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Calculus II - 15
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: =± ( )
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