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Unformatted text preview: 14.2 Derivatives and Integrals of Vector Functions Definition 14.2.1 Let r ( t ) be a vector-valued function. The vector valued function r ( t ) given by r ( t ) = lim t r ( t + h )- r ( t ) h is called the derivative. Geometric Interpretation: Definition 14.2.2 Whenever r ( t ) is defined and not equal to the zero vector, r ( t ) is called the tangent vector. In fact, whenever || r ( t ) || 6 = 0 , we can define the unit tangent vector T ( t ) as T ( t ) = r ( t ) || r ( t ) || . Theorem: Let r ( t ) = < f ( t ) ,g ( t ) ,h ( t ) > , with f , g , h , differentiable functions. Then the derivative r ( t ) is r ( t ) = < f ( t ) ,g ( t ) ,h ( t ) > . Example 14.2.1 Let r ( t ) = ln( t ) i + e- t j + t 3 k . 1. Find r ( t ) . 2. Find T ( t ) . 3. Find r (2) . 4. Find parametric equations of the tangent line to C when t = 2 . Example 14.2.2 Consider the vector function r ( t ) = < t 2 ,t 3 > ....
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