# lec09 - 8 Derivative as Rate of Change Lecture Notes P K...

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P. K. Lamm 09/29/11 (16:26) p. 1 / 4 Lecture Notes: 8. Derivative as Rate of Change These classnotes are intended to be supplementary to the textbook and are necessarily limited by the time allotted for classes. For full and precise statements of deﬁnitions and theorems, as well as material covering other topics and examples, please consult the textbook. 1. Rates of Change Let s ( t ) = the distance (in miles) traveled at time t (in hours) by an automobile going due east. Then the average velocity over a time period of length h , say [ t,t + h ], is the average rate of change of distance over the [ t,t + h ] time period. That is, average velocity over time period [ t,t + h ] = s ( t + h ) - s ( t ) ( t + h ) - t = s ( t + h ) - s ( t ) h If we then let h 0, we should get the instantaneous rate of change of distance at time t , i.e., instantaneous velocity at time t = lim h 0 s ( t + h ) - s ( t ) h = s 0 ( t ) . Thus, if s ( t ) denotes position of an object at time t , v ( t ) = s 0 ( t ) denotes the (instantaneous) velocity of the object at time t . Note that v ( t ) is a signed quantity, i.e., it can be positive or negative, so it has built into it some directional information. • | v ( t ) | denotes the speed of the object at time t . Speed is an absolute quantity, always nonnegative. a ( t ) = v 0 ( t ), or the rate of change of velocity at time t , denotes the acceleration of the object. Like velocity, acceleration is a signed quantity so directional information is included. Example 1.1: A particle moves along a horizontal line in such a way that its position in meters at time t seconds is s ( t ) = t 2 - 4 t + 3 , t 0 . 1. What is average velocity of the particle over the interval [3 , 4] seconds? s (4) - s (3) 4 - 3 = (4 2 - 4 · 4 + 3) - (3 2 - 4 · 3 + 3) 1 = 3 - 0 1 = 3 m/s . 2.

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## This note was uploaded on 04/02/2012 for the course MTH 132 taught by Professor Kihyunhyun during the Fall '10 term at Michigan State University.

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lec09 - 8 Derivative as Rate of Change Lecture Notes P K...

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