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Unformatted text preview: MIT OpenCourseWare http://ocw.mit.edu 18.01 Single Variable Calculus Fall 2006 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms . Lecture 1 18.01 Fall 2006 Unit 1: Derivatives A. What is a derivative? • Geometric interpretation • Physical interpretation • Important for any measurement (economics, political science, finance, physics, etc.) B. How to differentiate any function you know. d For example: e x arctan x . We will discuss what a derivative is today. Figuring out how to • dx differentiate any function is the subject of the first two weeks of this course. Lecture 1: Derivatives, Slope, Velocity, and Rate of Change Geometric Viewpoint on Derivatives Tangent line Secant line f(x) P Q x x + Δ x y Figure 1: A function with secant and tangent lines The derivative is the slope of the line tangent to the graph of f ( x ). But what is a tangent line, exactly? 1 Lecture 1 18.01 Fall 2006 • It is NOT just a line that meets the graph at one point. • It is the limit of the secant line (a line drawn between two points on the graph) as the distance between the two points goes to zero. Geometric definition of the derivative: Limit of slopes of secant lines PQ as Q P ( P fixed). The slope of PQ : → P Q (x +∆x , f(x +∆x )) (x , f(x )) Δ x Δ f Secant Line Figure 2: Geometric definition of the derivative lim Δ f = lim f ( x + Δ x ) − f ( x ) = f ( x ) Δ x Δ x Δ x Δ x →...
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This note was uploaded on 10/28/2010 for the course CHAP 12 taught by Professor Lebec during the Spring '10 term at Marlboro.
 Spring '10
 Lebec

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