Explanation Given information: The vector is given by, r ( t ) = 〈 e − t sin t , e − t cos t , e − t 〉 and The sphere is, x 2 + y 2 + z 2 = 4 Proof: If the path and the sphere intersect, there should be coordinates that satisfy both formulas. By the formula of the path, we can identify the x , y and z as follows. x = e − t sin t y = e − t cos t z = e − t By substituting these into the formula of the sphere, we can obtain the values of t at the intersection points. x 2 + y 2 + z 2 = 4 ( e − t sin t ) 2 + ( e − t cos t ) 2 + ( e − t ) 2 = 4 e − 2 t sin 2 t + e − 2 t cos 2 t + e − 2 t = 4 e − 2 t ( sin 2 t + cos 2 t + 1 ) = 4 e − 2 t ( 1 + 1 ) = 4 e − 2 t = 2 − 2 t = ln 2 t = − 1 2 ln 2 t = ln ( 1 2 ) There is only one value for t

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Author: Rogawski

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Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differentia…

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Chapter 14.1, Problem 12E

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To prove: That there is only one intersection point of the space curve r(t) and the sphere given and when t goes from −∞ to ∞ , the path travels inside the sphere from the outside.

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Chapter 14.1, Problem 11E

Chapter 14.1, Problem 13E

Chapter 14 Solutions

CALCULUS W/SAPLING ACCESS >IC<

Ch. 14.1 - Prob. 1PQ Ch. 14.1 - Prob. 2PQ Ch. 14.1 - Prob. 3PQ Ch. 14.1 - Prob. 4PQ Ch. 14.1 - Prob. 5PQ Ch. 14.1 - Prob. 6PQ Ch. 14.1 - Prob. 1E Ch. 14.1 - Prob. 2E Ch. 14.1 - Prob. 3E Ch. 14.1 - Prob. 4E

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To prove: That there is only one intersection point of the space curve r ( t ) and the sphere given and when t goes from − ∞ to ∞ , the path travels inside the sphere from the outside.

Solution Summary: The author proves that there is only one intersection point of the space curve r(t) and the sphere given.

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To prove: That there is only one intersection point of the space curve r(t) and the sphere given and when t goes from −∞ to ∞ , the path travels inside the sphere from the outside.

Want to see the full answer?

Chapter 14 Solutions