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Chap13sec2

# Chap13sec2 - 2sin,cos t t t =-r with the derivative vector...

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13.2 Derivatives and Integrals of Vector Functions The derivative of a vector function   ( ) ( ), ( ), ( ) t f t g t h t = r   is   '( ) '( ), '( ), '( ) t f t g t h t = r . Note that you  differentiate   each   component   separately.   For   example,   look   at   the   2-dimensional   space   curve   defined   by  ( ) 4sin( ), 2cos( ) t t t = - r . If  ( ) t r  is the position function of a particle, then  '( ) t r is the velocity function,  like the  velocity function in one dimension. Let's look at a geometric description of the derivative  '( ) t r . The derivative  '( ) t r  is tangent to the space curve  ( ) t r . The   figure   shown   above   is   the   curve
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Unformatted text preview: ( ) 2sin( ),cos( ) t t t = -r with the derivative vector '( ) 2cos( ), sin( ) t t t = --r shown a two points. The unit tangent vector is the derivative vector, denoted ( ) t T , is the derivative vector divided by its length. '( ) ( ) '( ) t t t = r T r Example: Find the unit vector ( ) t T for 2 1 ( ) , , tan t t t t t-=-r at t=1. Example: Find parametric equations for the tangent line to the curve 2 2 1, 1, 1 x t y t z t =-= + = + at (-1,1,1). Example: Find ( 29 ( 29 2 2 4 2 1 1 4 1 t t t dt +--- ∫ i j k-2-1 1 2-1-0.5 0.5 1...
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