This week was one where I didn’t learn anything new but got a lot of good practice working with and a better understanding of functions and their derivatives. I didn’t finish the unit Analyzing Functions but am working on the unit test (I’m only through 2/18 questions though 😔). Besides starting the unit test, all I got through this week was 1 video and 3 exercises, 2 of which I found fairly straightforward. For the most part I’m happy with my effort this week, although part of me thinks I could/should have done more. I’ll make up for it next week!

I picked up this week working through the optimization exercise from the section Solving Optimization Questions. I still found these questions quite confusing but eventually started to get the hang of them. It took my until Friday to get through the exercise which I was a bit disappointed about (to be fair though, something came up on Wednesday and I was only able to spend 5 minutes on KA). As well, by the time I passed the exercise I had more-or-less memorized the process to solve each question since, though they used different values each time I restarted the exercise, the questions had the same format every time I redid it. I also used Desmos for each question which I’m not sure if I was ‘supposed’ to do. Using it helped me to visualize the functions and get a more intuitive understanding of what they look like, however, so I think it was for the best that I used Desmos even though it may have been cheating.

After getting through that exercise, I worked through a section called Analyzing Implicit Relations which only had 1 video and 1 exercise in it. I found this section pretty simple to work through and it only took me 30-45 minutes to finish it. The questions had me come up with an equation for a line tangent to a curve that was either horizontal or vertical, i.e. the solutions were either x = a or y = b. This section had me use systems of equations (SoE) which really helped me understand the value of SoE’s and understand how they work more so than I did before. Here are 2 questions I worked through in this section, the first of which being from the single video in that section:

(Note: The equation x² + y⁴ + 6x = 7 is an equation for an ellipse that happens to be centered exactly on the x-axis. The question is asking me to use the derivative to find the equation for the horizontal line that is tangent to the top half of the ellipse.)

• Step 1 & 2 – Write out the equations for the derivative and the equation of the ellipse:
  • x² + y⁴ + 6x = 7
    • y⁴ = -x² – 6x + 7
  • Dy/dx = -2(x + 3)/4y³

• Step 3 – Given that when the derivative equals 0 the slope of the ellipse at that point will be 0 (a.k.a. the tangent line at that point will be horizontal), find where the numerator of the derivative equals 0:
  • Dy/dx = -2(x + 3)/4y³
    • 0 = -2(x + 3)/4y³
    • 0 = -2(x + 3)
    • 0 = (x + 3)
    • -3 = x
    • Therefore, the numerator equals 0 when x = -3.
• Step 4 – Input x = -3 back into the equation of the ellipse:
  • y⁴ = -x² – 6x + 7
    • y⁴ = -(-3)² – 6(-3) + 7
    • y⁴ = -(9) – (-18) + 7
    • y⁴ = -9 + 18 + 7
    • y⁴ = 16
    • y = (+/-)2
• Step 5 – Conclusion
  • Since we’re looking for where the horizonal line tangent to the ellipse is positive, the coordinates would be (-3, 2).

(Note: The question is asking me to find the equation for the line that is vertical to the curve in the equation, i.e. not where the function has a vertical asymptote, .)

• Step 1 – Write out the SoE and the derivative:
  • SoE:
    • xy² + 5xy = 50
    • -y(y + 5) ≠ 0 
      • (Note: The numerator has to have some value in order to be divided by 0 in the denominator for the tangent line to be completely vertical at that point.)
    • x(2y + 5) = 0
  • dy/dx = -y(y + 5)/x(2y + 5)
• Step 2 – Isolate y using the equation x(2y + 5) = 0:
  • 0 = x(2y + 5)
  • 0 = 2y + 5
  • -5 = 2y
  • -5/2 = y = -2.5
• Step 3 – Solve for x by inputting y = -2.5 into the first equation:
  • xy² + 5xy = 50
    • x(-5/2)² + 5x(-5/2) = 50
    • x(25/4) + (-25x/2) = 50
    • 25x/4 – 25x/2 = 50
    • 4 * (25x/4 – 25x/2) = 50 * 4
    • 25x – 50x = 200
    • -25x = 200
    • x = 200/-25
    • x = -8
• Step 4 – Conclusion:
  • The coordinates for the vertical line tangent to the curve of the equation xy² + 5xy = 50 are (-8, -2.5).

I began the unit test on Saturday and sadly only got through 2 questions. The good news is I got both questions correct but I was obviously disappointed I didn’t get further. The second question had to do with Mean Value Theorem and I couldn’t remember how it worked. I looked it up and remembered that MVT states that if there’s a continuous function between two points, there’ll be some point along the function where the derivative at that point is equal to the slope between the two end points. After looking it up, I essentially was able to figure out how to solve the question but got confused along the way and it took me way longer than it should have for me to input my final answer. As a reminder of what MVT is, here’s a photo and 2 notes from my Week 103 blog post that talk about it:

My layman’s explanation of MVT is that it states that when looking at a continuous function between a specific interval, [x₁, x₂] or [a, b], the slope between the 2 points of the interval, m, will be equal to the derivative of at least one point along the function, f(x₃) or f(c).

The key thing to remember is to set up the equations for MVT as: 

• f’(c) = f(b) – f(a)/b – a
  • “The derivative (a.k.a. the slope) at f’(c) is the same value as the slope between the two points f(a) and f(b). 

I’m hoping it won’t take me too long to get through the unit test of Analyzing Functions (1,440/1,800 M.P.). For the most part I feel confident that I will get through it quickly except if I run into a difficult optimization question. Once I pass the unit test, I’ll be ~63% of the way through Calculus 1 and will be moving on to the unit Integrals. From what I’ve heard on YouTube, I’m pretty sure integrals are integral (😏) to calculus so I’m looking forward to finally learning about them. First thing’s first though, gotta pass this unit test!