sec10_3

# sec10_3 - (1(3 2 π 3(2(3-2 π 3(3-3 2 π 3 Example 2.2...

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10.3 Polar Coordinates 1. 10.3 Polar Coordinates Remarks 1.1. (1) If r> 0 then the point is along the terminal side of the ray with angle θ . (2) If r< 0 then the point is along the terminal side of the ray with angle (3) For each point in the plane there is one and only one Cartesian coordinate representing that point. (4) For each point in the plane there are in±nitely many polar coordinates repre- senting that point. 1

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Section 10.3 Polar Coordinates 2 2. Conversions Converting from Polar to Cartesian: Given the point ( r, θ ), (1) x = (2) y = Converting from Cartesian to Polar: Given the point ( x, y ), (1) To fnd r use the equation r 2 = x 2 + y 2 (2) To fnd θ use the equation tan θ = y/x . (3) Make sure your choices For r and θ put the point in the correct quadrant. Example 2.1. Convert the Polar coordinates to Cartesian coordinates
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Unformatted text preview: (1) (3 , 2 π/ 3) (2) (3 ,-2 π/ 3) (3) (-3 , 2 π/ 3) Example 2.2. Convert the Cartesian coordinates to Polar coordinates (1) (-1 , √ 3) (2) (-1 ,-√ 3) 3. Polar Curves Example 3.1. Sketch θ = π/ 4 . Example 3.2. Find a polar equation for the curve with the Cartesian equation x 2 + y 2 = 1 . Example 3.3. Find a Cartesian equation for the curve with Polar equation r = 2 sin θ Section 10.3 Polar Coordinates 3 4. Tangents to Polar Curves Recall the slope of the tangent will be dy/dx . Example 4.1. (problem 66) r = e θ ,-π/ 2 < θ < π/ 2 (1) Find dy/dx (2) Find where the curve has horizontal tangents (3) Find where the curve has vertical tangents (4) Find where the curve is increasing/decreasing...
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sec10_3 - (1(3 2 π 3(2(3-2 π 3(3-3 2 π 3 Example 2.2...

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