I got my butt kicked on KA this week. My goal was to get through 3 videos and 5 exercises and I barely managed to get through 2 exercises. Both exercises had to do with polar coordinates which I have a very weak understanding of. They both also used what’s called ‘polar curves’ which I don’t understand at all… 😔 I passed the exercises by memorizing the steps/method needed to get to the correct solutions, but I didn’t really understand what was going on in the questions, particularly when it came to polar curves. By the end of the week, I slowly started to get a vague idea of what the questions were asking but I can tell I’m still far from intuitively understanding what’s going on. I’ve been in this position many times before however, where there’s a ton of fog surrounding what I’m working on but, the more I grind away at it, the more the fog will dissipate. It’s always frustrating trying to find my way through it though. 😠

Both exercises I worked through this week were from the section Polar Functions and the first exercise was called Differentiate Polar Functions. This exercise had questions where I was given a function for r(θ) and then I needed to input it into either x(θ) = r(θ)cos(θ) or into y(θ) = r(θ)sin(θ) and then differentiate both sides to find the rate of change of x at a given point. To be clear, I don’t know how polar coordinates work. I don’t understand why x(θ) = r(θ)cos(θ). I’m pretty sure that r(θ) is supposed to be the hypotenuse of a polar coordinate, i.e. the magnitude of the coordinate from the origin, so I don’t know why it’s multiplied by cos(θ). That would mean cos(θ) = x(θ)/r(θ) which would mean that x(θ) is the length of x as a function of θ. That could be exactly what’s going on or it could also be completely wrong. As of now, I have no idea and don’t understand. 🤬 Anyways…

Here are 3 questions from the first exercise:

Question 1  

Even though I don’t know what’s going on with polar coordinates, I’m definitely getting a lot better at working with them and finding their derivatives. I redid my notes for the question above but I actually got this question exactly correct, term for term, without any review. As you can see, I had to use the chain rule, some trig and some algebra to come up with the final expression, and I was pumped that I was able to work through it and get to the exact same answer as KA without really knowing what was happening. 🤷🏻‍♂️

Question 2

In this question, I was given r(θ) = θ²/5 and then asked to find the rate of change of x at point P on the graph. Since P is on the x-axis, and it’s the first time it crosses over the x-axis as the functions rotates counter-clockwise away from the origin, this means that θ = π. To find the rate of change of x at that point, the first thing you need to do is find the derivative of x(θ) = r(θ)cos(θ) which is the function for the x-coordinate as a function of θ. Once you find the expression for x’(θ), you then input π into the expression to output the change in x at that point, i.e. the rate of change of x at point P.

Question 3

It took me until Thursday to get through the first exercise and into the second exercise which was titled Tangents to Polar Curves. This exercise was very similar to the first except that once I found the derivative of the polar curve at a given value of θ, I then needed to use the derivative to come up with an equation for the line tangent to that point on the polar curve. This exercise had me use point-slope form to find the equation of the line tangent to the curve, which is another thing that I don’t understand right now. Even though I didn’t understand how to use point-slope form, most of the questions were multiple choice so I was able to eventually pass the exercise by making some calculated guesses based on what I was able to determine on my own and the four given answers. Here are two questions from that exercise:

Question 4

Even though I don’t really know what’s going on, I can tell that these questions are helping me get closer to understanding how and why parametric equations work. The first thing I did before working through this question was go onto Desmos and graph r(θ) = 2 + 6sin(θ) which produced the blue function you can see in the last screen shot. As you can see from my notes, I eventually worked out that the slope, a.k.a. dy/dx, a.k.a. (dy/dθ)/(dx/dθ), was equal to 2(3)^1/2. The red line in the screen shot is the tangent line to r(π/6) = 2 + 6sin(π/6) which the question didn’t ask me to find but I did just to help me better understand what’s going on. The equation for that line is y = (2√3)(x – (√3 + (3/2)√3)) + 5/2. Don’t ask me how I figured out that equation because it was mostly trial and error and I honestly don’t really know…

Question 5

This question did actually ask me to figure out what the equation was for the line tangent to r(θ) = θ at the point θ = π/2. The first steps to solve this question were the same as in the previous one which, after many attempts at this exercise, I eventually found to be pretty straightforward. Once I figured out the slope of the tangent line at θ = π/2, the next steps (which you can see at the top of the third page of my notes) was to input θ = π/2 into x(θ) = r(θ)cos(θ) and y(θ) = r(θ)sin(θ) which outputted the x-value and y-value at θ = π/2. (Again, I have no clue if I’m saying that correctly or if I’m even making any sense at all…) Lastly, you use some magic and input the values of x(π/2) and y(π/2) into the point-slope form equation which you can use some algebra on to come up with the final equation y = –2x/π + π/2.

The next 3 out of 4 sections in this unit in order are titled Area: Polar Regions (Single Curve), Area: Polar Regions (Two Curves), and Arc Length: Polar Curves. I’m hoping that as I work my way through these sections the fog around polar curves and coordinates will start to clear. I’m still hopeful that I’ll be able to get through this unit, Parametric Equations, Polar Coordinates, and Vector-Valued Functions (800/1,500 M.P.), before the end of October but it will probably be pretty close. It’s hard to tell but I think I’m over the hump of figuring out polar curves so hopefully it’s all downhill from here!