Homework #4
Instructions. This is due on October 24
at the beginning of class. It is worth 20 points. Late papers lose 3 points per
day late. You must work by yourself, but feel free to consult with me. Explain
what you did and why you did it: good explanations are essential! Your paper
should have an introduction, a main body and a conclusion. Start early, ask
questions and do well.
Introduction. Taylor series are used in
almost every field that uses mathematics. This assignment explores a common use
of Taylor series: finding a simple but accurate approximation.
Problems. Newton’s universal law of gravitation states that the weight of an object at altitude x miles above the surface of the earth is given by
w(x) = wR2 / (R+x)2
where w is the weight of the object at the surface of the earth and R is the radius of the earth (about 4000 miles). Show that
w(x) » w (1 - 2x/R )
for x near 0. Use this approximation to estimate how large x has to be to reduce weight by 1%.
Find the second degree Taylor polynomial for w(x) and use it to estimate how large x has to be to reduce weight by 1%. Compare your answer to the estimate obtained earlier. Is the difference significant? Do you think the estimate would change significantly if a third-degree polynomial was used?
Based on your calculations, answer the following questions. For events on earth, we often assume that weight is constant. How reasonable is this assumption? A “high-altitude” city like Mexico City has elevation 7500 feet. Would a 200-pound person’s weight be noticeably different in Mexico City? All of the above work assumes that the earth is a perfect sphere. Actually, the earth is an oblate spheroid that is fatter at the equator than at the poles. If the “radius” of the earth is 300 miles larger at the equator than at the poles, does altitude or latitude have more of an effect on weight?