a. Use Corollary 2 of the Mean Value Theorem for scalar functions to show that if two vector functions R₁(t) and R₂(t) have identical derivatives on an interval I, then the functions differ by a constant vector value throughout I.

b. Use the result in part (a) to show that if R(t) is any antiderivative of r(t) on I, then any other antiderivative of r on I equals R(t) + C for some constant vector C.

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COROLLARY 2 If f'(x) = g'(x) at each point x in an open interval (a, b), then there exists a constant C such that f(x) = g(x) + C for all x = (a, b). That is, fg is a constant function on (a, b).