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Unformatted text preview: MATH 20C Lecture 8 - Monday, October 11, 2010 A bit more about tangent lines. The parameter for the tangent line is different from the parameter of the curve itself. Take for instance the spiral ~ r ( t ) = h 3cos t, 3sin t, 4 t i . The velocity vector is ~v ( t ) = h- 3sin t, 3cos t, 4 i . At time π we have ~v ( π ) = h ,- 3 , 4 i and ~ r ( π ) = h- 3 , , 4 π i . Therefore the tangent line at t = π has equation ~ L ( θ ) = h- 3 , , 4 π i + θ h ,- 3 , 4 i = h- 3 ,- 3 θ, 4 π + 4 θ i . Note that at θ = 0 the line touches the curve, and then it goes away from it. Arc length s = distance travelled along trajectory. Since the rate of change of the distance is the speed, we have ds dt = speed = | ~v ( t ) | . Can recover length of trajectory by integrating ds/dt . So the length of the curve starting at time t 1 until time t 2 is s = Z t 2 t 1 | ~v ( t ) | dt. The distance traveled by a moving point along a curve starting at time t 1 until the current time is s ( t ) = Z t 1 | ~v (...
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- Spring '08
- Math, Distance, Parametric equation, Tangent lines to circles