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AP Calculus BC Monday, 11 November 2013

AP Calculus BC Monday, 11 November 2013. OBJECTIVE TSW use polar equations to find derivatives and tangent lines. ASSIGNMENT DUE WS Parametric Equations and Vectors  wire basket TODAY’S ASSIGNMENT Sec. 10.4: p. 737 (59-80 all) WS Review Sec. 10.2 – 10.4, Vectors LOOKING AHEAD

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AP Calculus BC Monday, 11 November 2013

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  1. AP Calculus BCMonday, 11 November 2013 • OBJECTIVETSW use polar equations to find derivatives and tangent lines. • ASSIGNMENT DUE • WS Parametric Equations and Vectors  wire basket • TODAY’S ASSIGNMENT • Sec. 10.4: p. 737 (59-80 all) • WS Review Sec. 10.2 – 10.4, Vectors • LOOKING AHEAD • Wednesday:TEST: Sec. 10.2 – 10.4, Vectors • Wednesday, 11/20:TEST: Indefinite Integration

  2. Sec. 10.4: Polar Coordinates and Polar Graphs

  3. Sec. 10.4: Polar Coordinates and Polar Graphs We've studied rectangular coordinate systems and parametric coordinate systems. We now turn to polar coordinate systems.

  4. Sec. 10.4: Polar Coordinates and Polar Graphs We fix a point O, called the pole (or origin), and construct from O an initial ray called the polar axis. Each point P in the plane can be assigned polar coordinates (r, θ), where r = directed distance from O to P θ = directed angle counterclockwise from the polar axis to ray OP.

  5. Sec. 10.4: Polar Coordinates and Polar Graphs With rectangular coordinates, each point (x, y) has a unique representation. With polar coordinates, this is not true. (r, θ) and (r, 2π + θ) represent the same point. (r, θ) and (–r, θ + π) represent the same point.

  6. Sec. 10.4: Polar Coordinates and Polar Graphs In general, (r, θ) = (r, θ + 2nπ) or (r, θ) = (–r, θ + (2n + 1)π), n Z.

  7. Sec. 10.4: Polar Coordinates and Polar Graphs How do you find the derivative of a polar equation ? If then we get and you have the following.

  8. Sec. 10.4: Polar Coordinates and Polar Graphs

  9. Sec. 10.4: Polar Coordinates and Polar Graphs Find the horizontal and vertical points of tangency of

  10. Special Polar Graphs Special Polar Graphs • Limacons Limacon with inner loop Cardiod Dimpled limacon Convex limacon

  11. Special Polar Graphs • Rose Curves • n petals if n is odd, 2n petals if n is even (n 2) n = 2 n = 5 n = 4 n = 3

  12. Special Polar Graphs Special Polar Graphs • Circles

  13. Special Polar Graphs Special Polar Graphs • Lemniscates

  14. Sec. 10.4: Polar Coordinates and Polar Graphs Find the horizontal and vertical points of tangency of

  15. Sec. 10.4: Polar Coordinates and Polar Graphs Theorem 10.12 Tangent Lines at the Poles If f ( ) = 0 and f ′( ) ≠ 0, then the line  =  is tangent at the pole to the graph of r = f ( ).

  16. Sec. 10.4: Polar Coordinates and Polar Graphs Find the equations of the tangent lines at the pole for

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