Rates of Change of Polar Functions (Lesson 8.5)
Unit 8 Day 5
CED Topic(s): 3.15
Analyze and interpret key features of polar functions including intervals of increasing/decreasing and extrema.
Find and interpret the average rate of change of a polar function.
Quick Lesson Plan
Congratulations! You’ve made it to the final lesson of the course!
This topic may be a new one for you (as it was for me) but it perfectly captures so much of the work we have done in this course. Instead of just talking about graphs of polar functions and their key features (a static approach), we can actually describe the polar function’s rate of change! How fast is the radius changing? When is the radius increasing? When is it decreasing?
In this activity, students work with a polar function whose graph is a rose. Question 1 activates students’ thinking about polar functions, rather than just equations, by asking them to identify the independent and dependent variable.
In questions 2 students are asked about the maximum value of r (which they are likely to know just from the equation and the patterns they noticed in the previous two lessons) but they also must identify the angle at which this maximum radius occurs. This now brings in the analytical aspect of figuring out what value of theta would produce an output of 5. We hope students are becoming ever more comfortable using the thought process we established in Lesson 7.3 for solving trigonometric equations. Note that the interval for theta is 0≤theta≤π. Students may misinterpret this to mean that 3theta must be in this range as well and not look for more than one solution in question 2, especially since the next angle is outside of the standard domain of [0,2π].
Question 3 asks about the minimum value of r and this one’s tricky! Because the radius is often thought of as a distance and distances can’t be negative, students may assume that the minimum value of r is 0! But identifying the range of the function as [-5,5] will help students see that the minimum radius is actually -5 (which represents a maximum distance from the origin, more on that later).
Question 4 asks where the function is equal to 0. Students have encountered this question countless times in this course. While the method for solving is the same, students must realize that an output of 0 no longer represents an x-intercept, it represents a point at the pole!
Using the information from questions 2-5, students can complete the table in question 6. Note that students record both the sign of the radius and whether the radius is increasing or decreasing. It may not be intuitive to students at which exact angles the radius changes from positive to negative or vice versa. The continuity of these functions which stems from the periodic behavior of the sine and cosine functions can give us a clue! Students don’t have formal notions of continuity but they can understand intuitively that the values aren’t jumping around, just like when moving the point on the unit circle in the Geogebra applet (Lesson 7.1 and 7.4), the lengths of the segments increased and decreased continuously. In order for the radius to change from positive to negative values or negative to positive values, it must pass through 0 (the pole). Identifying the zeros of the function, then, becomes a key strategy for determining where the function changes signs. Identifying the maximum and minimum values of the function becomes a key strategy for determining where the function changes from increasing to decreasing or vice versa.
Question 7 reviews ideas from Unit 1 about how relative extrema of a function can be identified based on the function’s increasing and decreasing behavior.
In question 8, students learn to distinguish between when a radius is increasing and when the distance from the origin is increasing. This is a tricky concept but we hope that by spiraling this idea throughout the course, students will have built up a robust conceptual understanding of how to interpret negative output values and negative rates of change. Finally, in question 9 students calculate an average rate of change. In the debrief, make sure students understand that this value represents a change in the radius per radian.
What is the range of this function? How do you know?
Is there another angle where the sine value is 1?
How will I know when to stop adding 2π to my angle?
How will you know the exact place (input, angle) where the function starts decreasing?
What does it mean that the radius is both positive and decreasing? What does it mean if the radius is both negative and decreasing?
What happens at the angles at the endpoints of these intervals?
What are the units of this rate of change? How do you know?
One key aspect of this lesson is being able to distinguish between when the radius is increasing and decreasing and when the distance from the origin is increasing and decreasing. It can be helpful to go through several scenarios to determine which combinations of positive and negative outputs and increasing and decreasing behavior results in an increasing distance and a decreasing distance from the origin. Students should not just memorize the table in the QuickNotes! Using a number line and thinking about one dimensional change is really helpful for establishing this idea. If you teach AP Calculus, you’ve probably noticed how this concept is identical to determining when a particle is moving toward or away from the origin.
In question 1e of the Check Your Understanding, note that the question asks where f is changing faster, not where f has a greater rate of change. This means we are looking for the rate of change with the greatest magnitude, not value.