1. 1 The Elegance of Mathematics Calculus for the 7th Grade Keynotes at California Mathcounts 2003 Dr. Zhiqin Lu University of California, Irvine 03/22/2003

2. 3/22/20032 The title is from Brain Greene’s best selling book The Elegant Universe To professor Greene, the universe is elegant. To us, mathematics is elegant.

3. 3/22/20033 Let’s begin with a problem Find the value of the following:

4. 3/22/20034 Idea?

5. 3/22/20035 Then we have

6. 3/22/20036 What is the observation here? In order to find the value of We write

7. 3/22/20037 Then we have Theorem This is the idea of Calculus!

8. 3/22/20038 Another Example Back-to-the-envelope calculation

9. 3/22/20039 One more Example 1+2+…+99+100=? Carl Friedrich Gauss solved this problem when he was six! (but probably he didn’t use our method.) Can you do it? I am sure.

10. 3/22/200310 What is Calculus? Calculus=Differentiation+Integration Differentiation (or subtraction) : Integration (or addition) : The fundamental theorem of calculus states the duality between differentiation and integration. Theorem If, then If a is the differentiation of b, Then b is the integration of a.

11. 3/22/200311 Go to Infinity

12. 3/22/200312 Theorem (Fundamental theorem of Calculus)

13. 3/22/200313 The idea I have just showed you can be used to prove the Fundamental Theorem of Calculus

14. 3/22/200314 Of course, we need to make the above setting precise. However, the basic duality between differentiation and integration roots in the very simple duality property between subtraction and addition. That is the elegance of mathematics. You may be able to discover your own theorem by enjoying and paying attention to the mathematics you are learning now.