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Introduction to Calculus

Sir Isaac NewtonGottfried Wilhelm

von Leibniz

Fathers of Calculus

Definition of Derivative

IntroductionLet'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 ?

           

                           

at f(x)function a of (a)f derivative The

.h

)a(f)ha(flim as defined is a x

0h

Tangent of a curve

Tangent

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

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).

An animation demonstrating the estimation of the slope

of the tangent by zooming in

Examples of Finding Derivatives

1.Find the derivative of a square root function at a point.

Solution

2.Find the derivative of a rational function at a point.

Solution.

Differentiability of a Function: A function is differentiable at x=a if

its derivative exists at x=a.Left-hand derivative Right-hand derivative

=

Functions which are not differentiable at x = 0

Group Discussion on Ex.1.7 Q.1

Group Discussion on Ex.1.7 Q.1

0

0.1

0.2

0.3

0.4

0.5

-4 -2 2 4x

0.5

1

1.5

2

2.5

-4 -2 0 2 4x

0

1

2

3

4

5

-4 -2 2 4x

Examples of Proving Differentiability

8.1 Is the function f(x) = |x| differentiable at x = 0? Proof

8.2 Is the function f(x) = |x3| differentiable at x = 0? Proof

Useful Websites

http://archives.math.utk.edu/visual.calculus/

http://www.ies.co.jp/math/products/

http://www-groups.dcs.st-and.ac.uk/~history

http://members.nbci.com/Sir_Lai/