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Consider the function \(f(x) = \ln(1+x)\).
  1. Find the Maclaurin polynomial of degree 3, \(P_3(x)\), for \(f(x)=\ln(1+x)\).
  2. Use this polynomial to approximate the value of \(\ln(1.2)\).
  3. The Lagrange form of the remainder term, \(R_n(x)\), is given by: $$ R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!}x^{n+1} $$ for some value \(c\) between \(0\) and \(x\). Write down the Lagrange form of the remainder \(R_3(x)\) for the approximation in part (b).
  4. By finding the maximum possible absolute value of \(R_3(0.2)\), determine the upper bound for the error in your approximation of \(\ln(1.2)\).

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