## Homework
1 (due 1/14)

CSC 431

We covered material from Sections 1.1 and 1.2 of the textbook. I also used
examples from

O'Leary's
Scientific Computing with Case Studies. Next week we will cover section 1.3
(with additional material from O'Leary and Goldbach's

paper on floating-point
arithmetic). Any programming/calculation on the homework you can do in any
of the standard programming environments (C/C++, Java, etc.) I will be using

Python in class. You typically won't need
special math software for the homework, though if you want to play around with
some, Section 1.4 of the book lists many options (including open-source
projects), and also be sure to check out

Sage
(which came into existence after the book).

For a quick calculus refresher, check out the

SOS math pages.

**Submission**: you can submit the homework by hardcopy
in class or by sending it to me as an email.

1. [Reading] Read
Sections 1.1 and 1.2 of the textbook.

2. [Errors, 10pt] Average human body temperate was first reported by
Wunderlich in 1861 as being 37ºC. When converted into Fahrenheit (multiply by
9/5 and add 32), you get 98.6ºF, and so average body temperate
is always reported such. However, Wunderlich's result only had two significant
digits (the 3 and the 7), the Fahrenheit temperature has 3.

- Assuming Wunderlich rounded
his average to the nearest integer degree Celsius, what is the range of
possible corresponding temperatures in Fahrenheit?
- Recent studies show that there really is a range of normal temperatures
with an average temperature of 98.2ºF. Using this as the true value,
what are the absolute and relative errors of 98.6ºF?

3. [Calculus, 30pt]

- [20pt]
Develop the Taylor series for f(x) = cos x with x
_{0} = 0. Show your
work in determining the series.
- [5pt] Use the first two terms of the series to evaluate cos 1. When
running it on the computer, only use 3 significant digits (this includes the
0 before the period, simulate arithmetic with three significant digits by
rounding each intermediate value you to compute to three significant digits
[eg multiply by 100 and do an integer round, then divide by 100]). Determine
(i) the total computational error, (ii) the rounding error, and (iii) the
truncation error.
- [5pt] Calculate the relative error of computing cos 1 using the first
two and using the first three terms of the Taylor series.

Marcus Schaefer

Last updated: January 8th 2010.