Calculus BC AP/Dual, Revised ©2018 Polar Area Day 4 Section 10.5C Calculus BC AP/Dual, Revised ©2018 viet.dang@humbleisd.net 4/24/2019 11:25 PM §10.5C: Polar AP Practice
Review Find the first derivative of 𝒓=𝟏+ 𝐜𝐨𝐬 𝜽 4/24/2019 11:25 PM §10.5C: Polar AP Practice
Example 1 Find the second derivative of 𝒓=𝟏+ 𝐜𝐨𝐬 𝜽 given that 𝒅 𝒅𝒚 𝒅𝒙 = 𝟑(𝟏+ 𝐜𝐨𝐬 𝜽) 𝐬𝐢𝐧 𝜽 +𝟐 𝐬𝐢𝐧 𝜽 𝐜𝐨𝐬 𝜽 𝟐 4/24/2019 11:25 PM §10.5C: Polar AP Practice
Example 1 Find the second derivative of 𝒓=𝟏+ 𝐜𝐨𝐬 𝜽 given that 𝒅 𝒅𝒚 𝒅𝒙 = 𝟑(𝟏+ 𝐜𝐨𝐬 𝜽) 𝐬𝐢𝐧 𝜽 +𝟐 𝐬𝐢𝐧 𝜽 𝐜𝐨𝐬 𝜽 𝟐 4/24/2019 11:25 PM §10.5C: Polar AP Practice
Your Turn Find the second derivative of 𝒓= 𝐬𝐢𝐧 𝜽 given that 𝒅 𝒅𝒚 𝒅𝒙 = 𝟐 𝐬𝐢𝐧 𝜽 𝐜𝐨𝐬 𝜽 𝐜𝐨𝐬 𝟐 𝜽− 𝐬𝐢𝐧 𝟐 𝜽 4/24/2019 11:25 PM §10.5C: Polar AP Practice
Example 2 Find the area between the two spirals of 𝒓=𝜽 and 𝒓=𝟐𝜽 for 𝟎≤𝜽≤𝟐𝝅. 4/24/2019 11:25 PM §10.5C: Polar AP Practice
Example 3 The graphs of the polar curves of 𝒓=𝟐 and 𝒓=𝟑+𝟐 𝐜𝐨𝐬 𝜽 are shown. The curves intersect when 𝜽= 𝟐𝝅 𝟑 and 𝜽= 𝟒𝝅 𝟑 . (calc allowed) Let 𝑹 be the region that is inside of the graph of 𝒓=𝟐 and also inside the graph of 𝒓=𝟑+𝟐 𝐜𝐨𝐬 𝜽 , as shaded in the figure. Find the area. A particle is moving with nonzero velocity along the polar curve given by 𝒓=𝟑+𝟐 𝐜𝐨𝐬 𝜽 has position of 𝒙 𝒕 ,𝒚 𝒕 at time 𝒕, with 𝜽=𝟎 with 𝒕=𝟎. This particle moves along the curve so that 𝒅𝒚 𝒅𝒕 = 𝒅𝒓 𝒅𝜽 . Find the value of 𝒅𝒓 𝒅𝒕 at 𝜽= 𝝅 𝟑 and interpret answer in terms of motion of the particle. For the particle described in part (b), 𝒅𝒚 𝒅𝒕 = 𝒅𝒓 𝒅𝜽 , find the value of 𝒅𝒚 𝒅𝒕 at 𝜽= 𝝅 𝟑 and interpret answer in terms of motion of the particle. 4/24/2019 11:25 PM §10.5C: Polar AP Practice
Example 3a The graphs of the polar curves of 𝒓=𝟐 and 𝒓=𝟑+𝟐 𝐜𝐨𝐬 𝜽 are shown. The curves intersect when 𝜽= 𝟐𝝅 𝟑 and 𝜽= 𝟒𝝅 𝟑 . Let 𝑹 be the region that is inside of the graph of 𝒓=𝟐 and also inside the graph of 𝒓=𝟑+𝟐 𝐜𝐨𝐬 𝜽 , as shaded in the figure. Find the area. 4/24/2019 11:25 PM §10.5C: Polar AP Practice
Example 3a The graphs of the polar curves of 𝒓=𝟐 and 𝒓=𝟑+𝟐 𝐜𝐨𝐬 𝜽 are shown. The curves intersect when 𝜽= 𝟐𝝅 𝟑 and 𝜽= 𝟒𝝅 𝟑 . Let 𝑹 be the region that is inside of the graph of 𝒓=𝟐 and also inside the graph of 𝒓=𝟑+𝟐 𝐜𝐨𝐬 𝜽 , as shaded in the figure. Find the area. 4/24/2019 11:25 PM §10.5C: Polar AP Practice
Example 3b The graphs of the polar curves of 𝒓=𝟐 and 𝒓=𝟑+𝟐 𝐜𝐨𝐬 𝜽 are shown. The curves intersect when 𝜽= 𝟐𝝅 𝟑 and 𝜽= 𝟒𝝅 𝟑 . (b) A particle is moving with nonzero velocity along the polar curve given by 𝒓=𝟑+𝟐 𝐜𝐨𝐬 𝜽 has position of 𝒙 𝒕 ,𝒚 𝒕 at time 𝒕, with 𝜽=𝟎 with 𝒕=𝟎. This particle moves along the curve so that 𝒅𝒚 𝒅𝒕 = 𝒅𝒓 𝒅𝜽 . Find the value of 𝒅𝒓 𝒅𝒕 at 𝜽= 𝝅 𝟑 and interpret answer in terms of motion of the particle. 4/24/2019 11:25 PM §10.5C: Polar AP Practice
