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chapter3.page12

# chapter3.page12 - Sum and Diﬀerence rule v t ± x t = v t...

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12 1. BRIEF INTRODUCTION TO VECTORS AND MATRICES The antiderivative R v ( t ) dt of an vector-valued function v ( t ) is a vector-valued function whose entries are the antiderivative of corresponding entries of v ( t ) . Example 2.3 . Find derivative of x ( t ) = 3t 2 - 5 sin( t ) Solution x 0 ( t ) = dx ( t ) dt = d dt 3t 2 - 5 sin( t ) = d dt (3t 2 - 5) d dt (sin( t )) = 6t cos( t ) a Example 2.4 . Find antiderivative of x ( t ) = 3t 2 - 5 sin( t ) Solution R x ( t ) dt = R 3t 2 - 5 sin( t ) dt = R (3t 2 - 5) dt R sin( t ) dt = t 3 - 5t + C 1 - cos( t ) + C 2 = t 3 - 5t - cos( t ) + C 1 C 2 a Theorem 2.1 . Suppose v ( t ) , x ( t ) , A ( t ) are differentiable vector- valued functions ( A ( t ) is matrix), and f ( t ) is differentiable scalar function. We have,
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Unformatted text preview: Sum and Diﬀerence rule:-[ v ( t ) ± x ( t )] = v ( t ) ± x ( t ) ,-R v ( t ) ± x ( t ) dt = R v ( t ) dt ± R x ( t ) dt. (2) Product rule:-[ f ( t ) v ( t )] = f ( t ) v ( t ) + f ( t ) v ( t ) ,-[ A ( t ) x ( t )] = A ( t ) x ( t ) + A ( t ) x ( t ) , Using Mathcad to ﬁnd derivative or antiderivative of a vector-valued function using Mathcad , you need to ﬁnd derivative or anti-derivative component wise as shown in the following screen shot,...
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