Polar Equation Questions (Type 9)

AP Questions Type 9: Polar Equations (BC Only)

Ideally, as with parametric and vector functions, polar curves should be introduced and covered thoroughly in a pre-calculus course. Questions on the BC exams have been concerned only with calculus ideas related to polar curves. Students have not been asked to know the names of the various curves (rose curves, limaçons, etc.). The graphs are usually given in the stem of the problem; students are expected to be able to determine which is which if more than one is given. Students should know how to graph polar curves on their calculator, and the simplest by hand. Intersection(s) of two graphs may be given or easy to find.

What students should know how to do:

  • Calculate the coordinates of a point on the graph,
  • Find the intersection of two graphs (e.g. to use as limits of integration).
  • Find the area enclosed by a graph or graphs: \displaystyle A=\frac{1}{2}\int_{{{{\theta }_{1}}}}^{{{{\theta }_{2}}}}{{{{{\left( {r\left( \theta \right)} \right)}}^{2}}d\theta }}
  • Use the formulas \displaystyle x\left( \theta \right)=r\left( \theta \right)\cos \left( \theta \right)\text{ and }y\left( \theta \right)=r\left( \theta \right)\sin \left( \theta \right)  to convert from polar to parametric form,
  • Calculate \displaystyle \frac{{dy}}{{d\theta }} and \displaystyle \frac{{dx}}{{d\theta }} (Hint: use the product rule on the equations in the previous bullet).
  • Discuss the motion of a particle moving on the graph by discussing the meaning of \displaystyle \frac{{dr}}{{d\theta }} (motion towards or away from the pole), \displaystyle \frac{{dr}}{{d\theta }} (motion in the vertical direction), and/or \displaystyle \frac{{dx}}{{d\theta }} (motion in the horizontal direction).
  • Find the slope at a point on the graph, \displaystyle \frac{{dx}}{{dx}}=\frac{{dy/d\theta }}{{dx/d\theta }}

When this topic appears on the free-response section of the exam there is no Parametric/vector motion question and vice versa. When not on the free-response section there are one or more multiple-choice questions on polar equations.

This question typically covers topics from Unit 9 of the CED.


Free-response questions:

  • 2013 BC 2
  • 2014 BC 2
  • 2017 BC 2
  • 2018 BC 5
  • 2019 AB 2

Multiple-choice questions from non-secure exams:

  • 2008 BC 26
  • 2012 BC 26, 91

Other posts on Polar Equations

Polar Basics

Polar Equations for AP Calculus

Extreme Polar Conditions

Polar Equations (Review 2018)


Revised March 12, 2021, April 8, 2022

Extreme Polar Conditions

Looking at the graphs of polar curves can be quite fascinating. Doing “calculus” kinds of things is different, yet the same as in Cartesian coordinates. A discussion on the AP Calculus Community here got me thinking about extreme values of polar functions.

The terms maximum and minimum here refers to the value of r(θ) which may be positive (when on the ray making an angle of θ with the polar axis), zero (at the pole), or negative (on the ray opposite to the one making an angle of θ with the polar axis). The distance from the pole to the point is |r(θ)|.  As Mark Howell pointed out in the thread linked above, extreme values of r(θ) lie on a circle or circles centered at the pole with radius of |r(θ)|, and finding the slope of tangent lines at the extremes is relatively easy, requiring no calculus: the slope of the ray is y/x, so the slope of the tangent at the extreme is –x/y. As with Cartesian coordinates, at extreme values \displaystyle dr/d\theta =0 since r(θ) is change from increasing to decreasing at this point (or vice versa). 

A quick look at the graph of simple polar functions seems to show obvious maximum values for r(θ), but a closer look reveals some complications.

The graph of \displaystyle r\left( \theta \right)=\sin \left( {2\theta } \right), shown below, appears to show 4 maximums. However, if we trace the graph, we find that these points are (1, π/4), (–1, 3π/4), (1, 5π/4) and (–1, 7π/4). Two of the values are maximums where r(θ) = 1 and two are minimums where r(θ) = –1.

Thus, points may be both maximums and minimums.

