14.4%20Prel%201-4,%20Ex%201%20-%209

14.4%20Prel%201-4,%20Ex%201%20-%209 - 500 C H A P T E R 14...

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500 CHAPTER 14 CALCULUS OF VECTOR-VALUED FUNCTIONS (ET CHAPTER 13) 14.4 Curvature (ET Section 13.4) Preliminary Questions 1. What is the unit tangent vector of a line with direction vector v = h 2 , 1 , 2 i ? SOLUTION A line with direction vector v has the parametrization: r ( t ) = −−→ OP 0 + t v hence, since 0 and v are constant vectors, we have: r 0 ( t ) = v Therefore, since k v k= 3, the unit tangent vector is: T ( t ) = r 0 ( t ) k r 0 ( t ) k = v k v k = h 2 / 3 , 1 / 3 , 2 / 3 i 2. What is the curvature of a circle of radius 4? The curvature of a circle of radius R is 1 R , hence the curvature of a circle of radius 4 is 1 4 . 3. Which has larger curvature, a circle of radius 2 or a circle of radius 4? The curvature of a circle of radius 2 is 1 2 , and it is larger than the curvature of a circle of radius 4, which is 1 4 . 4. What is the curvature of r ( t ) = h 2 + 3 t , 7 t , 5 t i ? r ( t ) parametrizes the line h 2 , 0 , 5 i + t h 3 , 7 , 1 i , and a line has zero curvature. 5. What is the curvature at a point where T 0 ( s ) = h 1 , 2 , 3 i in an arc length parametrization r ( s ) ? The curvature is given by the formula: κ ( t ) = k T 0 ( t ) k k r 0 ( t ) k In an arc length parametrization, k r 0 ( t ) 1forall t , hence the curvature is ( t ) =k T 0 ( t ) k . Using the given information we obtain the following curvature: =k h 1 , 2 , 3 ik= p 1 2 + 2 2 + 3 2 = 14 6. What is the radius of curvature of a circle of radius 4? The deFnition of the osculating circle implies that the osculating circles at the points of a circle, is the circle itself. Therefore, the radius of curvature is the radius of the circle, that is, 4. 7. What is the radius of curvature at P if P = 9? The radius of curvature is the reciprocal of the curvature, hence the radius of curvature at P is: R = 1 P = 1 9 Exercises In Exercises 1–6, calculate r 0 ( t ) and T ( t ) , and evaluate T ( 1 ) .
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14.4%20Prel%201-4,%20Ex%201%20-%209 - 500 C H A P T E R 14...

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