I had a bit of a pitiful week. Working through Applications of Derivatives, I only made it through 2 exercises and 5 videos, but I did get a lot of practice working through new types of questions that involved derivatives and started to feel a bit more confident with how to solve them. I still feel lost on these types of questions for the most part, but I also feel like I’m right on the brink of being able to intuitively understand them. It’s as if the map in my head is about 90% completed but I still have a few areas left to fill in to be able to see the whole picture. Once I do, I don’t actually think these questions will be that difficult to solve but, for now, I’m still struggling to understand what’s going on. 😔

The first exercise I worked through this week was titled Differentiate Related Functions and took me two days to get through. The questions in this exercise would give me some type of function where y(t) = x(t) (but more complicated than that), one of their derivatives (ex. dx(t)/d(t) = z), and then asked to find the derivative of the other variable when the former variable was equal to a certain amount (ex. “find dy(t)/d(t) when x is equal to w.”).

(I may not have explained that correctly but, as I said in the intro, I really don’t have a great understanding of how these questions work let alone how to explain them.)

Here’s an example from my notes:

• Givens:
  • Sin(y) = -5x
  • dy/dt = 10
  • y = -π
    
• Q. Find dx/dt @ y = -π.
  • (I don’t know how this works, but I know that dx/dt = dx/dy * dy/dt which I wrote at the top of the page and since I know that dy/dt = 10 that means I have to solve for dx/dy.)
    
• Step 1 – Differentiate both sides of the equation to find the derivative of x with respect to y (a.k.a. dx/dy):
  • Sin(y) = -5x 
    • d/dt[sin(y)] = d/dy[-5x]
    • cos(y) = -5 * dx/dy
    • dx/dy = cos(y)/-5
      
• Step 2 – Plug everything back into the equation dx/dt = dx/dy * dy/dt:
  • dx/dt = dx/dy * dy/dt
    • = cos(y)/-5 * 10
    • = cos(-π)/-5 * 10
    • = -2 * cos(-π)
    • = -2 * -1
    • = 2

In the next section, Solving Related Rates Problems, I worked through questions where I was given a shape that was expanding or contracting along with the values of a few other variables and had to use a specific formula for the volume, area, etc. of that shape to find the rate it was ‘growing’ or ‘shrinking’, so to speak. Here’s an example question from my notes:

(Note: I should have drawn the triangle in the opposite direction for it to make more sense. The question stated there were 2 cars going towards an intersection, one going at 30mph and the other going at 60mph, and then asked me to find the rate at which they were getting closer to each other at that exact moment, t₀. Since they’re both moving towards the intersection (you can think of the intersection as the origin of an x, y graph) the speed should be written as -30mph and -60mph which is why it would have made more sense for the triangle to be flipped into the ‘first quadrant’.)

• Q. What is the speed at t₀ at which the cars are getting closer to each other?
  • Givens:
    • x = 0.6 miles
    • x’(t₀) = -30mph
    • y = 0.8 miles
    • y’(t₀) = -60mph
    • C = √(x² + y²) (a.k.a. Pythags theory)
      • = 1 mile
        
  • Process – The goal here is to find the derivative of c’(t₀), i.e. the speed at which the hypotenuse is shrinking, so you need to use the Pythagorean Theorem where x, y, and c, are all functions of t and then use algebra to solve for c’(t₀):
    • c(t₀)² = x(t₀)² + y(t₀)²
      • d/dt[c(t₀)²] = d/dt[x(t₀)²] + d/dt[y(t₀)²]
      • 2c * dc(t₀)/dt = (2x * dx(t₀)/dt) + (2y * dy(t₀)/dt)
      • 2(1 mile) * dc(t₀)/dt = (2(0.6miles) * -30mph) + (2(0.8 miles) * -60mph)
      • 2 miles * dc(t₀)/dt = (1.2 miles * -30mph) + (1.6 miles * -60mph)
      • 2 miles * dc(t₀)/dt = (-36m²ph) + (-96m²ph)
      • 2 miles * dc(t₀)/dt = -132m²ph
      • dc(t₀)/dt = -132m²ph/2 miles
      • dc(t₀)/dt = -66mph

Right now, I find these questions pretty difficult to figure out. I have a hard time understanding how to set the formulas up properly and thinking of the variables used in the formulas as functions of time. I also find it hard knowing which derivatives need to be solved and how to go about solving for those derivatives. I got better at these questions as the week went on but I’m still struggling. I’m not going to go over the math, but here are a few questions from my notes that had me find the rate at which a circles’ area was contracting, a spheres’ surface area was expanding, and a cones’ volume was contracting:

I’m now 43% of the way through Applications of Derivatives (640/1500 M.P.). If I want to get through the unit before the end of the month (i.e. within the next 2 weeks), I’m going to need to get through a good chunk of material this coming week. I have 11 videos and 7 exercises left to get through so I’ll likely need to get through something like 7 videos and 5 exercises if I’m going to have a chance at finishing the unit by August. I’m really hoping that a) I can wrap my head around these related rates questions this week, and b) none of the upcoming sections are as difficult as what I’m working through on related rates. Please math god, be kind to me!! 🙏🏼🙏🏼🙏🏼