Homework #3
Instructions. This is due on October 4 at
the beginning of class. It is worth 20 points. Late papers lose 4 points if
turned in Monday, plus 3 points per additional day or partial 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 start
with an introduction (to save space, you do not need to repeat the
Feynman story) and finish with a conclusion. Start early, ask questions and do
well.
Introduction. Richard Feynman was a famous (Nobel
Prize-winning) physicist who delighted in the power of the human mind. The
story that this assignment is based on appears in the Feynman book Surely
You’re Joking Mr Feynman as well as in the biographical movie Infinity.
Feynman challenged a salesman to a calculating contest, Feynman’s brain against the leading technology of the day (an abacus). A neutral observer selected the problem, which was to compute the cube root of 1729.03. Feynman came up with 12.002 before the abacus user could get 12. In relating this story, Feynman admits to some luck in the choice of problem, because he knew that a cubic foot contains 1728 cubic inches. As for the decimals, Feynman explained, “I had learned in calculus that for small fractions, the cube root’s excess is one-third of the number’s excess. The excess, 1.03, is only one part in nearly 2000. So all I had to do is find the fraction 1/1728, divide by 3 and multiply by 12.” In this assignment, we will analyze this explanation.
Problems. Start by finding the cube root of 1729.03 on the TI-89. Assuming
that this is correct, how well did Feynman do?
The formal calculus way of approximating this number
is to find the tangent line to the curve
y = x1/ 3 at x = 1728. Write this in the
form y = 12 + (1/3)(1/144) (x-1728). Then compute the y-value that corresponds to x = 1729.03. Compare
your answer to the calculator’s value.
Feynman basically did an informal version of the above procedure.
First, explain why Feynman’s knowledge of the cubic inches conversion factor
told him that the answer is slightly greater than 12. Note that this is the
first step in your calculation above. Feynman’s explanation says that “the cube
root’s excess is one-third of the number’s excess.” Identify which part of the
second step in the calculation is glossed over in this explanation. In his more
detailed explanation, this part shows up when Feynman takes 1/1728 and
multiplies by 12. Indicate how 12/1728 simplifies. Feynman says, “all I
had to do is find the fraction 1/1728, divide by 3 and multiply by 12.” Could
you do this in your head? I don’t think that Feynman did, either. He mentions
that the excess is only one part in nearly 2000. Replacing 1/1728 with 1/2000,
explain how to quickly divide by 3 and multiply by 12. What do you get if you
start with 12 and add this result? Finally, we know this approximation is off
because we rounded 1/1728 to 1/2000. Is the exact answer larger or smaller than
this approximation?