Polar Curves can be really fun. While working on the idea above, I explored some other curves. Try some yourself using Desmos, GeoGabra, or another graphing app with sliders. Shown below are members of the family of polar curves \displaystyle r\left( \theta \right)=A\cos \left( {B\theta } \right)+C\sin \left( {D\theta } \right). The domain is extended to \displaystyle 0\le \theta \le 20\pi . Notice:

  • How very slight changes in the parameters give very different looking graphs
  • Other values give far less “organized” curves
  • In the third graph, the maximums and minimums on the irregular part of the curve closest to the pole

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Polar Equations for AP Calculus

A recent thread on the AP Calculus Community bulletin boards concerned polar equations. One teacher observed that her students do not have a very solid understanding of polar graphs when they get to calculus. I expect this is a common problem. While ideally the polar coordinate system should be a major topic in pre-calculus courses, this is sometimes not the case. Some classes may even omit the topic entirely. Getting accustomed to a new coordinate scheme and a different way of graphing is a challenge.  I remember not having that good an understanding myself when I entered college (where first-year calculus was a sophomore course). Seeing an animated version much later helped a lot. 

This blog post will discuss the basics of polar equations and their graphs. It will not be as much as students should understand, but I hope the basics discussed here will be a help. There are also some suggestions for extending the study of polar function as the end.

Instead of using the Cartesian approach of giving every point in the plane a “name” by giving its distance from the y-axis and the x-axis as an ordered pair (x,y), polar coordinates name the point differently. Polar coordinates use the ordered pair (r, θ), where r, gives the distance of the point from the pole (the origin) as a function of θ, the angle that the ray from the pole (origin) to the point makes with the polar axis, (the positive half of the x-axis).

Start with this Desmos graph. It will help if you open it and follow along with the discussion below. The equation in the example is \displaystyle r(\theta )=2+4\sin (\theta ) You may change this to explore other graphs. (Because of the way Desmos graphs, you cannot have a slider for θ; the a-slider will move the line and the point on the graph. r(a) gives the value of r(θ).

  • Notice that as the angle changes the point at varying distance from the pole traces a curve. 
  • Move the slider to π/6. Since sin(π/6) = 0.5, r(π/6) = 4. The red dot is at the point (4, π/6). Move the slider to other points to see how they work. For example, θ = π/2 gives the point (6,π/2).
  • When the slider gets to θ = 7π/6, r = 0 and the point is at the pole. After this the values of r are negative, and the point is now on the ray opposite to the ray pointing into the third and fourth quadrants. The dashed line turns red to remind you of this.
  • As we continue around, the point returns to the origin at θ = 11π/6, then values are again positive. 
  • The graph returns to its starting point when θ = 2π. Note (2,0) is the same point as (2, 2π).
  • Even though this is the graph of a function, some points may be graphed more than once and the vertical line test does not apply. 
  • If we continued around, the graph will retrace the same path. This often happens when the polar function contains trig functions with integer multiples of θ.
    • This does not usually happen if no trig functions are involved – try the spiral r = θ.
    • If you enter non-integer multiples of θ and extend the domain to large values, vastly different graphs will appear, often making nice designs. Try \displaystyle r\left( \theta \right)=2+4\sin \left( {1.3\theta } \right) for \displaystyle 0\le \theta \le 20\pi . This is an area for exploration (if you have time).

In pre-calculus courses several families of polar graphs are often studied and named. For example, there are cardioids, rose curves, spirals, limaçons, etc. The AP Exams do not refer to these names and students are not required to know the names. The exception is circles which have the following forms where R is the radius: θ=R, r = Rsin(θ) or r = Rsin(θ)

To change from polar to rectangular for use the equations x=r\cos \left( \theta  \right) and  y=r\sin \left( \theta  \right). This is simple right triangle trigonometry (draw a perpendicular from the point to the x-axis and from there to the pole). 

To change from rectangular to polar form use  r=\sqrt{{{{x}^{2}}+{{y}^{2}}}} and  \displaystyle \theta =\arctan \left( {\tfrac{y}{x}} \right)

AP Calculus Applications

There are two applications that are listed on the AP Calculus Course and Exam Description: using and interpreting the derivative of polar curves (Unit 9.7) and finding the area enclosed by a polar curve(s) (Units 9.8 and 9.9).

