Explanation Given: r ( t ) = 〈 e − t sin t , e − t cos t , e − t 〉 x 2 + y 2 + z 2 = 4 Formula used: sin 2 θ + cos 2 θ = 1 sin ( − t ) = − sin t cos ( − t ) = cos t log m n = n log m log e = 1 Calculation: r ( t ) = 〈 e − t sin t , e − t cos t , e − t 〉 ⇒ x = e − t sin t ⇒ y = e − t cos t ⇒ z = e − t If curve intersects the sphere, then the points of curve will satisfy the equation of sphere. So, we will substitute the points of curve in equation of sphere. 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 ) + e − 2 t = 4 ⇒ e − 2 t ( 1 ) + e − 2 t = 4 [ ∵ sin 2 θ + cos 2 θ = 1 ] ⇒ e − 2 t + e − 2 t = 4 ⇒ 2 e − 2 t = 4 ⇒ e − 2 t = 2 Apply log on both sides, we get ⇒ log ( e &

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ISBN: 9781319050740

Author: Jon Rogawski, Colin Adams, Robert Franzosa

Publisher: W. H. Freeman

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Chapter 13.1, Problem 12E

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Show that the path given by r(t)=〈e−tsint,e−tcost,e−t〉 intersects the sphere x2+y2+z2=4 once, travelling from outside the sphere to inside as t goes form −∞ to ∞.

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Chapter 13.1, Problem 11E

Chapter 13.1, Problem 13E

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Show that the path given by r ( t ) = 〈 e − t sin t , e − t cos t , e − t 〉 intersects the sphere x 2 + y 2 + z 2 = 4 once, travelling from outside the sphere to inside as t goes form − ∞ to ∞ .

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Show that the path given by r(t)=〈e−tsint,e−tcost,e−t〉 intersects the sphere x2+y2+z2=4 once, travelling from outside the sphere to inside as t goes form −∞ to ∞.

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