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# 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: =± ( )