Calculus II - 15

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Stewart Calculus Section 10.1

Stewart Calculus Section 10.1

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Transcript

  • 1. 10.1 Parametric Equations Curves can be defined by functions: = ( ) = + ··· But they cannot describe all general curves.
  • 2. Parametric equations: = ( ), = ()Ex: = , = + , = , = , ···
  • 3. Parametric curve: = , = + ,
  • 4. Parametric curve: = , = + ,
  • 5. Parametric curve: = , = + ,
  • 6. Parametric curve: = , = + ,
  • 7. Parametric curve: = , = ,
  • 8. Parametric curve: = , = ,
  • 9. Parametric curve: = , = ,
  • 10. Parametric curve: = , = ,
  • 11. More examples:= + , = + ,
  • 12. More examples: = = , + +
  • 13. More examples:= + , = + ,
  • 14. More examples: = =
  • 15. Find parametric equations for the circleswith center ( , ) and radius . y r (h,k) o x
  • 16. Find parametric equations for the circleswith center ( , ) and radius . y r (h,k) o x = + = +
  • 17. Eliminate the parameter to find the Cartesianequation of the curve. = , = ,
  • 18. Eliminate the parameter to find the Cartesianequation of the curve. = , = , +( ) = y (0,3) o x
  • 19. Eliminate the parameter to find the Cartesianequation of the curve. = , = , +( ) = y s pos sible! N ot alway (0,3) o x
  • 20. 10.2 Calculus with parametric Curves = ( ), = ()TangentsAreasArc LengthArea of Surfaces of Revolution
  • 21. Tangents:
  • 22. Tangents: If = , =
  • 23. Tangents: If = , = = =
  • 24. Tangents: If = , = = = !! =
  • 25. Ex: Find the tangents of the curve at ( , ): = , =
  • 26. Ex: Find the tangents of the curve at ( , ): = , =At ( , ), = ± . = = =±
  • 27. Ex: Find the tangents of the curve at ( , ): = , =At ( , ), = ± . = = =±The equations of the tangents are: =± ( )