First of all, Zach Teitler pointed me to this game that you may find interesting. A human ship (that you control) is trying to avoid a plasma cannon operated by a robot. There are three levels, depending on whether the robot only has access to your position, or to your position and velocity, or to your position, velocity, and acceleration.
Now, this set is due Monday, April 15, at the beginning of lecture.
- L’Hôpital’s rule: Compute the following limits:
, , , and .
- Use the mean value theorem to solve Problem 47 in Chapter 11 from Spivak’s book.
- Apply Newton’s method to solve (with six digits of accuracy) the equation .
- Recall that in Newton’s method, we produce a sequence of approximations to a solution of the equation , where each new approximation is related to the previous one via . Recall as well that for some between and .
Now consider the particular case where , so . As first approximation, take .
Begin by checking that for all .
Find a number such that for all . In particular, for any , this means that .
Find a number such that for all . In particular, for any , this means that . Conclude that . Note that the right hand side is a constant , it does not depend on at all.
Check that we could have picked so that this . If the particular you first used do not satisfy this inequality, check that you can pick different values so that this is now true. (In fact, I expect you will get that is significantly smaller that , though or even works for the next part.)
Conclude from this that each essentially duplicates the numbers of accurate digits. How large should be if we want to agree with for at least digits?
- Use that to approximate and . Compare with the actual values.