2. Bit Portfolio of Lai Chi Keung

3. Introduction to Calculus Sir Isaac Newton Gottfried Wilhelm von Leibniz Fathers of Calculus

4. Definition of Derivative Introduction Let's think the slope of the secant line that goes through two points (a,f(a)) and (a+h,f(a+h)) on y = f(x). The slope is (f(a+h)-f(a))/h. When we make h smaller, what does (f(a+h)-f(a))/h approach ?

5. Tangent of a curve Tangent Derivative of a function at a point is in fact the slope of the tangent at that point.

6. A man is riding on the surf. We set f(x) as the curve of the wave. Observe the slope of the surfboard. The trace of the slope is the derivative of f(x).

7. An animation demonstrating the estimation of the slope of the tangent by zooming in

8. Examples of Finding Derivatives 1.Find the derivative of a square root function at a point. Solution Solution 2.Find the derivative of a rational function at a point. Solution.

9. Differentiability of a Function: A function is differentiable at x=a if its derivative exists at x=a. Left-hand derivative Right-hand derivative =

10. Functions which are not differentiable at x = 0 Group Discussion on Ex.1.7 Q.1

11. Examples of Proving Differentiability 8.1 Is the function f(x) = |x| differentiable at x = 0?ProofProof 8.2 Is the function f(x) = |x 3 | differentiable at x = 0?ProofProof

12. Useful Websites http://archives.math.utk.edu/visual.calcu lus/ http://www.ies.co.jp/math/products/ http://www-groups.dcs.st- and.ac.uk/~history http://members.nbci.com/Sir_Lai/