<p>We're doing Taylor's Theorem in class and are supposed to determine the accuracy of an approximation. We have to determine the accuracy (basically, find the error value) of the following:</p>
<p>arcsin(.4) = 0.4 + [(0.4)^3]/3!</p>
<p>Please give VERY detailed solution. I have no idea what I'm doing, just following the book's example. After finding the 4th derivative (for the remainder), I don't know what to do, or rather, don't really understand what I'm doing and why it's being done. Understanding's key and it's just not clear.</p>
<p>Please, please, please help. Thank you.</p>
<p>Cheers!</p>
<p>Wow we're on the exact same thing...and if I knew I'd definitely share...but I'm thoroughly confused as well.</p>
<p>hope someone will come through, heh.</p>
<p>isn't this called euler's method? or is that something else... taylor, newton, euler... it all blends together after a while... but if it is it has something to do with recursives... where the value of x1 with a chosen value for x is put into deriv of f(x) of x + x over and over... sorry i couldn't be of more help</p>
<p>arcsin(.4) = 0.4 + [(0.4)^3]/3!</p>
<p>Ok. The real value you can punch into your calculator as arcsin (.4)
The approximation you have is correct. Subtract the difference.</p>
<p>It should look like:</p>
<p>Error = arcsin (.4) - .4 - [(0.4)^3]/3! = .0008501794.</p>
<p>The more terms you use, the more accurate the approximation will be.</p>
<p>The following formula is helpful: Error = l Real - Approximation l</p>
<p>The formula you were using is the Remainder Estimation for an Alternating Series. Picture those series as a pendulum that is losing speed with every swing. You add a term, subtract a term, add a term...until you reach the real value (when the pendulum is stopped). The reason the next term is the largest possible remainder is because after that term you subtract a smaller term, and then add an even smaller term, such that the addition and subtraction can never rise above the n+1 term. We finished the book 3 weeks ago so if you have any other questions IM me @ dwintz02. Hopefully that is what you were looking for!</p>