### Calculus research project

Calculus research project

Calculus Project:

You will choose one of the eleven project below. The project does not have a minimum page requirement. If you can complete the project in one page, all the power to you.

Project guidelines:

1. You have a lot of flexibility in choosing and completing your project of choice.

2. You will need to cite any references using the APA style format. Please refer to the following two websites:

http://citationmachine.net/

http://owl.english.purdue.edu/owl/resource/560/01/

3. The project due date is listed on the tracking calendar.

Please contact me if you have any questions.

Calculus projects – choose one

(mostly stolen from David Richeson, Dickinson College)

1. (History) Biographical sketch of Newton’s life and Newton’s contribution to the Calculus. You have a lot of freedom in this project. Your paper should include each of the following.

(a) Give biographical information about Newton’s life such as when and where he lived, what he was like as a person, etc.

(b) Give information about his major scientific accomplishments.

(c) Discuss his role in the development of calculus. Be as concrete as possible.

(d) Compare Newton’s view of calculus to our modern view.

(e) You may want to discuss the Newton-Leibniz controversy over the invention of calculus.

2. (History) Biographical sketch of Leibniz’s life and Leibniz’s contribution to the Calculus. You have a lot of freedom in this project. Your paper should include each of the following.

(a) Give biographical information about Leibniz’s life such as when and where he lived, what he was like as a person, etc.

(b) Give information about his major academic interests and accomplishments.

(c) Discuss his role in the development of calculus. Be as concrete as possible.

(d) Compare Leibniz’s view of calculus to our modern view.

(e) You may want to discuss the Newton-Leibniz controversy over the invention of calculus.

3. (History) Newton and Leibniz calculus controversy. You have a lot of freedom in this paper. Here are some topics you should include.

(a) Discuss Newton’s contribution to calculus.

(b) Discuss Leibniz’s contribution to calculus.

(c) Discuss how their view of the calculus differs from today’s view.

(d) Discuss the controversy over the invention of calculus.

(e) What was the opinion at the time? What is the opinion today?

4. (Related rates) The falling ladder paradox. The article The falling ladder paradox,by Paul Scholten and Andrew Simpson (College Math. Journal, Jan. 1996) investigates the classic falling ladder problem and the paradoxical situation that seem inevitable.

(a) State and solve the classic falling ladder problem (you may assume that the ladder is 40 ft long and that the base is moving at a rate of 5ft/sec).

(b) What is the paradox?

(c) Find the location at which the top of the ladder is traveling at light speed.

5. (Physics) How not to land at Lake Tahoe. The article How Not to Land at Lake Tahoe!, by Richard Barshinger (Amer. Math. Monthly, May 1992) creates a model of a descending airplane for various situations. He is very sketchy with the details of his argument. Rewrite his analysis including all of the details so that the argument easy to follow.

6. (Physics, optimization) Refraction and reflection of light: Snell’s law

(a) What is Fermat’s principle of optics?

(b) Suppose a light is shined from point P to point Q by reflecting it off of a mirror. P is located a units from the mirror and Q is located b units from the mirror, the horizontal distance between P and Q is L. Find the point where the light reflects off the mirror. Show that the angle of incidence equals the angle of reflection.

(c) Give an interesting example with actual numbers for a, b and L.

(d) State and prove Snell’s law. In particular, suppose a light is shined from point P to point Q where P is located in the air a units above the pool of water (or any fluid), Q is located b units below the surface of the water, and the horizontal distance between P and Q is L. Assume that v1 is the velocity of the light in air, v2 is the velocity of light in water and that the beam of light makes angles A1 and A2 with the vertical (y-axis) in the air and water respectively. Find a relationship between v1, v2, A1, and A2.

(e) Be sure to define the index of refraction and the refraction angle in your discussion.

(f) €Give an interesting example with actual numbers for a, b and L.

7. (Optimization) Do dogs know calculus? For this project you may wish to consult the article Do dogs know calculus? by Timothy Pennings (College Math. Journal, May 2003).

(a) Suppose a dog can run with a velocity vl on the land and can swim with a velocity of vw. You and the dog are standing on the shore of a lake and you throw a ball in the water. The ball lands a feet off shore and b units to your right. What route should the dog take to reach the ball fastest?

(b) Pick reasonable values for vl, vw, a and b and compute the time for this optimal path. Also, compute the time for the direct (swimming only) route and the “right angle” route.

8. (Optimization) A new wrinkle on an old folding problem. This project is based on the article A new wrinkle on an old folding problem by Greg Frederickson (College Math. Journal, 2003).

(a) State and prove the classic “box folding” problem. That is, given a piece of paper of length l and width w, find the largest box that can be made by cutting squares out of the corners.

(b) Discuss the history of this problem.

(c) Reproduce the results of the paper. The author is very sketchy with the details of his argument. Rewrite his analysis including all of the details so that the argument easy to follow.

9. (History) History of calculus in Egypt, Greece, and India. It’s generally agreed that Newton and Leibniz invented calculus, but other people came close, some of them long before and far away from those two.

(a) Give a brief history of Egyptian (e.g., the Moscow mathematical papyrus, Alhazen), Greek (e.g., Eudoxus, Archimedes), and Indian (e.g, Narayana et al.) contributions to calculus.

(b) In what sense is it true that each of these groups “knew calculus”?

(c) In what sense is it false? (Some possible starting points for this project are Joseph’s book The Crest of the Peacock and several articles by David Bressoud. Some history of math or history of calculus books have a little information on the subject.)

10. (History) Hyperbolic functions and their history. We skipped the section on the hyperbolic functions. In this project, you’ll fill us in.

(a) Define the hyperbolic functions and discuss their properties, practical uses, and derivatives.

(b) How are they like the trig functions? How do they differ?

(c) Discuss their origin and history. (One possible source is Barnett’s article “Enter, Stage Center: The Early Drama of the Hyperbolic Functions” (Math. Magazine, 77 #1).)

11. (History) Women in calculus. A great place to start on this project is Dr. Riddle’s website, Biographies of Women Mathematicians, http://www.agnesscott.edu/lriddle/women/women.htm .

(a) Discuss some of the contributions that women have made to the development of calculus.

(b) What factors kept women from contributing more? (Why are almost none of the results in your textbook named after women?)

(c) To what extent are these factors still in existence? (You may want to interview students and faculty about their experiences).