Since calculus is concerned with motion, AP Students should be able to analyze polar curves for how things are changing:

  • The rate of change of r away from or towards the pole is given by  \displaystyle \frac{{dr}}{{d\theta }}
  • The rate of change of the point with respect to the x-direction is given by  \displaystyle \frac{{dx}}{{d\theta }} where \displaystyle x=r\cos \left( \theta \right) from above.
  • The rate of change of the point with respect to the y-direction is given by  \displaystyle \frac{{dy}}{{d\theta }} where \displaystyle y=r\sin \left( \theta \right)from above.
  • The slope of the tangent line at a point on the curve is \displaystyle \frac{{dy}}{{dx}}=\frac{{dy/d\theta }}{{dx/d\theta }}. See 2018 BC5 (b)

Area

 \displaystyle \underset{{\Delta \theta \to 0}}{\mathop{{\lim }}}\,\sum\limits_{{i=1}}^{\infty }{{\tfrac{1}{2}}}{{r}_{i}}^{2}\Delta \theta =\tfrac{1}{2}\int_{{{{\theta }_{1}}}}^{{{{\theta }_{2}}}}{{{{r}^{2}}d\theta }}

CAUTION: In using this formula, we need to be careful that the curve does not overlap itself. In the Desmos example, the smaller loop overlaps the larger loop; integrating from 0 to 2π counts the inner loop twice. Notice how this is handled by considering the limits of integration dividing the region into non-overlapping regions: 

  • The area of the outer loop is  \displaystyle \tfrac{1}{2}\int_{{-\pi /6}}^{{7\pi /6}}{{{{{(2+4\sin (\theta ))}}^{2}}d\theta }}\approx 35.525
  • The area of the inner loop is  \displaystyle \tfrac{1}{2}\int_{{7\pi /6}}^{{11\pi /6}}{{{{{(2+4\sin (\theta ))}}^{2}}d\theta }}\approx 2.174
  • Integrating over the entire domain gives the sum of these two:  \displaystyle \tfrac{1}{2}\int_{0}^{{2\pi }}{{{{{(2+4\sin (\theta ))}}^{2}}d\theta }}=12\pi \approx 37.699. This is not the correct area of either part. 

This problem can be avoided by considering the geometry before setting up the integral: make sure the areas do not overlap. Restricting r to only non-negative values is often required by the fine print of the theorem in textbooks, but this restriction is not necessary when finding areas and makes it difficult to find, say, the area of the smaller inner loop of the example. Here is another example: 

\displaystyle r\left( \theta \right)=\cos \left( {3\theta } \right). Between 0 and \displaystyle 2\pi this curve traces the same path twice.

Polar Equation Questions (Type 9)

AP  Questions Type 9:  Polar Equations (BC Only)

Ideally, as with parametric and vector functions, polar curves should be introduced and covered thoroughly in a pre-calculus course. Questions on the BC exams have been concerned only with calculus ideas related to polar curves. Students have not been asked to know the names of the various curves (rose curves, limaçons, etc.). The graphs are usually given in the stem of the problem; students are expected to be able to determine which is which if more than one is given. Students should know how to graph polar curves on their calculator, and the simplest by hand. Intersection(s) of two graph may be given or easy to find.

What students should know how to do:

  • Calculate the coordinates of a point on the graph,
  • Find the intersection of two graphs (to use as limits of integration).
  • Find the area enclosed by a graph or graphs: Area =\displaystyle A=\tfrac{1}{2}\int_{{{\theta }_{1}}}^{{{\theta }_{2}}}{(r(}θ\displaystyle ){{)}^{2}}dθ
  • Use the formulas x\left( \theta  \right)\text{ }=~r\left( \theta  \right)\text{cos}\left( \theta  \right)~~\text{and}~y\left( \theta  \right)\text{ }=~r(\theta )\text{sin}\left( \theta  \right)~  to convert from polar to parametric form,
  • Calculate \displaystyle \frac{dy}{d\theta } and \displaystyle \frac{dx}{d\theta } (Hint: use the product rule on the equations in the previous bullet).
  • Discuss the motion of a particle moving on the graph by discussing the meaning of \displaystyle \frac{dr}{d\theta } (motion towards or away from the pole), \displaystyle \frac{dy}{d\theta } (motion in the vertical direction), and/or \displaystyle \frac{dx}{d\theta } (motion in the horizontal direction).
  • Find the slope at a point on the graph, \displaystyle \frac{dy}{dx}=\frac{dy/d\theta }{dx/d\theta }.

