170 – Homework 5

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:

\displaystyle \lim_{x\to0}\frac{e^x-\sin x-1}{x^2}, \displaystyle \lim_{x\to+\infty}x\cos\bigl(\frac\pi2+\frac1x\bigr), \displaystyle\lim_{x\to1}(2-x)^{\tan(\pi x/2)}, and \displaystyle \lim_{x\to\frac\pi2^-}\frac{\tan x}{\ln\left(\frac\pi2 -x\right)}.

  • 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 x^3+3x-2=0.
  • Recall that in Newton’s method, we produce a sequence x_1,x_2,x_3,\dots of approximations to a solution \alpha of the equation f(x)=0, where each new approximation x_{n+1} is related to the previous one x_n via \displaystyle x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}. Recall as well that \displaystyle |\alpha-x_{n+1}|=\frac12\left|\frac{f''(c_n)}{f'(x_n)}\right| |\alpha-x_n|^2 for some c_n between x_n and \alpha.

Now consider the particular case where f(x)=x^2-10, so \alpha=\sqrt{10}. As first approximation, take x_1=3.

Begin by checking that x_n\ge 3 for all n\ge 1.

Find a number A>0 such that {}|f'(t)|\ge A for all t\ge3. In particular, for any n, this means that {}|f'(x_n)|\ge A.

Find a number B>0 such that {}|f''(t)|\le B for all t\ge 3. In particular, for any n, this means that {}|f''(c_n)|\le B. Conclude that \displaystyle \frac12\left|\frac{f''(c_n)}{f'(x_n)}\right|\le \frac{B}{2A}. Note that the right hand side is a constant C, it does not depend on n at all.

Check that we could have picked A,B so that this C\le 1/2. If the particular A,B 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 C is significantly smaller that 1/2, though 1/2 or even 1 works for the next part.)

Conclude from this that each x_{n+1} essentially duplicates the numbers of accurate digits. How large should n be if we want x_n to agree with \sqrt{10} for at least 100 digits?

  • Use that f(x)\approx f(a)+(x-a)f'(a) to approximate \sin(.1) and \sqrt{9.1}. Compare with the actual values.

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