# lec01 - 1. Rates of Change and Tangents P. K. Lamm Lecture...

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1. Rates of Change and Tangents P. K. Lamm 08/30/11 (16:04) p. 1 / 5 Lecture Notes: 1. Rates of Change and Tangents 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 Example 1.1: Suppose a student is driving on an expressway from his home to the university, and that after about 30 minutes on the road one can approximate the distance he’s traveled from home using the following function, y = f ( t ) = 120 - 40( t - 2) 2 , where f ( t ) = distance from his home, in mi (miles), at time t, t = time, in hr (hours), for t . 5 . At one hour into clocking the trip, he notices a police car in the rearview mirror following him. What speed (velocity) is the student traveling at the moment he notices the police car? The speed of the car at time t = 1 hr is an instantaneous velocity which occurs at that one particular instant in time. To understand how to compute this velocity, we’ll ﬁrst compute the average velocity of the car over increasingly smaller intervals of time near t = 1. For example, the average velocity of the car over the time period [1 ,t 1 ] hr, for some t 1 > 1, is found using Δ y Δ t = change in distance change in time = f ( t 1 ) - f (1) t 1 - 1 mph . (1) In this case, f (1) = 120 - 40(1 - 2) 2 = 80 mi . The value of

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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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lec01 - 1. Rates of Change and Tangents P. K. Lamm Lecture...

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