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handout 13-2

handout 13-2 - Math 200 Spring 2010 Handout 6 Section 13-2...

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Math 200, Spring 2010 Handout 6 Section 13-2 Calculus of Vector-Valued Functions Definition: Limit of a Vector-Valued Function A vector-valued function ( ) t r approaches the limit v as t approaches 0 t if and ( ) 0 lim 0 t t t = r v . In this case, we write ( ) 0 lim t t t = r v Theorem 1: Vector-Valued Limits Are Computed Componentwise A vector-valued function ( ) ( ) ( ) ( ) , , t x t y t z t = r approaches a limit as 0 t t if and only if each component approaches a limit, that is, ( ) ( ) ( ) ( ) 0 0 0 0 lim lim , lim , lim t t t t t t t t t x t y t z t = r 1. Evaluate the following limits. a. limsin 2 cos tan 4 t t t t π + + i j k b. 0 1 1 lim 4 1 t t e t t t + + + i j k Continuity of Vector-Valued Functions A vector-valued function ( ) ( ) ( ) ( ) , , t x t y t z t = r is continuou s 0 t if ( ) ( ) 0 0 lim t t t t = r r . By Theorem 1, ( ) t r is continuous at 0 t if and only if the components ( ) x t , ( ) y t , and ( ) z t are continuous at 0 t . Derivative of a Vector-Valued Function The derivative r of a vector function is defined as ( ) ( ) ( ) 0 lim h t h t d t dt h + = = r r r r if this limit exists. T The vector ( ) t r is called the tangent vector to the curve defined by r at the point P. Theorem 2: Vector-Valued Derivatives Are Computed Componentwise

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handout 13-2 - Math 200 Spring 2010 Handout 6 Section 13-2...

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