1. Calculus BC AP/Dual, Revised ©2018
Polar Area Day 4
Section 10.5C
Calculus BC AP/Dual, Revised ©2018
4/24/ :25 PM
§10.5C: Polar AP Practice

2. Review Find the first derivative of 𝒓=𝟏+ 𝐜𝐨𝐬 𝜽 4/24/2019 11:25 PM
§10.5C: Polar AP Practice

3. 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/ :25 PM
§10.5C: Polar AP Practice

4. 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/ :25 PM
§10.5C: Polar AP Practice

5. 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/ :25 PM
§10.5C: Polar AP Practice

6. Example 2
Find the area between the two spirals of 𝒓=𝜽 and 𝒓=𝟐𝜽 for 𝟎≤𝜽≤𝟐𝝅.
4/24/ :25 PM
§10.5C: Polar AP Practice

7. 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/ :25 PM
§10.5C: Polar AP Practice

8. 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/ :25 PM
§10.5C: Polar AP Practice

9. 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/ :25 PM
§10.5C: Polar AP Practice

10. 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/ :25 PM
§10.5C: Polar AP Practice

11. 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/ :25 PM
§10.5C: Polar AP Practice

12. 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/ :25 PM
§10.5C: Polar AP Practice

13. 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/ :25 PM
§10.5C: Polar AP Practice

14. 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/ :25 PM
§10.5C: Polar AP Practice

15. 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/ :25 PM
§10.5C: Polar AP Practice

16. 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/ :25 PM
§10.5C: Polar AP Practice

17. 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/ :25 PM
§10.5C: Polar AP Practice

18. 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/ :25 PM
§10.5C: Polar AP Practice

19. 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/ :25 PM
§10.5C: Polar AP Practice