Example 3b The graphs of the polar curves of 𝒓=𝟐 and 𝒓=𝟑+𝟐 𝐜𝐨𝐬 𝜽 are shown. The curves intersect when 𝜽= 𝟐𝝅 𝟑 and 𝜽= 𝟒𝝅 𝟑 . (b) A particle is moving with nonzero velocity along the polar curve given by 𝒓=𝟑+𝟐 𝐜𝐨𝐬 𝜽 has position of 𝒙 𝒕 ,𝒚 𝒕 at time 𝒕, with 𝜽=𝟎 with 𝒕=𝟎. This particle moves along the curve so that 𝒅𝒚 𝒅𝒕 = 𝒅𝒓 𝒅𝜽 . Find the value of 𝒅𝒓 𝒅𝒕 at 𝜽= 𝝅 𝟑 and interpret answer in terms of motion of the particle. 4/24/2019 11:25 PM §10.5C: Polar AP Practice
Example 3c The graphs of the polar curves of 𝒓=𝟐 and 𝒓=𝟑+𝟐 𝐜𝐨𝐬 𝜽 are shown. The curves intersect when 𝜽= 𝟐𝝅 𝟑 and 𝜽= 𝟒𝝅 𝟑 . (c) For the particle described in part (b), 𝒅𝒚 𝒅𝒕 = 𝒅𝒓 𝒅𝜽 , find the value of 𝒅𝒚 𝒅𝒕 at 𝜽= 𝝅 𝟑 and interpret answer in terms of motion of the particle. 4/24/2019 11:25 PM §10.5C: Polar AP Practice
Example 3c The graphs of the polar curves of 𝒓=𝟐 and 𝒓=𝟑+𝟐 𝐜𝐨𝐬 𝜽 are shown. The curves intersect when 𝜽= 𝟐𝝅 𝟑 and 𝜽= 𝟒𝝅 𝟑 . (c) For the particle described in part (b), 𝒅𝒚 𝒅𝒕 = 𝒅𝒓 𝒅𝜽 , find the value of 𝒅𝒚 𝒅𝒕 at 𝜽= 𝝅 𝟑 and interpret answer in terms of motion of the particle. 4/24/2019 11:25 PM §10.5C: Polar AP Practice
Example 3c The graphs of the polar curves of 𝒓=𝟐 and 𝒓=𝟑+𝟐 𝐜𝐨𝐬 𝜽 are shown. The curves intersect when 𝜽= 𝟐𝝅 𝟑 and 𝜽= 𝟒𝝅 𝟑 . (c) For the particle described in part (b), 𝒅𝒚 𝒅𝒕 = 𝒅𝒓 𝒅𝜽 , find the value of 𝒅𝒚 𝒅𝒕 at 𝜽= 𝝅 𝟑 and interpret answer in terms of motion of the particle. 4/24/2019 11:25 PM §10.5C: Polar AP Practice
Example 4 A curve is drawn in the 𝒙𝒚-plane and is described by the equation in polar coordinates 𝒓=𝜽+𝐬𝐢𝐧 𝟐𝜽 for 𝟎≤𝜽≤𝝅. Using the calculator: Find the area bounded by the curve and 𝒙-axis Find the angle 𝜽 that corresponds to the point on the curve with 𝒙-coordinate 𝒙=−𝟐. For 𝝅 𝟑 <𝜽< 𝟐𝝅 𝟑 , 𝒅𝒓 𝒅𝜽 is negative. What does your answer tell you about 𝒓? What does it tell you about the curve? At what angle 𝜽 is the interval 𝟎≤𝜽≤ 𝝅 𝟐 the curve farthest away from the origin. Justify. 4/24/2019 11:25 PM §10.5C: Polar AP Practice
Example 4a A curve is drawn in the 𝒙𝒚-plane and is described by the equation in polar coordinates 𝒓=𝜽+𝐬𝐢𝐧 𝟐𝜽 for 𝟎≤𝜽≤𝝅. Using the calculator: Find the area bounded by the curve and x-axis 4/24/2019 11:25 PM §10.5C: Polar AP Practice
Example 4b A curve is drawn in the 𝒙𝒚-plane and is described by the equation in polar coordinates 𝒓=𝜽+𝐬𝐢𝐧 𝟐𝜽 for 𝟎≤𝜽≤𝝅. Using the calculator: b) Find the angle 𝜽 that corresponds to the point on the curve with 𝒙-coordinate 𝒙=−𝟐. 4/24/2019 11:25 PM §10.5C: Polar AP Practice
Example 4c A curve is drawn in the 𝒙𝒚-plane and is described by the equation in polar coordinates 𝒓=𝜽+𝐬𝐢𝐧 𝟐𝜽 for 𝟎≤𝜽≤𝝅. For 𝝅 𝟑 <𝜽< 𝟐𝝅 𝟑 , 𝒅𝒓 𝒅𝜽 is negative. What does your answer tell you about 𝒓? What does it tell you about the curve? 𝒅𝒓 𝒅𝜽 is how fast the radius is changing with respects to rotation. Since 𝒅𝒓 𝒅𝜽 is negative it is away from the pole/origin. 4/24/2019 11:25 PM §10.5C: Polar AP Practice
Example 4d A curve is drawn in the 𝒙𝒚-plane and is described by the equation in polar coordinates 𝒓=𝜽+𝐬𝐢𝐧 𝟐𝜽 for 𝟎≤𝜽≤𝝅. At what angle 𝜽 is the interval 𝟎≤𝜽≤ 𝝅 𝟐 the curve farthest away from the origin. Justify. 4/24/2019 11:25 PM §10.5C: Polar AP Practice