This preview shows pages 1–2. Sign up to view the full content.
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 we have ~v ( ) = h , 3 , 4 i and ~ r ( ) = h 3 , , 4 i . Therefore the tangent line at t = has equation ~ L ( ) = h 3 , , 4 i + h , 3 , 4 i = h 3 , 3 , 4 + 4 i . Note that at = 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
This note was uploaded on 04/28/2011 for the course MATH 20C taught by Professor Helton during the Spring '08 term at UCSD.
 Spring '08
 Helton
 Math

Click to edit the document details