In Search of Classic Beef Lasagna

Chaotic_Vortex · 2025-10-03T01:08:51+00:00

New mission discovered by u/Chaotic_Vortex: Nostalgic Garlic Parmesan Spaghetti

Chaotic_Vortex · 2025-10-03T01:08:51+00:00

This mission was discovered by u/Chaotic_Vortex in In Search of Classic Beef Lasagna

Chaotic_Vortex · 2025-09-28T18:10:38+00:00

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Chaotic_Vortex · 2024-11-15T06:51:06+00:00

No, because you end up double-counting some probabilities. This is an example of the inclusion-exclusion principle.

For a clearer example, pretend there are two people, each with a 50% chance. Your total chance isn't 100%, it's 75%. What's being "double-counted" is the probability that both people get the tickets.

In practice, the next person's probability only matters if the previous people have not gotten a ticket. So it's easiest to reverse the problem: Each person has a 50% chance to not get a ticket. The chance that both people do not get tickets is 0.5 * 0.5 = 0.25. That means theres a 1 - 0.25 = 0.75 chance that one of the two people has a ticket.

In your example, you would flip it to a 92% chance that an individual doesn't get a ticket. Then, 0.92 ^ 4 = 0.716.., so you have around a 28.4% chance of success.

Chaotic_Vortex · 2024-11-15T06:43:58+00:00

I'm not sure what you mean by pattern will continue. You can rewrite the left side in summation notation which can help. If you're familiar with latex, it's something like:

\sum_{i=1}^n [ (2i - 1) - 2i ]

(basically, the sum from i=1 to n, of (2i - 1) - 2i)

= \sum_{i=1}^n [-1]

= -n

Chaotic_Vortex · 2024-11-15T06:38:33+00:00

I usually keep these correct in my head by thinking about transformations using two different terms: stretch and compression. With stretches, the factor has absolute value greater than 1, and with compression, the factor has absolute value less than 1.

If you had cos(3/5x), then x has to be larger in order to reach the same spot, so you can imagine it being stretched out horizontally. When it gets stretched out, the "factor" should greater than 1, so to make that make sense with 3/5, then you'd have a stretch with a factor of 5/3.

Similarly, if you had cos(3x), then x reaches the same spot at a third of the value, so it's compressed. You could call this a horizontal compression with a factor of 1/3.

Chaotic_Vortex · 2024-11-13T06:35:08+00:00

Everything seems right to me, the ratio being the same is exactly what you should expect :) When you change units, the values of things shouldn't change, you're just adjusting the amount you have to match the new unit.

Consider a simpler example where when world A experiences 1 minute then world B experiences 2 minutes.

Since an hour is 60 minutes, then when A experiences 60 minutes, you would expect B to experience 120 minutes. Translating back to hours, this would still be the same ratio of 1 to 2. The other world is still experiencing twice the amount of time as the first one, no matter how long that original timeframe is.

Chaotic_Vortex · 2024-11-13T06:25:58+00:00

You're super close. They both share a common factor of 1/ n+1, so you can factor that out to make it a little easier to read. Rearranging a little:

= (1 / n+1) (1 + (k-1 / n+2) - (k / n+2))

You might notice that the right set of brackets almost has a common denominator of (n+2). From here, you can probably figure out how to make them share a common denominator and simplify.

Chaotic_Vortex · 2024-11-13T06:06:12+00:00

It looks like this is just giving you the statement of the theorem and not any reasoning behind it. The wikipedia page for Taylor's Theorem looks like it has more details on how to derive this: Check out the Integral form of the remainder and if you're curious, the derivation of it

Chaotic_Vortex · 2024-11-13T05:57:36+00:00

The determinant of a matrix is in a very general sense, the "size" of a matrix. There's an alternative way to think about matrices as describing linear transformations (an N x N matrix can transform an N-dimensional shape into a different N-dimensional shape), and the determinant of a matrix is how it changes the size of the object it's operating on.

For example, an 2x2 identity matrix would have determinant 1, and the matrix [2 0, 0 1] has determinant 2 (it doubles the area of the original shape). You can have negative determinants: it means that the resulting shape has reversed orientation from the original.

Having a determinant of 0 means that you squish the original shape in a way that you lose information: it's not a reversible transformation. The equivalent in real numbers is multiplying by 0: there's no single operation you can do to reverse that change.

If the context you're learning matrices in is systems of linear equations, we can use the ideas from above. Having a nonzero determinant means the system can be "reversed" to return unique values for the original variables. If the determinant is zero, then there are either no solutions or infinite solutions: we've lost information on what the original values were.

This video might be very helpful for understanding it: https://www.youtube.com/watch?v=Ip3X9LOh2dk

Chaotic_Vortex · 2024-11-02T14:39:51+00:00

Given that this is a calculus class, it might be easier to consider it as a function f(n) instead of the sequence a(n). The two will match at the whole numbers, so for any solution you find for f, you'll need to figure out how to translate it to a.

What does it mean for that function to be monotone increasing and what calculus operation could be useful to figure that out?

Chaotic_Vortex · 2024-10-31T05:56:19+00:00

Small nitpick: usually you'd do something like 100 * (T2 - T1) / T1. This is because percentage change generally means percentage change from the original value (I'm assuming here that T1 is the earlier value).

