Polar Coordinates: Example 12: Limaçons Analytical Proof: Part 2: Question B
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Example 12: Limaçons Analytical Proof: Part 2: Question B
a) In Example 11 the graphs suggest that the Limaçon r = 1 + c sin θ has an inner loop when |c| > 1.
Prove that this is true, and find the values of θ that correspond to the inner loop.
b) Also from Example 11, it appears that the Limaçon loses its dimple when c = 1/2.
Prove this.
Recall from Example 11
r = 1 + c sin θ
Solution to b)
b) For 0 < c < 1, the dimple (if it exists) is characterized by the fact that y has a local maximum at θ = 3π/2.
To determine if there is a local maximum we can use the Second Derivative Test:
Thus we need to find the c values where dy/dθ = 0 and d2y/dθ2 < 0 at θ = 3π/2.
Similarly for -1 < c < 0, y only has a local minimum at θ = π/2 (indicting a dimple) for c < -1/2.