bounds

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Intros
Lessons
  1. Approximating Functions with Taylor Polynomials and Error Bounds

    i) Taylor Polynomials and the Error Term

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Examples
Lessons
  1. Approximate ln 2 using the 3'rd degree Taylor Polynomial. Find the error term.
    1. Find the 4th degree Taylor Polynomial centred around a=0a=0 of f(x)=exf(x)=e^x. Then approximate e2e^2.
      1. Find the 2nd degree Taylor Polynomial centred around a=1a=1 of f(x)=(x+1)f(x)=\sqrt{(x+1)} and the error term where x[0,2]x \in [0,2].
        1. Show that f(x)=exf(x)=e^x can be represented as a Taylor series at a=0a=0.
          1. Show that f(x)=cos?xf(x)= \cos ?x can be represented as a Taylor series at a=0a=0.
            Topic Notes
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            To approximate a function with a Taylor Polynomial of n'th degree centred around a=0a=0, use
            f(x)f(a)+f(a)(xa)+f"(a)(xa)22!++fn(a)(xa)2n!f(x) \approx f(a) + f'(a)(x-a)+\frac{f^{"}(a)(x-a)^2}{2!}+ \cdots + \frac{f^n (a)(x-a)^2}{n!}

            where Pn(x)=f(a)+f(a)(xa)+f"(a)(xa)22!++fn(a)(xa)2n!P_n (x) = f(a)+f'(a)(x-a)+\frac{f^{"}(a)(x-a)^2}{2!}+ \cdots + \frac{f^n (a)(x-a)^2}{n!} is the Taylor Polynomial.

            To find the difference between the actual value and your approximated value, look for the error term, which is defined as
            Rn(x)=fn+1(z)(xa)n+1(n+1)!R_n(x)=\frac{f^{n+1}(z)(x-a)^{n+1}}{(n+1)!}

            Note that adding your Taylor Polynomial with your error would give you the exact value of the function. In other words,
            f(x)=Pn(x)+Rn(x)f(x)=P_n(x)+R_n(x)