Quantifying the accuracy of approximations

In this activity we approximate the values of several different alternating series using the Alternating Series Estimation Theorem.

1. Use the fact that $\sin(x) = x - \frac{1}{3!}x^3 + \frac{1}{5!}x^5 - \cdots$ to estimate $\sin(1)$ to within $0.0001$. Do so without entering “$\sin(1)$” on a computational device. After you find your estimate, enter “$\sin(1)$” on a computational device and compare the results.

2. Recall our recent work with $\int_0^1 e^{-x^2} \, dx$ which can be expressed as the series . Use this…

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…

In this exercise we consider the definite integral from two different perspectives.

1. First, find the Taylor series for $\frac{4}{1+x^2}$ and use it to evaluate as an infinite series of real numbers.

2. Observe that your result in (a) is an alternating series. Estimate the value of that alternating series to within $0.01$. How many terms of the series are needed to do so?

3. Recall that $\frac{d}{dx}[\arctan(x)] = \frac{1}{1+x^2}$. Use this fact and the First Fundament…