# 11 - ELEMENTARY MATHEMATICS W W L CHEN and X T DUONG c W W...

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P ( x 0 , y 0 ) Q ( x 1 , y 1 ) x yy = f ( x ) ELEMENTARY MATHEMATICS W W L CHEN and X T DUONG c ° W W L Chen, X T Duong and Macquarie University, 1999. This work is available free, in the hope that it will be useful. Any part of this work may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, with or without permission from the authors. Chapter 11 INTRODUCTION TO DIFFERENTIATION 11.1. Tangent to a Curve Consider the graph of a function y = f ( x ). Suppose that P ( x 0 ,y 0 ) is a point on the curve y = f ( x ). Consider now another point Q ( x 1 1 ) on the curve close to the point P ( x 0 0 ). We draw the line joining the points P ( x 0 0 ) and Q ( x 1 1 ), and obtain the picture below. Clearly the slope of this line is equal to y 1 y 0 x 1 x 0 = f ( x 1 ) f ( x 0 ) x 1 x 0 . This chapter was written at Macquarie University in 1999.

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P ( x 0 , y 0 ) x yy = f ( x ) P ( x 0 , y 0 ) x f ( x ) 11–2 W W L Chen and X T Duong : Elementary Mathematics Now let us keep the point P ( x 0 ,y 0 ) fxed, and move the point Q ( x 1 1 ) along the curve towards the point P . Eventually the line PQ becomes the tangent to the curve y = f ( x ) at the point P ( x 0 0 ), as shown in the picture below. We are interested in the slope oF this tangent line. Its value is called the derivative oF the Function y = f ( x ) at the point x = x 0 , and denoted by d y d x ¯ ¯ ¯ ¯ x = x 0 or f 0 ( x 0 ) . In this case, we say that the Function y = f ( x ) is di±erentiable at the point x = x 0 . Remark. Sometimes, when we move the point Q ( x 1 1 ) along the curve y = f ( x ) towards the point P ( x 0 0 ), the line does not become the tangent to the curve y = f ( x ) at the point P ( x 0 0 ). In this case, we say that the Function y = f ( x ) is not di±erentiable at the point x = x 0 . An example oF such a situation is given in the picture below. Note that the curve y = f ( x ) makes an abrupt turn at the point P ( x 0 0 ).
x y P ( x 0 , y 0 ) Q ( x 1 , y 1 ) y = x 2 x y 4 -2 -4 y x 2 Chapter 11 : Introduction to Diferentiation 11–3 Example 11.1.1. Consider the graph of the function y = f ( x )= x 2 . Here the slope of the line joining the points P ( x 0 ,y 0 ) and Q ( x 1 1 ) is equal to y 1 y 0 x 1 x 0 = f ( x 1 ) f ( x 0 ) x 1 x 0 = x 2 1 x 2 0 x 1 x 0 = x 1 + x 0 . It follows that if we move the point Q ( x 1 1 ) along the curve towards the point P ( x 0 0 ), then the slope of this line will eventually be equal to x 0 + x 0 =2 x 0 . Hence for the function y = f ( x x 2 , we have d y d x ¯ ¯ ¯ ¯ x = x 0 = f 0 ( x 0 )=2 x 0 . In particular, the tangent to the curve at the point (1 , 1) has slope 2 and so has equation y x 1, whereas the tangent to the curve at the point ( 2 , 4) has slope 4 and so has equation y = 4 x 4.

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11–4 W W L Chen and X T Duong : Elementary Mathematics Example 11.1.2. Consider the graph of the function y = f ( x )= x 3 . Here the slope of the line joining the points P ( x 0 ,y 0 ) and Q ( x 1 1 ) is equal to y 1 y 0 x 1 x 0 = f ( x 1 ) f ( x 0 ) x 1 x 0 = x 3 1 x 3 0 x 1 x 0 = x 2 1 + x 1 x 0 + x 2 0 .
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## This note was uploaded on 10/19/2010 for the course MATHEMATIC Math123 taught by Professor Goh during the Spring '10 term at UCLA.

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11 - ELEMENTARY MATHEMATICS W W L CHEN and X T DUONG c W W...

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