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Questions and Answers
Who were the two mathematicians who independently developed calculus?
Who were the two mathematicians who independently developed calculus?
- Blaise Pascal and Rene Descartes
- Albert Einstein and Stephen Hawking
- Isaac Newton and Gottfried Leibniz (correct)
- Galileo Galilei and Johannes Kepler
What was the main difference between Newton's and Leibniz's approach to calculus?
What was the main difference between Newton's and Leibniz's approach to calculus?
- Newton and Leibniz used different notations to represent calculus.
- Newton considered variables changing with time while Leibniz thought of them as sequences of infinitely close values. (correct)
- Newton thought of variables as sequences of infinitely close values while Leibniz considered them changing with time
- Newton and Leibniz had the same approach to calculus.
Which notation was better suited for generalizing calculus to multiple variables and is widely used today?
Which notation was better suited for generalizing calculus to multiple variables and is widely used today?
- Riemann's notation
- Cauchy's notation
- Newton's notation
- Leibniz's notation (correct)
What are the three stages in the development of calculus?
What are the three stages in the development of calculus?
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What replaced infinitesimals in the development of calculus?
What replaced infinitesimals in the development of calculus?
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Who reformulated calculus in terms of limits?
Who reformulated calculus in terms of limits?
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What did putting calculus on a logical footing allow mathematicians to do?
What did putting calculus on a logical footing allow mathematicians to do?
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Study Notes
- Calculus was independently developed by Isaac Newton and Gottfried Leibniz.
- Newton considered variables changing with time while Leibniz thought of them as sequences of infinitely close values.
- Leibniz's notation was better suited for generalizing calculus to multiple variables and is widely used today.
- The development of calculus can be described in three stages: Anticipation, Development, and Rigorization.
- Infinitesimals were used in the development of calculus but were later replaced by a notion of quantities being "close" to others.
- Cauchy, Weierstrass, and Riemann reformulated calculus in terms of limits.
- Putting calculus on a logical footing allowed mathematicians to better understand and extend its results.
- Calculus is often taught in a backwards order, beginning with limits and defining the derivative and integral in the modern way.
- The historical development of calculus involved the use of infinite processes to find areas under curves and maximize certain quantities.
- Bishop Berkeley's criticisms of calculus focused mathematicians on a logical clarification of the subject.
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