1. Sherin Stanley, Sophia Versola
Polar- Calc Review
Sherin Stanley, Sophia Versola

2. Classwork:
Multiple Choice 1-8 → Kahoot
FRQ #9 → Independent Work
Homework:
FRQs #10 & #11

3. Card 69: Common Polar Graphs/Equations
Rose Curves:
r=a(cos/sin)(n𝜃)
Period- 𝝿 if n is odd, 2𝝿 if n is even
If n is odd, then n represents the number of petals
If n is even, then n multiplied by 2 represents the number of petals
SYMMETRY:
Cos- symmetrical to the x-axis
Sin- symmetrical to the y-axis
Cos OR sin, NOT cos divided by sin

4. Card 69 (cont.) Limacons: period=2𝝿 Convex Limacon
Aka dimpled limacon
Convex Limacon
Same general formula
a/b>2

5. Card 69 (cont.) Circles and Lemniscates: Period=𝝿
The distance between the pole and the tip is the a value in your lemniscate
Lemniscate formulas-
Circle formulas-

6. Card 70: Polar Conversion Formulas

7. Card 71: Power reducing formulas

8. Card 72: Finding Polar Area
If f is continuous and non-negative on the interval [a,b], where 0 < b - a < 2𝝿, then the area of the region bounded by the graph of r=f(𝜃) between radial lines 𝜃=a and 𝜃=b is:
Figure out the formula of the polar graph given (sometimes the formula itself will be given to you)
Find bounds (in-depth step by step on the next slide)
Plug into the Area formula and integrate
Sometimes it’s helpful to find the area of half of a region and then multiplying it by 2.
To find the area enclosed by two polar curves, sub in the outer curve squared minus the inner curve squared into r.

9. Card 72 (cont.) Determining Radial Lines:
Sketch the region of the area you’re trying to solve for
Draw a radial line from the pole to the boundary curve
Find the interval of values in which the radial line sweeps out the region R
*Set R equal to zero in order to find one or both of your radial lines.
There’s usually multiple ways to solve polar area problems, so don’t feel discouraged when you don’t see your answer as an answer choice.

10. Polar Area Example
What is equal to the area of the region inside the polar curve r=2cos𝜃 and outside the polar curve r=cos𝜃?

11. Card 73: Arc Length
*This will not be on the AP Exam, but you still need to know it for the BC test*
Arc Length Formula →
s =
Steps for Solving:
Establish your interval [a,b] and your polar equation r(𝜃)
Derive r(𝜃)
Substitute your given polar equation r and r’ into the Arc Length formula above
Using the interval that is given, evaluate the integral to find the Arc Length
(simple substitution, usually calc active)

12. Card 74: ∫teps to finding dy/dx
Establish the given 𝜃 and polar function r(𝜃)
Using the equations y=rsin𝜃 and x=rcos𝜃, substitute your given polar function into r to find your x and y equations
Find dx/d𝜃 and dy/d𝜃
dy/dx=(dy/d𝜃)/(dx/d𝜃)
Substitute your given value for 𝜃 into the dy/dx equation to find your slope

13. Card 75: Tangent Lines of Polar Equations
Steps:
Establish your r(𝜃), your r value, and your 𝜃 value
In order to write a linear function for the tangent line, use the equations x=rcos𝜃 and y=rsin𝜃, and your r and 𝜃 values to find x and y
Substitute your x and y into the tangent line equation, leaving dy/dx to be found
Find dy/dx
Sub dy/dx into the tangent line equation

14. Card 76: Interpreting Motion
Rules to remember:
dr/d𝜃 > 0 → the particle is moving away from the origin/pole; the radius is getting larger
dr/d𝜃 < 0 → the particle is moving toward the origin/pole; the radius is getting smaller
dr/d𝜃 = 0 → the particle’s position might be at a maximum or minimum distance from the origin/pole

15. Interpreting Motion Example
A particle moving with a nonzero velocity along the polar curve given by r=4- 2cos𝜃 has position (x(t),y(t)) at time t, with 𝜃=0 when t=0. This particle moves along the curve so that dr/dt=dr/d𝜃. Find the value of dr/dt at 𝜃=5π/3 and interpret your answer in terms of the motion of the particle.

16. Time for a Kahoot!