When this topic appears on the free-response section of the exam there is no Parametric/vector motion question and vice versa. When not on the free-response section there are one or more multiple-choice questions on polar equations.

Free-response questions:

  • 2013 BC 2
  • 2014 BC 2
  • 2017 BC 2
  • 2018 BC 5
  • 2019 AB 2

Multiple-choice questions from non-secure exams:

  • 2008 BC 26
  • 2012 BC 26, 91

Other posts on Polar Equations

Polar Basics

Polar Equations for AP Calculus

Extreme Polar Conditions

Polar Equations (Review 2018)


 

 

 

 

Revised March 12, 2021


 


This question typically covers topics from Unit 9 of the 2019 CED .


Schedule of future posts for reviewing for the 2019 Exams

Exams for AP Calculus are Tuesday May 5, 2020 at 08:00 local time

NOTE: The type number I’ve assigned to each type DO NOT correspond to the 2019 CED Unit numbers. Many AP Exam questions have parts from different Units. The CED Unit numbers will be referenced in each post.

Tuesday February 25 – AP Exam Review 2020 
Friday, February 28 – Reviewing Resources 2020
Tuesday March 3, 2020: Rate and accumulation questions (Type 1) 
Friday March 6, 2020: Linear motion problems (Type 2) 
Tuesday March 10, 2020: Graph analysis problems (Type 3)
Friday March 13, 2020: Area and volume problems (Type 4)
Tuesday March 17, 2020: Table and Riemann sum questions (Type 5)
Friday March 20, 2020: Differential equation questions (Type 6)
Tuesday March 24, 2020: Other questions (Type 7)
Friday March 27, 2020: Parametric and vector questions (Type 8) BC topic
Tuesday March 31, 2020: Polar equations questions (Type 9) BC Topic
Friday April 3, 2020: Sequences and Series questions (Type 10) BC Topic


 

 

 

 

 

 

 

Parametric Equations and Vectors

In BC calculus the only application parametric equations and vectors is motion in a plane. Polar equations concern area and the meaning of derivatives. See the review notes for more detail and an outline of the topics. (only 3 items here)

Motion Problems: Same Thing Different Context (11-16-2012)

Implicit Differentiation of Parametric Equations (5-17-2014)

A Vector’s Derivative (1-14-2015)

Review Notes 

Type 8: Parametric and Vector Equations (3-30-2018) Review Notes

Type 9: Polar Equation Questions (4-3-2018) Review Notes

Roulettes 

This is a series of posts that could be used when teaching polar form and curves defined by vectors (or parametric equations). They might be used as a project. Hopefully, the equations that produce the graphs will help students understand these topics. Don’t let the names put you off. Except for one post, there is no calculus here.

Rolling Circles  (6-24-2014)

Epicycloids (6-27-2014)

Epitrochoids (7-1-2014) The most common of these are the cycloids.

Hypocycloids and Hypotrochoids  (7-7-2014)

Roulettes and Calculus  (7-11-2014)

Roulettes and Art – 1  (7-17-2014)

Roulettes and Art – 2 (7-23-2014)

Limaçons (7-28-2014)


The College Board is pleased to offer a new live online event for new and experienced AP Calculus teachers on March 5th at 7:00 PM Eastern.

I will be the presenter.

The topic will be AP Calculus: How to Review for the Exam:  In this two-hour online workshop, we will investigate techniques and hints for helping students to prepare for the AP Calculus exams. Additionally, we’ll discuss the 10 type questions that appear on the AP Calculus exams, and what students need know and to be able to do for each. Finally, we’ll examine resources for exam review.

Registration for this event is $30/members and $35/non-members. You can register for the event by following this link: http://eventreg.collegeboard.org/d/xbqbjz


 

 

 

 

 


 

Polar Equations

Ideally, as with parametric and vector functions, polar curves should be introduced and covered thoroughly in a pre-calculus course. Questions on the BC exams have been concerned with calculus ideas related to polar curves. Students have not been asked to know the names of the various curves (rose, curves, limaçons, etc.). The graphs are usually given in the stem of the problem, but students should know how to graph polar curves on their calculator, and the simplest by hand.

