Explanation Given: The Fundamental Theorem of Calculus I from single-variable, that is, if f is continuous on an open interval I and a be in I, then d d x ∫ a x f ( s ) d s = f ( x ) . Proof: Let r ( t ) = 〈 x ( t ) , y ( t ) , z ( t ) 〉 be continuous on an open interval I, and a be in I. So, x ( t ) , y ( t ) and z ( t ) must be continuous on open interval I, and the following results must be true. d d t ∫ a t x ( s ) d s = x ( t ) , d d t ∫ a t y ( s ) d s = y ( t ) , d d t ∫ a t z ( s ) d s = z ( t )

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Author: Jon Rogawski, Colin Adams, Robert Franzosa

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Chapter 13.2, Problem 70E

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To prove that second part of the Fundamental Theorem of Calculus for Vector-Valued Functions that if r(t) is continuous on an open interval I, and a be in I, then ddt∫atr(s)ds =r(t) by using the Fundamental Theorem of Calculus II from single-variable calculus.

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Chapter 13.2, Problem 69E

Chapter 13.2, Problem 71E

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Calculus: Early Transcendentals

Ch. 13.1 - Prob. 1PQ Ch. 13.1 - Prob. 2PQ Ch. 13.1 - Prob. 3PQ Ch. 13.1 - Prob. 4PQ Ch. 13.1 - Prob. 5PQ Ch. 13.1 - Prob. 6PQ Ch. 13.1 - Prob. 1E Ch. 13.1 - Prob. 2E Ch. 13.1 - Prob. 3E Ch. 13.1 - Prob. 4E

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To prove that second part of the Fundamental Theorem of Calculus for Vector -Valued Functions that if r ( t ) is continuous on an open interval I, and a be in I, then d d t ∫ a t r ( s ) d s = r ( t ) by using the Fundamental Theorem of Calculus II from single-variable calculus.

Solution Summary: The author explains the Fundamental Theorem of Calculus for Vector-Valued Functions. If r(t) is continuous on an open interval I, and a be in I

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To prove that second part of the Fundamental Theorem of Calculus for Vector-Valued Functions that if r(t) is continuous on an open interval I, and a be in I, then ddt∫atr(s)ds =r(t) by using the Fundamental Theorem of Calculus II from single-variable calculus.

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