One common approach in this course will be estimating an answer by adding together a series of terms. In some

cases, we will be adding a certain number of terms until a series converges. The last term that is included gives

one estimate of the error. The next term $($i$.$e$.$ the first term not used$)$ provides another estimate of the error.

In this problem, consider the infinite "Maclaurin" series for $cosx($with $x$ in radians$),$

$cosx=1-\frac{x^2}{2}!+\frac{x^4}{4}!-\frac{x^6}{6}!+\frac{x^8}{8}!$cdots

Starting with the simplest version, $cosx=1,$ add terms one at a time to estimate $cos\frac{}{5}$ using eqn $1.$ Afer each

new term is added, compute the true and approximate fraction relative errors. Reminder: the true error compares

numeric estimate to what you can figure out $($e$.$g$.$ cos function on a calculator or spreadsheet$)$ for $\frac{}{5}$rad$=36.$

The approximate error compares the amount that you added in the more recent term to your current numeric

estimate.

Compute the results that you obtain for $1,2,3,4,$ and $5$ terms.

SUGGESTION and plea: use a computer program, as practice.

Why?? The idea of using Excel or Matlab is to make the calculation not very repetitive for you. One goal of

the course is learning how to make your life easier when using computer tools. Think about how you can

write a simple formula and then cut and paste $($Excel$)$ or write sets of simple expressions $($Matlab$).$

You will learn in a thermodynamics class that the liquid$-$phase side of Raoult's Law $y_{i}P=x_i{P}_i^{\text{s}\text{a}\text{t}}$ is often

written as $x_i{}_i{P}_i^{\text{s}\text{a}\text{t}},$ where the activity coefficient ${}_i$ corrects for deviations from ideal solution behavior. In

some cases the activity coefficient is expressed mathematically as

$ln=(a_0-\frac{a_1}{T})(1+(c_0+\frac{c_1}{T})\frac{x_i}{1-x_i})$

where $a_0,a_1,c_0,$ and $c_1$ are constants, $T$ is absolute temperature, and $x_i$ is mole fraction. For this case, derive

an expression for the uncertainty in the vapor pressure $y_{i}P$ as a function of the uncertainty in the mole fraction

and temperature, assuming that the vapor pressure is calculated using an Antoine equation. Assume that the

uncertainties in the Antoine equation parameters and the $ln$ parameters are negligible.