Transcript Sample questions - Purdue University
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Study Guide
Who, when and in which publication, first squared the parabola?
• Does Archimedes consider the exposition in “The Method” a proof? Why or why not?
Who, when and in which publication first squared “higher” parabolas of the form x=py n .
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?
• • • • 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 x=py n ?
State Cavalieri's principle.
State Archimedes Theorem of the squaring of the parabola!
Show how to derive from this the integral of the function y=px 2 from a to b!
What principle does Archimedes use in “The method” to square the parabola?
• • • • Discuss Cavalieri’s paradox.
What do the terms, all lines, all squares and all cubes refer to in the proof of the squaring of x=py 3 .
Give a modern rendition of Cavalieri's proof.
Give and discuss Zeno’s paradoxes.
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Study Guide
What were the three main inputs into Leibniz’ theory of calculus?
• What was Berkeley’s critique on Leibniz’ calculations? Was it justified?
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?
• • • What is was the controversy between Leibniz and Newton? How did it end?
How does Cauchy define infinitesimals?
Show that x m is continuous in the style of Cauchy!
Calculate d(xy) a la Leibniz!
Give Leibniz' original version of the fundamental theorem of calculus.
• Derive the results of Cavalieri with the methods of Cauchy!
• How does Cauchy define integration?
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?
• • • Compare Cauchy’s definition to Riemann’s 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?