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Do Now

Do Now. Find the area of the given sector:. Slopes and Planar Areas of Polar Curves. Section 10.6a. Finding the Slope of a Polar Curve. The slope of is given by . Observe…. First, conceptualize the curve parametrically:. The derivative (Chain Rule!):.

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Do Now

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  1. Do Now Find the area of the given sector:

  2. Slopes and Planar Areas of Polar Curves Section 10.6a

  3. Finding the Slope of a Polar Curve The slope of is given by . Observe… First, conceptualize the curve parametrically: The derivative (Chain Rule!): Product Rule!

  4. Finding the Slope of a Polar Curve provided at . The curve has a 1. horizontal tangent at a point where and , 2. vertical tangent at a point where and , (Note: If both derivatives are zero, no conclusion can be drawn without further investigation)

  5. Finding the Slope of a Polar Curve Practice Problem: Find the slope of the given curve at the indicated point.

  6. Finding the Slope of a Polar Curve Practice Problem: Find the slope of the given curve at the indicated point. Undefined Check these results graphically!

  7. Finding the Slope of a Polar Curve Practice Problem: Find the slope of the given curve at the indicated point. Check these results graphically!

  8. Finding the Slope of a Polar Curve Practice Problem: Find the lines tangent to the given rose curve at the pole. Start with a graph… is zero at Tangent Lines:

  9. Area in the Polar Plane The new challenge: How to find the area of a polar region TOS… Break it into smaller sectors! Area of a typical sector:

  10. Area in the Polar Plane The new challenge: How to find the area of a polar region TOS… Break it into smaller sectors! Area of regionOTS is approximately:

  11. Area in the Polar Plane Area of regionOTSis approximately: Letting get infinitely small (summing the areas of infinitely many sectors…) The areaof the region between the origin and the curve , , is

  12. Area in the Polar Plane What about the area between two polar curves?: The area of region , , is

  13. Guided Practice Find the area of the region in the plane enclosed by the cardioid: First, check out the graph…

  14. Guided Practice Find the area of the region in the plane enclosed by the cardioid: Confirm this answer numerically!

  15. Guided Practice Find the area of the region that lies inside the circle and outside the cardioid . First, check out the graph…

  16. Guided Practice Find the area of the region that lies inside the circle and outside the cardioid .

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