Stewart Calculus Section 10.1

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 circles
with center ( , ) and radius .
y
r
(h,k)
o x

16. Find parametric equations for the circles
with center ( , ) and radius .
y
r
(h,k)
o x
= +
= +

17. Eliminate the parameter to find the Cartesian
equation of the curve.
= , = ,

18. Eliminate the parameter to find the Cartesian
equation of the curve.
= , = ,
+( ) =
y
(0,3)
o x

19. Eliminate the parameter to find the Cartesian
equation of the curve.
= , = ,
+( ) =
y s pos sible!
N ot alway
(0,3)
o x

20. 10.2 Calculus with
parametric Curves
= ( ), = ()
Tangents
Areas
Arc Length
Area 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:
=± ( )