I think I fell short this week of working on KA for >9 hours but I’m happy with the progress I made. I got through 9 videos and 4 exercises, started to get a better feel for how to solve parametric equations, and got a slightly better understanding of what they are. Like last week, I still didn’t get much of a grasp on why parametric equations work the way they do or what they represent. My general understanding is that two parametric equations will use the same input to output separate but related values. An example I gave last week was they might input time into an x-equation and y-equation to output the respective x and y coordinates. Even though I don’t really know what’s going on, I’m getting better at solving the questions and I’m assuming, like always, it will all eventually click for me the more I work through these questions.

I started this week continuing through the section titled Second Derivatives of Parametric Equations. I didn’t find the questions from the exercise too difficult to solve once I memorized out the formula, but I don’t really know what the application of second derivative parametric equations would be. Nonetheless, this is the formula for a parametric 2^nd derivative:

Looking at this formula now, I don’t really know what’s going on in it. It looks like you find dx and square it (i.e. 1/dx * 1/dx = (dx)²) and do more-or-less the same thing with dy. Like I said, once I figured out how to use the formula, the questions were simple to solve. Here are two example questions:

Question 1

As you can see, I made a note at the bottom of the second page or my notes that states that, after working with negative exponents for close to a million years, I somehow didn’t realize that the coefficient in front of a variable doesn’t move with the variable if the variable is being raised to a negative exponent… I honestly don’t know how I made it this far without knowing that and feel like a bit of a dolt. At least I know it now though! …🤷🏻‍♂️

Question 2

I found that all of the questions I worked through this week, including the question above, required me to use and gave me a lot of good practice with the power and chain rules in calc, fractions, and using exponents and exponent rules. As a reminder to myself, here’s an example of why the additive and multiplicative exponent rules work:

• Exponent Addition
  • 2² * 2³
    • = (2 * 2) * (2 * 2 * 2)
      • = (4) * (8)
      • = 32
        • = 2⁵
    • = 2 * 2 * 2 * 2 * 2
      • = 32
        • = 2⁵
    • = 2^{2 + 3}
      • = 2⁵
  • 2³ / 2²
    • = (2 * 2 * 2) / (2 * 2)
      • = (2 * 2 * 2) / (2 * 2)
        • = 2¹
    • = 2^{3 – 2}
      • = 2¹
• Exponent Multiplication
  • (2²)³
    • = (2 * 2) (2 * 2) (2 * 2)
      • = (4) * (4) * (4)
      • = 64
        • = 2⁶
    • = 2^{2 * 3}
      • = 2⁶

A few of the questions that came up in the exercise from this section asked me to find the derivative of, for example, 2^x. I had a vague recollection that I needed to somehow use ln(x) and e^x to find the derivative for these terms where the variable is the exponent but I had to go back and review them to figure it out. The positive thing is that their solution seems much more simple to me now than it did when I first learned it. Here’s a page from my notes that covers how to solve the derivative for these types of terms:

As a brief reminder, since 2³ = 8, then log₂(8) = 3, which means that 2^log_2(8) = 8. Similarly, e^ln(x) = x. Using log rules, log₂(a^b) = log₂(a) * b. Combining these rules, you find the derivative of 2^x as follows:

• d/dx[2^x]
  • = d/dx[e^{ln(2^x)}]
  • = d/dx[e^{ln(2) * x}]
  • = d/dx[a(b(x))]
    • a(x) = e^x
    • a’(x) = e^x
    • b(x) = ln(2) * x
    • b’(x) = ln(2)
  • = a’(b(x)) * b’(x)
  • = e^{ln(2) * x} * ln(2)
  • = e^{ln(2^x)} * ln(2)
• = 2^x * ln(2)

The steps to solve these derivatives seem very simple and clear to me now; 1) raise the term 2^x to e^lnx; 2) use log rules to bring out the variable; 3) use the chain rule; 4) reverse the log rules; 5) take the term that was being raised by e^lnx back out of e^lnx. Boom. Here’s a question that I worked through where I had to do this:

Question 3

I started on the following section, Parametric Arc Curve Length, on Thursday. Again, this was a section that I didn’t really understand that well but was able to work my through. To start, here’s a screen shot from one of the KA vids which went through the formula to solve the length of a parametric function which was a pretty similar to formula to what’s used to solve for the length of a regular (?) function:

The thing that confuses me is that the function squiggles around and at certain points multiple coordinates of the function have the same x-coordinate. As far as I remember, functions aren’t allowed to do that. I think what’s going on is that this type of function is actually indicating the POSITION of something and could be expressed using two other functions which indicate the x-position across time and the y-position across time. Here’s a messy page from my notes that goes through what I think’s happening:

Regardless of if I’m right or wrong about that, I was able to memorize the formula to solve the questions from the exercises in this section. Here’s a question I worked through:

The last section I worked through this week was titled Vector-Valued Functions and once again I didn’t really understand what was going on. My understanding of vector-valued functions is that they’re (x, y) coordinates where x and y are each functions unto themselves which, in the case of the questions from this exercise, were functions of time. What this means is that the (x, y) coordinates can be thought of as (x(t), y(t)). If you find the derivative of both x(t) and y(t) then you know the velocity of the object being tracked and the 2^nd derivative of both x(t) and y(t) will tell you the objects acceleration. Here are two questions I worked through:

Question 5

Question 6

I’m now 27% of the way through Parametric Equations, Polar Coordinates, and Vector-Valued Functions (400/1,500 M.P.). I oddly have only 9 videos and 10 exercises left to get through before starting the unit test. (I say “oddly” because I don’t think I’ve ever had more exercises left in a unit than videos.) I think If I work hard there’s a good chance I can get through this unit by the end of October which would be great! Come to think of it, my plan is to celebrate Halloween by getting through this unit, celebrate Christmas by getting through the following unit, Series (0/2,000 M.P.), and celebrate New Years by finishing off Calc.2. Everyone knows that the best way to celebrate holidays is to celebrate by learning math… Right? 🤓