# Lect12 - Defining the Derivative Consider the following two...

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92.131 Lecture 12 1 of 12 Ronald Brent © 2009 All rights reserved. Defining the Derivative Consider the following two problems: The rate-of-change problem: Let the function ) ( t D give the distance, in miles, traveled by a car up to t hours. Find ) 1 ( D , the instantaneous speed (the rate-of-change of distance with respect to time) at time 1 = t , or after one hour. The tangent-line problem: Let ) ( x f y = be a function. Find ) 1 ( f , the slope of the line tangent to the graph of ) ( x f at 1 = x .

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92.131 Lecture 12 2 of 12 Ronald Brent © 2009 All rights reserved. The Velocity Problem: The following data is for an object traveling in a straight line. t (hours) 0 . 00 . 51 . 01 . 5 Distance Traveled (miles) 0 10 20 30 40 50 60 70 Suppose we wish to find out how fast the object is going when t = 1 hour. Let’s look at a blowup of the graph about this point in time.
92.131 Lecture 12 3 of 12 Ronald Brent © 2009 All rights reserved. t (hours) 0.90 0.95 1.00 1.05 1.10 Distance Traveled (miles) 20 25 30 35 40 Samples of the object’s distance are taken every 3 minutes. If we approximate the speed using the interval [0.95, 1.0] we obtain 58 05 . 0 1 . 27 30 speed ous instantane = mph. Using the interval [1.0, 1.05] we approx. 62 05 . 0 30 1 . 33 speed ous instantane = .

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