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This test consists of 5 multiple choice questions, 5 short answer questions, and 10 short essay questions.

## Multiple Choice Questions

**1. What did Newton's calculus involve?**
**(a)** Determining the area under a curve. **(b)** Proving the existance of pi. **(c)** Proving the cubic equation. **(d)** Determining the volume of a sphere.

**2. In the Bernoulli's time, what was the current definition of a series?**
**(a)** The sum of a finite series of terms. **(b)** The infinite sum of a convergent series. **(c)** The finite sum of a divergent series. **(d)** The sum of a never-ending series of terms.

**3. What series was Euler most famous for?**
**(a)** 1 + 1/2 + 3/4 + 4/5 . . . **(b)** 1 + 1/2 + 1/6 + 1/10 + 1/15 . . . **(c)** 1 + 1/4 + 1/9 + 1/16 . . . + 1/k² . . . **(d)** 1 + 1/2³ + 1/3³ + 1/4³ . . . 1/k³ . . .

**4. In what area was Gauss especially interested?**
**(a)** The proof of the infinite series. **(b)** The elements of number theory. **(c)** The elements of geometry. **(d)** The circumference of Earth.

**5. What did Cantor suspect about transfinite cardinals?**
**(a)** That the sum of the series is infinite. **(b)** That there are transfinite cardinals much less than c. **(c)** That the sum of the series is finite. **(d)** That there are transfinite cardinals even greater than c.

## Short Answer Questions

**1.** Which of the following is a series that the Bernoullis proposed did not converge on a finite sum?

**2.** On who's work did Euler base his number theory?

**3.** How did Gauss feel about his best work?

**4.** Which of the following was a major part of Gauss' work in mathematics?

**5.** What didn't Euler attempt?

## Short Essay Questions

**1.** Describe Newton's days in Cambridge and what he eventually came to discover.

**2.** Explain why Eulers sum of Ï€Â²/6 was in some ways surprising.

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

**4.** Explain how Gottfried Leibniz was able to publish his method of calculus.

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

**6.** What was Gauss's major unpublished achievement in geometry?

**7.** Who was Georg Cantor, and what was significant about his work in mathematics?

**8.** Describe some of Gauss's work.

**9.** Describe Cantor's difficult personal life.

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

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