Study Guide

1
Study Guide

• Who, when and in which publication, first squared
  the parabola?
• Who, when and in which publication first squared
  higher parabolas of the form xpyn.
• Who is credited with the invention of calculus?
• When was it invented?
• Who wrote the first modern textbook of calculus?
  When was it written and what is it called?
• State Archimedes Theorem of the squaring of the
  parabola!
• Show how to derive from this the integral of the
  function ypx2 from a to b!
• What principle does Archimedes use in The
  method to square the parabola?

• Does Archimedes consider the exposition in The
  Method a proof? Why or why not?
• What is the history and content of the text The
  method?
• What method does Archimedes use in his
  mathematical proof to square the parabola? Who
  invented this method and when?
• How did Cavalieri state his result on the
  integration of xpyn?
• State Cavalieri's principle.
• Discuss Cavalieris paradox.
• What do the terms, all lines, all squares and all
  cubes refer to in the proof of the squaring of
  xpy3.
• Give a modern rendition of Cavalieri's proof.
• Give and discuss Zenos paradoxes.

2
Study Guide

• What were the three main inputs into Leibniz
  theory of calculus?
• In what sense are forming partial sums and
  differences of sequences inverses to each other.
  How is this related to the fundamental theorem?
• What is d(x/y) in Leibniz terms?
• Calculate d(xy) a la Leibniz!
• Give Leibniz' original version of the fundamental
  theorem of calculus.
• When and where did it appear?
• Give the modern version of the fundamental
  theorem.
• How is the modern version related to Leibniz'
  version?
• What is are the assignable and inassignable
  triangles in Leibniz proof of the fundamental
  theorem and what role do they play?

• What was Berkeleys critique on Leibniz
  calculations? Was it justified?
• What is was the controversy between Leibniz and
  Newton? How did it end?
• How does Cauchy define infinitesimals?
• Show that xm is continuous in the style of
  Cauchy!
• Derive the results of Cavalieri with the methods
  of Cauchy!
• How does Cauchy define integration?
• Compare Cauchys definition to Riemanns
  definition.
• Who formulated the theory of integration is which
  is used today by mathematicians?
• What does the work of Robinson say about the use
  of infinitesimals? When did he do this work?