History of Calculus, Test 1

3 October 2000

Possible answers

1. What are the main problems that calculus is used to solve? Finding areas enclosed by curves, such as the area of the circle, parabola, spiral and cycloid. Finding tangent lines to curves. Calculating rates of change. Maximization and minimization problems.

2. What contributions did Archimedes make toward solving these problems? Squaring the circle: the area of a circle is the same as a right triangle with one side the radius and the other the circumference. The volume of a sphere is 2/3 that of the circumscribed cylinder. Quadrature of the parabola: the area of a segment of a parabola is equal to 4/3 the area of the biggest inscribed triangle. The area enclosed by one turn of the spiral is 1/3 that of the circle going through the last point. Calculated approximations to the length of the circumference of a circle (p).

3. What is the binomial theorem? How did Newton extend it?

   (a+b)n = an + nan-1b + n(n-1) 2! an-2b2 + n(n-1)(n-2) 3! an-3b3 + ¼ + nabn-1 + bn

   where n is a positive integer. Newton extended this theorem to fractional and negative n, in which case the sum on the right does not stop because n(n-1)(n-2)¼ is never equal to zero. You get an infinite series on the right.

4. Simmons suggests that "Fermat was the true creator of [differential calculus] as early as 1629, more than a decade before either Newton or Leibniz was born." Why did he say that? Fermat's method for finding maxima and minima was to write down the equation

   é ê ë f(x+E) - f(x) E ù ú û E = 0  = 0

   which is essentially setting the derivative of f, computed from scratch, equal to zero. He was also the first to come up with the "calculus definition" of the tangent line to a curve, and developed methods for drawing them.

5. Describe the curve known as a cycloid. Discuss the problem of calculating its area. The cycloid is traced about by a point on the rim of a wheel as the wheel rolls along the ground. Galileo conjectured that the area enclosed by one arch of the cycloid was approximately, but not exactly, equal to three times the area of the wheel. Roberval, using Cavalieri's principle, showed that the area was equal to three times the area of the wheel.

6. What is Cavalieri's principle? What good is it? What does it have to do with calculus? Who was Cavalieri? See page 23 for statements of the two principles. They are used to calculate areas and volumes by comparing a figure of unknown area or volume with one of for which it is known. The idea of calculating an area by slicing it up into lines is the fundamental idea of the integral calculus. Cavalieri (1598-1647) studied with Galileo, was a professor of mathematics at the University of Bologna, and one of the most influential mathematicians of his time.