2007-11-15
Today's nonsense
- Guy named "Eric" is observing today.
- Math club movie night tonight. "The Princess Bride".
- Whenever random answers are being spewed out, Spiller can't just stop the onslaught when the right answer is produced, so people just keep going.
- Four or five students in this class have gone through speech therapy (Dakota, Katie, Avery?, Bobby?, Bryan, ...)
Taylor's theorem and we had this remainder and now we will try to deal with this more. Use Taylor's theorem to determine the accuracy of the approximation.
2) arctan(0.5) is approximately 0.5 - (0.5)^3 / 3
F^(4) (x) = -24x(x^2 - 1) / (1+x^2)^4 ..... ............................ R_3 of (x) < = | F^(4) at some z * x^4 all over 4! | .... What's the range of z? Z falls between 0 and 1 .. so 0 < z < 0.5 (which is x). So now you'd have to take the derivative of the fourth derivative, and then use the number line test. Set your window from 0 to 0.5 and then plot the fourth derivative and then look for the maxima on that interval. So in this case it's something like .......... 4.6686. ........ R_3 (x) < = 0.01216.
ex) Determine the degree of the Maclaurin polynomial required so that the error in the function at the specified value of x is less than 0.0001.
a) f(x) = cosx Approximate f(0.5) .
"What this is going on here". Maclaurin polynomial, centered at zero, to approximate cosine function, and substite in point five for x, and we want it in within 1 ten thousandth. What degree do we need to go up to? What does n need to be?
f^(n+1) of (x) will always be < = 1 ...
R_n of (x) < = | f^(n+1) of (z) * x^(n+1) / (n+1)! | < 0.0001
And then we already know the worst case scenario for f^(n+1) of (x) as described above. So plug that "1" in for f^(n+1) ... and you then have to figure out what value for that statement makes it true etc. Look at your table values and find the first time that you come up with a value less than 0.1.
n=5
2) h(x) = e^x Approximate h(1.2) within 1 ten thousandth of the actual value.
n=8
Determine n for error < 0.001
g) f(x) = e^(2x) ... Approximate f(1.5)