Chapter 11: Q. 7 (page 859)

$Let r (t) = <x (t), y (t), z (t)> . Provide a definition for the continuity of the vector function r at a point c in the domain of r .$

Short Answer

Expert verified

$r (t) = <cost, sint, tant> is a continuous vector function at \frac{π}{4} which is in the domain of r (t) .$

Step by step solution

Step 1. Given information

$r (t) = <x (t), y (t), z (t)>$

Step 2. Result

$The function r is said to be continuous at a point c in the domain of r if \lim_{t \to c} r (t) = r (c) . That is, \lim_{t \to c} <x (t), y (t), z (t)> = \lim_{t \to c} x (t) i + \lim_{t \to c} y (t) j + \lim_{t \to c} z (t) k = x (c) i + y (c) j + z (c) k For example : let r (t) = <cost, sint, tant> and let c = \frac{π}{4} is in the domain of r (t) . Then \lim_{t \to c} r (t) = \lim_{t \to \frac{π}{4}} <cost, sint, tant> = \lim_{t \to \frac{π}{4}} cost i + \lim_{t \to \frac{π}{4}} sint j + \lim_{t \to \frac{π}{4}} tant k = \cos \frac{π}{4} i + \sin \frac{π}{4} j + \tan \frac{π}{4} k = \frac{1}{\sqrt{2}} i + \frac{1}{\sqrt{2}} j + 1 (k) = \frac{\sqrt{2}}{2} i + \frac{\sqrt{2}}{2} j + k Thus r (t) = <cost, sint, tant> is a continuous vector function at \frac{π}{4} which is in the domain of r (t) .$

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$Let r (t) = <x (t), y (t), z (t)> . Provide a definition for the continuity of the vector function r at a point c in the domain of r .$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Result

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Letr(t)=x(t),y(t),z(t).Provideadefinitionforthecontinuityofthevectorfunctionratapointcinthedomainofr.

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Step by step solution

Step 1. Given information

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$Let r (t) = <x (t), y (t), z (t)> . Provide a definition for the continuity of the vector function r at a point c in the domain of r .$