Stitz-Zeager_College_Algebra_e-book

# The existence and uniqueness of the additive inverse

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Unformatted text preview: 6 cos(2θ)). We shade the region below. y y θ= π 6 x r = 3 and r = 6 cos(2θ ) x (r, θ) : 3 ≤ r ≤ 6 cos(2θ), 0 ≤ θ ≤ π 6 11.5 Graphs of Polar Equations 815 3. From Example 11.5.2 number 2, we know that the graph of r = 2 + 4 cos(θ) is a lima¸on c π π whose ‘inner loop’ is traced out as θ runs through the given values 23 to 43 . Since the values r takes on in this interval are non-positive, the inequality 2 + 4 cos(θ) ≤ r ≤ 0 makes sense, and we are looking for all of the points between the pole r = 0 and the lima¸on as θ ranges c ππ over the interval 23 , 43 . In other words, we shade in the inner loop of the lima¸on. c r y 6 4 θ= 2π 3 2 x 2π 3 π 2 4π 3 π 3π 2 2π θ= θ 4π 3 −2 π (r, θ) : 2 + 4 cos(θ) ≤ r ≤ 0, 23 ≤ θ ≤ 4π 3 4. We have two regions described here connected with the union symbol ‘∪.’ We shade each in turn and ﬁnd our ﬁnal answer by combining the two. In Example 11.5.3, number 1, we found that the curves r = 2 sin(...
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