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This test consists of 5 short answer questions, 10 short essay questions, and 1 (of 3) essay topics.

Short Answer Questions

1. Who encourages Newton during his studies at Cambridge?

2. What did Newton's calculus involve?

3. What series was Euler most famous for?

4. What was true when Euler used n = 5 in the statement 2²ⁿ + 1?

5. Which of the following was NOT a field in which Isaac Newton made enormous advances?

Short Essay Questions

1. Summarize in a few sentences, what types of number sets did Cantor prove to be denumerable and non-denumerable.

2. Describe what the Bernoullis discovered about series, and give an example.

3. Why did Euler start working on the sum of series?

4. Explain why Eulers sum of π²/6 was in some ways surprising.

5. Describe some of Gauss's work.

6. Describe the connection between Fermat and Euler's work.

7. Describe some of the characteristics of Leonhard Euler, and what made him successful.

8. What was Gauss's major unpublished achievement in geometry?

9. Describe what mathematical and artistic movements are focused on in the second half of the 19th century.

10. What was Euler able to prove about 2²ⁿ + 1? Why was this a great accomplishment?

Essay Topics

Write an essay for ONE of the following topics:

Essay Topic 1

Write a three part essay on the lives and work of both Euler and Gauss.

Part 1) What was similar about Euler and Gauss from an early age? What characteristics did they have in common?

Part 2) Give some examples of great theorems proposed by Euler and by Gauss. Do you think their work had anything in common mathematically?

Part 3) What could be considered Euler's greatest work? What could be considered Gauss's greatest work? How did both mathematicians advance the study of mathematics?

Essay Topic 2

Debate in essay format one of the following controversies:

a) Newton versus Leibniz.

b) Cardano versus Tartaglia.

Explain a defense for each side of the controversy, then propose your opinion with your final conclusions.

Essay Topic 3

Write a three part essay to describe third century Alexandra.

Part 1) Describe the geographic location of Alexandria. Why did its location affect its scholars?

Part 2) Name and describe the work of a few scholars in third century Alexandria.

Part 3) Who was Heron? How did his achievements improve mathematics?

(see the answer keys)

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