The assignment is based on taking points off from 10 points a graded problem
and adding a base score of 20 to get a total of 100 points. Since 8(b) was
done in class, I did not grade it.
The assignment is based on taking points off from 10 points a graded problem
and adding a base score of 20 to get a total of 100 points. Since 8(b) was
done in class, I did not grade it.
The textbook uses a tilde over characters as the computational approximation
to a number. I cannot do that so easily in HTML, so I am using a boldface
x instead of x-tilde.
| Problem | Answer and grading |
| 1(b) | Ey = y - y = 350 |
| Ry = Ey/y = 3.5532x10-3 | |
| 10-3/2 < Ry < 10-2/2 ; hence d = 2 significant digits | |
| -1 for E or R wrong, -2 for d wrong, -3 for d missing. | |
| 1(c) | Ez = .000008 |
| Rz = 1.1764x10-1 | |
| 10-1/2 < Rz < 100/2 ; hence d = 0 significant digits | |
| -1 for E or R wrong, -2 for d wrong, -3 for d missing. | |
| 3(b) | p1 + p2 = 34.442 (5 significant digits) |
| p1 p2 = .85392 (5 significant digits) | |
| -1 each wrong sum, product, or digits. | |
| 4(b) | ( ln( 2 + .00005 ) - ln( 2 ) ) / .00005 = .00002499969 / .00005 = .49999... |
| Loss of significance in the subtration and division. The .00005 could be questioned as to a previous loss of significance, leading one to wonder just how many digits of accuracy there really are in this problem (1, 5, 7?). | |
| -1 wrong number, -2 for wrong or no explanation. | |
| 5(b) | ( x2 + 1 )1/2 - x for large x... There are multiple answers to this one, none of which really does the job down to the last ulp. |
| (a) csc(a) - 1 for a = tan-1x (or sec(a) - x). | |
| This assumes that someone write sec, csc, and/or tan-1 functions which are accurate to the last bit. This is probably true these days on some hardware platforms, but... | |
| (b) ( ( x2 + 1)1/2 + x )-1. | |
| Addition like this and squaring can cause overflows and division is notorious for losing half the precision. This is not as good as (a). | |
| -5 for an unacceptable function, -2 for not simplying enough. | |
| 5(c) | abs( cos( x/2 ) ). |
| -5 for an unacceptable function, -2 for not simplying enough, -1 for missing absolute value. | |
| 8(a) | ( p+q+r) - (p+q+r) = Ep+Eq+Er. |
| Hence the error perturbations add. Note that the errors can cancel under certain round off schemes. | |
| -4 for wrong expression, -2 for missing explanation. | |
| 8(c) | ( pqr) - (pqr) = Rp+Rq+Rr. |
| Hence the relative error perturbations add. | |
| Alternately, you can look at the expansion directly and derive an acceptable explanation based on the relative sizes of the error perturbations and the multipliers. | |
| -4 for wrong expression, -2 for missing explanation. |
Cheers,
Craig C. Douglas