Homework 5

Due April 30

  1. Exercise 5.4 (page 219) in our text.
  2. Exercise 5.5 (page 219).
  3. What is the relation between Del Ferro's method and Cardono's? Why were they separate solutions then and the same solution now?
  4. Wikipedia (http://en.wikipedia.org/wiki/Binomial_theorem#Newton.27s_generalised_binomial_theorem) has a reasonable short discussion of Newton's binomial theorem for fractional exponents. Prove that his infinite series converges for $(1+x)^{1/2}$ when $|x| < 1$. How many terms would you need to get 10 decimal place accuracy for $\sqrt{1/2} = \sqrt{2}/2$? (This is really a calculus 2 problem - estimating the error in a series expansion - but you've all had calculus 2).
  5. Figure out or look up an "ice-cream cone" proof for the focus-directrix definition of a parabola.
  6. Use modern tools (calculus and analytic geometry) to prove some theorems on parabolas that Archimedes found in Euclid and used in his quadrature. In particular, referring to Figure 3.5 (page 121) in our text:

Show that

\begin{equation} EB = BD \end{equation}

and that

\begin{equation} MO:OP = CA:AO \end{equation}

Note: If you see why Equation (1) is a consequence of Equation (2) you need only prove Equation (2). If you can find a way to show that Equation (2) follows from Equation (1) then you only have to prove (1), which might be easier.

  1. (numbering should continue, not restart at 1). Find a formula for the sum of the first $n$ cubes using (Leibniz') idea: find a function $f$ such that $f(n+1)-f(n) = n^3$ (an "antiderivative" for $n^3$ ).
  2. Having found the right answer in the previous problem, prove it's the right answer in the old boring way, using the standard induction argument. You should see that you need exactly the same formal algebra, just deployed in a different place in the argument.
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