This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 (...
View
Full Document
 Spring '08
 Helton
 Math, Distance, Parametric equation, Tangent lines to circles

Click to edit the document details