What students should know how to do:

  • Calculate the coordinates of a point on the graph,
  • Find the intersection of two graphs (to use as limits of integration).
  • Find the area enclosed by a graph or graphs: Area =\displaystyle A=\tfrac{1}{2}\int_{{{\theta }_{1}}}^{{{\theta }_{2}}}{(r(}θ\displaystyle ){{)}^{2}}dθ
  • Use the formulas x\left( \theta  \right)\text{ }=~r\left( \theta  \right)\text{cos}\left( \theta  \right)~~\text{and}~y\left( \theta  \right)\text{ }=~r(\theta )\text{sin}\left( \theta  \right)~  to convert from polar to parametric form,
  • Calculate \displaystyle \frac{dy}{d\theta } and \displaystyle \frac{dx}{d\theta } (Hint: use the product rule on the equations in the previous bullet).
  • Discuss the motion of a particle moving on the graph by discussing the meaning of \displaystyle \frac{dr}{d\theta } (motion towards or away from the pole), \displaystyle \frac{dy}{d\theta } (motion in the vertical direction) or \displaystyle \frac{dx}{d\theta } (motion in the horizontal direction).
  • Find the slope at a point on the graph, \displaystyle \frac{dy}{dx}=\frac{dy/d\theta }{dx/d\theta }.

This topic appears only occasionally on the free-response section of the exam instead of the Parametric/vector motion question. The most recent on the released exams were in 2007,  2013, 2014, and 2017. If the topic is not on the free-response then 1, or maybe 2 questions, probably finding area, can be expected on the multiple-choice section.

Shorter questions on these ideas appear in the multiple-choice sections. As always, look over as many questions of this kind from past exams as you can find.


Here are the few post I have written on polar curves and polar graphing.

Limaçons   A discussion of how polar curves are graphed

Back in the summer of 2014 I got interested in some polar equations and wrote a series of post on them which include some gifs showing how they are graphed. They are nothing that will appear on the AP exams. You can use them as enrichment if you like.

Rolling Circles

Roulettes and Calculus

Epitrochoids

Epicycloids

Hypocycloids and Hypotrochoids

Roulettes and Art – 1  

Roulettes and Art – 2


 

 

 


 

Polar Curves (Type 9 for BC only)

Ideally, as with parametric and vector functions, polar curves should be introduced and covered thoroughly in a pre-calculus course. Questions on the BC exams have been concerned with calculus ideas related to polar curves. Students have not been asked to know the names of the various curves (rose, curves, limaçons, etc.). The graphs are usually given in the stem of the problem, but students should know how to graph polar curves on their calculator, and the simplest by hand.

What students should know how to do:

  • Calculate the coordinates of a point on the graph,
  • Find the intersection of two graphs (to use as limits of integration).
  • Find the area enclosed by a graph or graphs: Area =\displaystyle A=\tfrac{1}{2}\int_{{{\theta }_{1}}}^{{{\theta }_{2}}}{(r(}θ\displaystyle ){{)}^{2}}dθ
  • Use the formulas x\left( \theta  \right)\text{ }=~r\left( \theta  \right)\text{cos}\left( \theta  \right)~~\text{and}~y\left( \theta  \right)\text{ }=~r(\theta )\text{sin}\left( \theta  \right)~  to convert from polar to parametric form,
  • Calculate \displaystyle \frac{dy}{d\theta } and \displaystyle \frac{dx}{d\theta } (Hint: use the product rule on the equations in the previous bullet).
  • Discuss the motion of a particle moving on the graph by discussing the meaning of \displaystyle \frac{dr}{d\theta } (motion towards or away from the pole), \displaystyle \frac{dy}{d\theta } (motion in the vertical direction) or \displaystyle \frac{dx}{d\theta } (motion in the horizontal direction).
  • Find the slope at a point on the graph, \displaystyle \frac{dy}{dx}=\frac{dy/d\theta }{dx/d\theta }.

This topic appears only occasionally on the free-response section of the exam instead of the Parametric/vector motion question. The most recent on the released exams were in 2007,  2013, 2014, and 2017. If the topic is not on the free-response then 1, or maybe 2 questions, probably finding area, can be expected on the multiple-choice section.

Shorter questions on these ideas appear in the multiple-choice sections. As always, look over as many questions of this kind from past exams as you can find.


Next post:

Tuesday April 4: For BC Sequences and Series.

Friday April 7, 2017 The Domain of the solution of a differential equation.


 

 

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