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Imaginary axis imaginary axis 3i i 2i z 4cis i 1 w 0

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Unformatted text preview: e origin. Not only do we reach the origin when θ = 23 , a π theorem from Calculus5 states that the curve hugs the line θ = 23 as it approaches the origin. 5 The ‘tangents at the pole’ theorem from second semester Calculus. 11.5 Graphs of Polar Equations 803 r y 6 θ= 2π 3 4 2 x 2π 3 4π 3 π π 2 3π 2 2π θ −2 π On the interval 23 , π , r ranges from 0 to −2. Since r ≤ 0, the curve passes through the π origin in the xy -plane, following the line θ = 23 and continues upwards through Quadrant IV 6 Since |r | is increasing from 0 to 2, the curve pulls away from towards the positive x-axis. the origin to finish at a point on the positive x-axis. r y 6 θ= 2π 3 4 2 x 2π 3 π 2 4π 3 π 3π 2 2π θ −2 π Next, as θ progresses from π to 43 , r ranges from −2 to 0. Since r ≤ 0, we continue our π graph in the first quadrant, heading into the origin along the line θ = 43 . 6 Recall that one way to visualize plotting polar coordinates (r, θ) with r < 0 is to start the rotation from the left π side of the pole - in this case, the negative x-axis. Rotating between 23 and π radians from the negative x-axis in 2π this case determines the r...
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