Approximating functions with Taylor polynomials and error bounds

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  1. Approximating Functions with Taylor Polynomials and Error Bounds

    i) Taylor Polynomials and the Error Term

  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
            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

            Note that adding your Taylor Polynomial with your error would give you the exact value of the function. In other words,