Taylor’s Theorem and Taylor Polynomials

Taylor’s Theorem gives us a bound on the error that would result from using a Taylor polynomial \(P_n(x)\) to calculate the approximate value of a function \(f(x)\) at a given value.

Use this result to answer the following questions for our next lesson.

1. Consider the Taylor polynomials for \(e^x\), centred at \(a=0\).
   1. Using the fact that \(e^x\) is an increasing function, and \(e<3\), find a value of \(n\) such that \(|R_n(1)|<10^{-5}\).
   2. Hence, determine the value of \(e\) accurate to 4 decimal places.
2. 1. Generate the Taylor polynomial of degree \(3\) at \(x=0\) for the function \(f(x)=\ln(x+1)\).
   2. Hence, calculate an approximate value for \(\ln(1.1)\). Give a bound on the error of your approximation.