Topic 3: Early Methods for Finding Tangents

The history of calculus is fascinating! Do you know that the first person who formulated explicitly the ideas of limits and derivatives was Sir Isaac Newton in the 1660s? However, Newton acknowledged that "If I have seen further than other men, it is because I have stood on the shoulders of giants."

These giants were Pierre Fermat (1601 - 1665) and Newton's teacher at Cambridge, Isaac Barrow (1630 - 1677). Newton was familiar with the methods that these men used to find tangent lines, and their methods played a role in Newton's eventual formulation of calculus (quoted from Stewart's Calculus book).

In this topic, I want you to do some research online about:
1) Fermat's life and career (25 points)
2) Fermat's method in finding tangent lines (25 points)
3) Barrow's life and career (25 points)

Barrow's method in finding tangent lines is more complicated. However, you are welcome to try to understand this method and describe it here for 25 extra-credit points.

Please post your comments to briefly:
- describe each of the above items. It is important to give credit to the websites that your comments refer to. Please copy their links and paste them in your comments! Otherwise, half of your grade will be deducted.
- compare Fermat's method for finding tangents and the method we have learned in class (25 points).

The deadline to submit your comments is a week from now (i.e., Thursday, 09/27). Late submission will be graded zero.

THE DEADLINE IS EXTENDED TO THURSDAY (10/04) (i.e., a week from now).

PLEASE WRITE YOUR NAME AT THE BOTTOM OF YOUR COMMENTS FOR GRADING.

Topic 2: Limits and Continuity

Since we are learning about limits and continuity, I want to see how much you have understood these concepts. Please post your comments to answer the following questions:

1) In the 1st problem of the homework assignment of Sections 2.1 and 2.2, why are the average velocities and the instantaneous velocity at time t = 2s NEGATIVE?

2) Geometrically, what is "the line tangent to a curve" at a particular point?

3) What is the difference between the statements "f(a) = L" and "lim_{x->a} f(x) = L"?

4) Assume that f(1) = -5 and f(3) = 5. Does there have to be a value of x, between 1 and 3, such that f(x) = 0? (Think about the Intermediate Value Theorem).

5) Using the point-slope form, derive the equation of the secant line going through two points A(1,0) and B(2,1) and having the slope m = 1. Show me two equations:
- one derived from choosing A as the point in the point-slope form,
- the other derived from choosing B as the point in the point-slope form.
Are they different?

The deadline to submit your comments is a week from now (i.e., Thursday, 09/13). Each question weights 20 points. Late submission will be graded zero.

PLEASE WRITE YOUR NAME AT THE BOTTOM OF YOUR COMMENTS FOR GRADING.