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Unformatted text preview: Lecture 13 Derivatives and rates of change (contd) (Relevant section from Stewart, Sixth Edition: Section 2.7) In the previous lecture, we examined very briefly the following situation from Physics: A particle is moving in one dimension, e.g., along the xaxis, and its position as a function of time is denoted by the function x ( t ), as plotted below. slope = x ( t 1 ) t 1 t 2 t tangent line at t = t 1 y = x ( t ) x ( t 1 ) x ( t 2 ) y x t If we consider a time interval [ t 1 ,t 2 ] for t 1 < t 2 , then the quotient, x ( t 2 ) x ( t 1 ) t 2 t 1 = x t , (1) defines the average velocity of the particle over the time interval [ t 1 ,t 2 ]. We then supposed that the time t 2 was allowed to approach t 1 in an effort to obtain a better estimate of the rate of change of position with respect to time of the particle at t = t 1 . In the limit, provided that it exists, we have lim t 2 t 1 x ( t 2 ) x ( t 1 ) t 2 t 1 = lim t x t = x ( t 1 ) . (2) This is the derivative of x ( t ) at t = t 1 which is the instantaneous rate of change of the position with respect to time of the particle at t 1 . Once again, as you well know, this is the velocity of the particle at time t 1 , denoted as v ( t 1 ). Historically, the idea of letting the time t 2 approach t 1 in an effort to describe the motion of the particle over smaller and smaller time represented a monumental jump in thought which led to 87 the development of Calculus. Some more comments on this procedure will be given in this weeks tutorials. We now return to do some mathematics. The derivative as a function Given a function f ( x ), we now define its derivative at a general point x (as opposed to an anchor point a ): f ( x ) = lim h f ( x + h ) f ( x ) h , (3) provided that the limit exists. The derivative is now a function of x , i.e., we could write, g ( x ) = f ( x ) , (4) to emphasize that it is a new function. Lets apply this idea to some simple functions. Example 1: The worlds simplest function f ( x ) = C , where C is a constant. (You know the answer to this, but lets do it anyway.) The Newton quotient of f at an arbitrary x is f ( x + h ) f ( x ) h = 1 1 h = 0 . (5) As a result, f ( x ) = lim h f ( x + h ) f ( x ) h = lim h 0 = 0 . (6) Example 2: The worlds nexttosimplest function f ( x ) = x . (Yes, once again, you know the answer.) The Newton quotient of f at an arbitrary x is f ( x + h ) f ( x ) h = ( x + h ) x h = h h = 1 . (7) As a result, f ( x ) = lim h f ( x + h ) f ( x ) h = lim h 1 = 1 . (8) Example 3: The worlds nexttonexttosimplest function f ( x ) = x 2 . The Newton quotient of f at an arbitrary x is f ( x + h ) f ( x ) h = ( x + h ) 2 x 2 h = 2 xh + h 2 h = 2 x + h. (9) 88 As a result, f ( x ) = lim h f ( x + h ) f ( x ) h = lim h (2 x + h ) = lim h 2 x + lim h h = 2 x. (10) Example 4:...
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 Fall '08
 SPEZIALE
 Calculus, Derivative

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