In case you were wondering, calculus Optimization questions are hard. 😔 I spent the entire week working through the section Solving Optimization Questions in the unit Analyzing Functions and only managed to get through 4 videos and tried (but failed) to get through the single exercise. That said, I wrote out 47 pages worth of notes so I don’t think my lack of progress was due to a lack of effort. A funny real-life analogy that came to me this week regarding my progress through calc happened when a friend of mine was talking about him speaking another language. He told me that he can speak Portuguese but, when the person he’s speaking with begins to use more complicated words, he’s able to understand what they’re saying but can’t speak that way himself. Going through the optimization questions, I felt the same way. I couldn’t solve the questions on my own but I understood everything Sal did as he worked through them. I feel like I’m a-ways-away from being able to solve questions like this on my own, but I suppose the silver lining is that I can at least understand the math behind the solutions.

Since I don’t really understand how these types of optimization questions work, I can’t give much of an explanation about them but, in place of that, here are 2 example questions I worked through this week:

            Question 1:

1. Step 1 – Write out the givens and unknowns:
   • Base = x²
   • Height = ?
   • Volume = 4³
2. Step 2 – Create function for Surface Area:
   • S(x) = S(base) + (4 * S(sides))
     • Base = x²
     • Sides = x * height
   • S(x) = x² + (4 * x * h)
3. Step 3 – Using Volume formula, V = l * w * h, where l * w = b, solve for h in terms of x:
   • V = l * w * h 
     • V = b * h
     • 4³ = x² * h
     • 4³/x² = h
4. Step 4 – Replace h in surface area function with 4/x^2:
   • S(x) = x² + (4 * x * h)
     • = x² + (4 * x * 4/x²)
     • = x² + (16/x)
5. Step 5 – Find derivative of S(x) in order to find critical points:
   • S(x) = x² + 16/x
     • = x² + 16x^-1
   • S’(x) = d/dx[x² + 16x^-1]
     • = 2x – 16x^-2
   • S’(x) = 0 = 2x – 16x^-2
     • 0 * x² = (2x – 16x^-2) * x²
     • 0 = 2x³ – 16
     • 16 = 2x³
     • 8 = x³
     • 2 = x
6. Step 6 – Input x = 2 into S(x):
   • S(2) = (2)² + 16/2
     • = 4 + 8
     • = 12
7. Step 7 – Find second derivative and see if S’’(2) is positive or negative to determine if x = 2 is min/max point:
   • S’(x) = 2x – 16x^-2
   • S’’(x) = d/dx[2x – 16x^-2]
     • = 2 + 32x^-3
   • S’’(2) = 2 + 32/(2)³
     • = 2 + 32/8
     • = 2 + 4
     • = 6
   • (Note: since S’’(2) is positive, the function at x = 2 is concave up and therefore point S(2) = 12 is a minimum point. This means the smallest surface area possible of the aquarium occurs when the sides of the base, x, are equal to 2.
8. (Note: It makes more sense to reverse Step 7 and Step 6 and check to see if the function is concave up or down at x = 2.)
   

Question 2:

1. Step 1 – Write out both functions and the function for their difference:
   • f(x) = x + 2
   • g(x) = x²
   • j(x) = (x + 2) – x² 
     • (Note: on the final screen shot where I graphed the functions, the difference between f(x) and g(x) is represented by the purple function. You can see that the purple function is only positive between the interval -1 ≤ x ≤ 2 where f(x) is above g(x). This is the interval where f(x) has a greater value than g(x) so, when subtracting g(x) from f(x), that’s the only interval where h(x) is positive.)
2. Step 2 – Finding derivative of j(x):
   • j(x) = (x + 2) – x²
   • j’(x) = d/dx[(x + 2) – x²]
     • = 1 – 2x
3. Step 3 – Find “0”’s of j’(x) in order to find min/max point:
   • j'(x) = 0 = 1 – 2x
     • –1 = –2x
     • ½ = x
4. Step 4 – input x = ½ into j(x):
   •  j(½) = (½ + 2) – ½² 
     • = 2.5 – 0.25
     • = 2.25

This week I learned two new BEDMAS principles that I thought were worth mentioning:

• Factor any number out of a fraction:
  • You can factor any number out of a fraction by simply multiplying the denominator by the number you’re factoring out while leaving the numerator as it is. For example, if you wanted to factor and 8 out of 6/35:
    • 6/35 = 8 * (6/(35 * 8))
      • = 8(6/260)
      • = 48/260
      • = 6/35
  • This seems very obvious to me now, but before this I thought you could only factor a number out of a fraction if the numerator and denominator were evenly divisible by that number (ex. 3/6 = 3(1/2)).
• Factor out a variable from a term in a polynomial that doesn’t have the variable in it:
  • Example, factoring out a 4/x² from the binomial (2 + 4x²):
    • (Note: since the term 2 doesn’t have a 1/x² in it, the key is to multiply the binomial by x²/x² which is fine because that’s the same thing as multiplying it by 1.)
    • (2 + 4x²) = (2 + 4x²) * x²/x²
      • = ((2x²/x²) + (4 x⁴/x²))
      • = 4/x² ((2/4)x² + x⁴)
      • = 4/x² ((1/2)x² + x⁴)

As a final side note, I tried and failed the Algebra 1 Course Challenge about 6 or 7 times this week. 😔 I thought I was going to breeze through it but there were a handful of questions that came up which had terms/formulas/concepts that I couldn’t remember. I came to the sad conclusion that I’m going to have to go back and review a that subject for a few days before attempting the challenge again. I’m annoyed about this because 1) it’s going to take up time that I could be putting towards learning calc, and 2) I feel like I should already know this stuff considering I passed the course challenge 2 years ago with a 100% score. (To be fair, I find the arithmetic/algebra needed in the questions very easy to do, I just don’t remember some of the formulas/terms.) As annoyed as I am that I’ll have to spend time reviewing Algebra 1, I think it will be good for me to go back and look over this stuff to fill in the gaps in my knowledge. 

I’m assuming I’m not going to get through Analyzing Functions (1250/1800 M.P.) by the end of the month. Isy guess is it will take me at least a few days to get through the optimization exercise and will still have 2 videos and 1 exercise remaining. I really want to get a solid grasp on how/why these optimization questions work so I’m ok with having it take longer than I anticipated. I’m guessing I should get through the unit by the end of the second week of October. Even though it’s taking a lot longer than I thought it was going to, I’m really happy that I’m beginning to understand how derivatives work and understand why they’re useful. 😬🤓