B revision chapter 51 polar coordinates c application

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b) Revision - Chapter 5.1 Polar Coordinates. c) Application - Obtain slopes. - Obtain area. - Obtain length. d) Formula & Theorem - Eqn. (5.5) (5.8) - Eqn. (5.9) (5.13) - Eqn. (5.14) (5.15) e) Exam Question - Example 5.9 5.10 - Example 5.11 5.13 - Example 5.14 F IGURE 5.12: Where are the horizontal and vertical tangents to this cardioid? (Example 5.9) 5.2 Calculus of Polar Curves In this section, you will see how to find slopes, areas, and lengths of polar curves ) ( f r . 5.2.1 Slope The slope of a polar curve ) ( f r is given by dy/dx , not by d df r / ' . To see why, think of the graph of f as the graph of the parametric equations cos ) ( cos f r x , sin ) ( sin f r y . If f is a differentiable function of , then so are x and y , and when 0 / d dx , we can calculate dx dy / from the parametric formula sin ) ( cos cos ) ( sin ) cos ) ( ( ) sin ) ( ( / / f d df f d df f d d f d d d dx d dy dx dy Definition Slope of the Polar Curve ) ( f r , sin ) ( cos ) ( ' cos ) ( sin ) ( ' | ) , ( f f f f dx dy r provided 0 / d dx at ) , ( r . We can see from the above equation and its derivation that the curve ) ( f r has a 1. Horizontal tangent at a point where 0 / d dy and 0 / d dx . 2. Vertical tangent at a point where 0 / d dx and 0 / d dy . If both derivatives are zero, no conclusion can be drawn without further investigation, as illustrated in Example 5.9. Example 5.9: Finding Horizontal and Vertical Tangents Find the horizontal and vertical tangents to the graph of the cardioid cos 1 r , 2 0 . Solution 5.9: The graph in Figure 5.12 suggests that there are at least two horizontal and three vertical tangents. The parametric form of the equation is . sin cos sin sin ) cos 1 ( sin , cos cos cos ) cos 1 ( cos 2 r y r x We need to find the zeros of d dy / and d dx / . (5.5) (5.6) (5.7) EXAMPLE DEMO FORMULA THEORY IMPORTANT
Chapter 5: Polar Coordinates 5-9 KE17103 Multivariable Calculus Chapter 5: Polar Coordinates 5-9 F IGURE 5.12: Where are the horizontal and vertical tangents to this cardioid? (Example 5.9) (a) Zeros of d dy / in ] 2 , 0 [ : ). cos 1 )( cos 2 1 ( cos 2 cos 1 cos ) cos 1 ( cos cos sin cos 2 2 2 2 2 d dy . 3 / 4 , 3 / 2 0 cos 2 1 2 , 0 0 cos 1 Thus, 0 / d dy in 2 0 if 2 , 3 / 4 , 3 / 2 , 0 . (b) Zeros of d dx / in ] 2 , 0 [ : . sin ) 1 cos 2 ( sin cos 2 sin d dx . 2 , , 0 0 sin 3 / 5 , 3 / 0 1 cos 2 Thus, 0 / d dx in 2 0 if 2 , 3 / 5 , , 3 / , 0 .