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Math118_S10_W11

Math118_S10_W11 - Math118(M Kohandel-Outline(week 11 11 1...

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Math118 (M. Kohandel) ---------------------------------------------------------------------------------------------------------------------------- 1 Outline (week 11) 11. 1 Parametric equations (9.1) Definition Tangents, areas and arc-lengths 11. 2 Polar coordinates (9.2)

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Math118 (M. Kohandel) ---------------------------------------------------------------------------------------------------------------------------- 2 11. 1 Parametric equations So far we have described curves by giving y as a function of x , i.e. ) ( x f y , or x as a function of y , i.e. ) ( y g x , or by an implicit relation between x and y , i.e. 0 ) , ( y x F . In this week, we discuss another method to describe a function. Curves defined by parametric equations: Suppose a particle moves along a curve C . The x and y coordinate of the particle are functions of time t ; then we write ) ( t f x and ) ( t g y . In this case, x and y are both given as functions of time the third variable t (called parameter). Definition: A curve is defined parametrically if it is given in the form, t t g y t f x , ) ( , ) ( Note that in general the parameter t is necessary time, and in fact we can use another letter. Example 11.1: Sketch the curve defined by the parametric equation t t x 2 2 and 1 t y . Each value of t gives a point on the curve: 0 , 3 , 1 3 , 0 , 2 2 , 1 , 1 1 , 0 , 0 , , y x t The arrows on the curve show the direction when t increases. Note that eliminating t gives: 1 ) 2 ( 3 4 ) 1 ( 2 ) 1 ( 2 2 2 y y y y y x . In this example t was not restricted. Now consider:
Math118 (M. Kohandel) ---------------------------------------------------------------------------------------------------------------------------- 3 4 0 : 1 , 2 2 t t y t t x : ) 1 , 0 ( 0 t initial point : ) 5 , 8 ( 4 t terminal point Example 11.2: Sketch 2 0 : sin , cos t t y t x . 0 , 1 , 2 1 , 0 , 2 / 3 0 , 1 , 1 , 0 , 2 / 0 , 1 , 0 , , y x t The point ) , ( y x moves on the circle in the counterclockwise direction. Eliminating t : 1 2 2 y x . What happens if 2 0 : 2 cos , 2 sin t t y t x ?

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Math118_S10_W11 - Math118(M Kohandel-Outline(week 11 11 1...

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