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s11wk06su - Math 23 B. Dodson Week 3a Homework: 13.3...

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Unformatted text preview: Math 23 B. Dodson Week 3a Homework: 13.3 curvature (using differentiation, formula(9)) 14.2 limits 14.3 partial derivatives, 2nd order deriv Week 3a Homework: (continued) 13.3 curvature Problem 13.3.16 Use formula (9) to find the curvature of vector r ( t ) = < t 2 , 2 t, ln t > . Solution: We start with part (a), find the unit tangent vector T and principal unit normal vector N. We compute vector r = < 2 t, 2 , 1 t > and | vector r | 2 = 4 t 2 + 4 + 1 t 2 = (2 t + 1 t ) 2 , so | vector r | = 2 t + 1 t (since t > , is used for ln t ). 2 . We also simplify 1 | vector r | = t 2 t 2 + 1 , then vector T = 1 | vector r | vector r = t 2 t 2 + 1 < 2 t, 2 , 1 t > . Using the product rule, vector T = parenleftbigg t 2 t 2 + 1 parenrightbigg < 2 t, 2 , 1 t > + t 2 t 2 + 1 ( < 2 t, 2 , 1 t > ) = parenleftbigg (2 t 2 + 1) t (4 t ) (2 t 2 + 1) 2 parenrightbigg < 2 t, 2 , 1 t > + t 2 t 2 + 1 parenleftbigg < 2 , , 1 t 2 > parenrightbigg = parenleftbigg 1 (2...
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This note was uploaded on 11/24/2011 for the course MATH 23 taught by Professor Yukich during the Spring '06 term at Lehigh University .

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s11wk06su - Math 23 B. Dodson Week 3a Homework: 13.3...

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