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.