Activity 8.6.3

In this activity we apply the Lagrange Error Bound to quantify the accuracy of several different approximations.

1. Use the degree $10$ Taylor polynomial (centered at $a = 0$) of $f(x) = e^x$ to estimate the value of $e^2$. What is the maximum error of your estimate, according to the Lagrange Error Bound? How does this compare to the actual error between $e^2$ and $T_{10}(2)$ as reported by a computer algebra system?

2. Use a degree $n$ Taylor polynomial (centered at $a = 0$) of $f(x) = \cos(x)$ to estimate the value of $\cos(1)$ within an accuracy of $0.00000001$. According to the Lagrange Error Bound, what value of $n$ is needed to achieve this accuracy? What is the resulting approximate value of $\cos(1)$?

3. Recall that for $f(x) = \ln(1+x)$, its Taylor series centered at $a = 0$ is given by $$\begin{gathered} \ln(1+x) = x - \frac{1}{2}x^2 + \frac{1}{3}x^3 - \cdots + (-1)^{n-1}\frac{1}{n}x^n + \cdots \end{gathered}$$ and that the $n^{\text{th}}$ derivative of $f(x) = \ln(1+x)$ is given by $$\begin{gathered} f^{(n)}(x) = \frac{(-1)^{n-1}(n-1)!}{(1+x)^n} \end{gathered}$$. If we want to estimate $f(0.5) = \ln(1.5)$ to within an accuracy of $0.0001$, what value of $n$ is needed to achieve this accuracy from computing $T_n(0.5)$, according to the Lagrange Error Bound?