Lecture_8 - CALCULUS II Spring 2009 LECTURE 8 This lecture...

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Unformatted text preview: CALCULUS II Spring 2009 LECTURE 8 This lecture provides a definition of a derivative of a vector-valued function definition of a tangent line to parametric curve models for tangent lines to parametric curves parametric curves that can have more than one tangent line at a given point a parametric model for a cycloid and its tangent lines 1 Recall We recall from Lecture 7. a vector is a point in R 2 p + ( q- p ) ,- < < models a line in R 2 ( r cos( t ) ,r sin( t )) , t < 2 models a circle in R 2 End of Recall 2 Tangent Lines to Parametric Curves Derivatives of Vector-Valued Functions . Working informally, we ex- plore the notion of a derivative of the vector-valued function r ( t ). (It is understood that t belongs to some real interval.) One chooses an arbitrary time (value of the parameter), say t , and computes the (vector) difference quotient r ( t + t )- r ( t ) t = ( x ( t + t ) , y ( t + t ) )- ( x ( t ) ,y ( t )) t = ( x ( t + t )- x ( t ) , y ( t + t )- y ( t ) ) t = x ( t + t )- x ( t ) t , y ( t + t )- y ( t ) t . One notes that the two components of this last vector are difference quo- tients, identical to those used in defining a derivative of a real-valued func- tion. Thus, if x ( t ) and y ( t ) exist, this last expression has a limit as t approaches zero. Specifically, one has lim t x ( t + t )- x ( t ) t , y ( t + t )- y ( t ) t = ( x ( t ) ,y ( t )) . Hence the value of lim t r ( t + t )- r ( t ) t is exactly what one would hope for, that is, lim t r ( t + t )- r ( t ) t = ( x ( t ) ,y ( t )) . In words, one says that the derivative of a vector-valued function is the derivatives of its components. 1 Based on this observation one makes the following definition and comment. 1 The equality lim t x ( t + t )- x ( t ) t , y ( t + t )- y ( t ) t = ( x ( t ) ,y ( t )) is appealing; it is correct. But it needs more attention than we have given. 3 Definition . Let the vector-valued function r ( t ), be defined on some real interval, I and let t I . One denotes the derivative of r ( t ), at t , by r ( t ) and defines it by r ( t ) = lim t r ( t + t )- r ( t ) t provided the limit exists. 2 Comment. If x ( t ) and y ( t ) are real-valued, differentiable functions, such that r ( t ) = ( x ( t ) ,y ( t )) , t I , then for each t I one has r ( t ) = ( x ( t ) ,y ( t )) . This is, in fact, the working definition of r ( t ). EXAMPLE. For the parametric curve r ( t ) = ( x ( t ) ,y ( t )) = ( t 3 ,t ) ,- < t < and for- < t < , find an (explicit) expression for r ( t )....
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This note was uploaded on 02/09/2009 for the course MATH 1020 taught by Professor Ecker during the Spring '08 term at Rensselaer Polytechnic Institute.

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Lecture_8 - CALCULUS II Spring 2009 LECTURE 8 This lecture...

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