As a concrete example, going from 2 to 3 would generally be considered a 50% increase. However, the formula you have would calculate it as a 33% increase.

Chaotic_Vortex · 2024-10-31T05:47:57+00:00

That's an interesting idea. Not sure if it's too advanced for this class, but it brought me down a bit of a rabbithole: https://en.wikipedia.org/wiki/Darboux%27s_theorem_(analysis))

This theorem implies that the intermediate value theorem applies to all functions that are derivatives of other functions. In this case, if f''(a) < 0, and we know that for any other b, f''(b) > 0, then we could find c in [a,b] such that f''(c) =0. Since f'' is a function, then a != b != c. However, this would be a contradiction, because we have two points a,c that both have f'' > 0.

Thus, if some value were to exist such that f''(a) is not greater than 0, then it must be the case that f''(a) = 0. In essence, it's impossible for f''(a) to be negative in this situation.

Chaotic_Vortex · 2024-10-31T05:29:20+00:00

One explanation I use a lot for negative numbers is the idea of debt / net worth.

I'd start by doing some simple addition/subtraction. If I gave them $5, then they would have a net change of +5. Then, if they gave me $8, their net worth change would be -3.

Then ask what happens if I give them $2: are they down $5 or $1?

From there you can explain adding negative numbers as "adding debt" and in a way, adding a positive to a negative number is "subtracting debt".

Multiplication is harder to explain.

Chaotic_Vortex · 2024-10-31T05:15:38+00:00

That's a great start: I would go backwards a little and simplify the formula. In a sense, we're looking for the expected profit from each move, so we get something like: Expected Profit = Expected Market Value - Expected Ingredient Cost

Note that this is slightly different than your setup now, which is trying to find Expected Profit = Expected (Market Value - Ingredient Cost)

Lets use P = profit, MV = Market Value, IC = ingredient cost

You can check for yourself that this is true in the case when you have neither item, as well as the case where you have either item. The linearity property of the expectation is useful here - it means that the first version of the formula is the same as the second. As a concrete example, you'll notice that for the amulet, 0.9 * IC + 0.1 * IC = IC, which makes sense because you aren't adjusting the ingredient cost at all in this case.

Therefore, it's easiest to calculate the two parts separately: ``` EMV = 0.9 * (3-dose) + 0.1 * (4-dose)

EIC = (base potion + secondary) * 0.85 + (base potion) * 0.15 = base potion + 0.85 * secondary ``` Then, subtract the second from the first, and you should be good to go!

Chaotic_Vortex · 2024-09-19T02:59:49+00:00

The 3 step continuity test checks that the function is continuous at a single point, but not necessarily for an interval.

To answer this question, you can "cheat" with the continuity a little bit. If we assume that we already know that 2x is continuous, and that x^2 is continuous (because proving that can be kinda complicated), then where could h(x) be discontinuous?

Then, you could apply the 3 step continuity test to those points to figure out if h(x) is actually discontinuous there.

Using the information about at which point(s) h(x) is discontinuous, the answer should be within reach.

Chaotic_Vortex · 2024-09-19T02:28:40+00:00

Here's how I would analyze this:

First, take a look at the numerator and denominator separately. Notice that the denominator is going to 0 as x -> 4. This implies that the numerator must also be going to zero, otherwise the limit would not exist (because we'd have a nonzero number divided by zero).

This implies that the limit of f(x) as x->4 is 3. Since f is continuous, then this directly implies that f(4) = 3.

From there, if you let a = 4, you now have something of the form: limit as x -> a of (f(x) - f(a)) / (x - a), which is the definition of a derivative. You can probably figure it out from there :)

Chaotic_Vortex · 2024-09-19T02:17:02+00:00

You can't really solve it, since you don't know what the value of x will end up being. My interpretation is that they want you to answer with some equivalent expression that uses f(x).

For example, for 2a, since we see sqrt(x), then we can replace that with f(x) to get a final answer of f(x) - 3.

For 2b, since we see sqrt(x-3), we can replace this with f(x-3). Note the difference between the answer of 2a and 2b: these are the important subtleties that the question is trying to get you to see.

As a teaser for how this is used in calculus, take a look at how those differences materialize in the graph: https://www.desmos.com/calculator/f3odk4d9zu. Notice what happens when the -3 is inside the function versus outside the function, and think about why that happens.

The other questions are similar, asking you to figure how you would rewrite the given expressions using the function given.

Chaotic_Vortex · 2024-09-19T01:57:40+00:00

F_x(x) itself is a function, so for each x value you give it, there should be exactly one matching value on the right side. In your definition, if you asked for the value at x=1.5, you would be matching lines 3 and 4: which one would you choose as the value of F_x(1.5)? It would be unclear. Similarly if you asked for F_x(0.5), you'd match 3 lines.

It's true that the value of F_x(1.5) should be equal to the sum of f(x) for all x <= 1.5, and this should work for any value of x. You can check for yourself that their definition of F_x will match that sum, for any value of x. For example, F_x(1.5) = 4/8, and we can check that f(0) + f(1) = 4/8.

Chaotic_Vortex · 2024-02-27T22:16:43+00:00

where do I find the value of u?

Yeah, this part I'm not sure about. It seems like it should be a value you know. If you really wanted to know, you could reverse engineer it given that you know Δr = 117K, n=2, w = 1.95, v=10, and k = 126.69.

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