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Tutor profile: Benjamin C.

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Benjamin C.
Experienced tutor for math, economics, and music theory!
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Questions

Subject: Music Theory

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Question:

What is harmonic function?

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Benjamin C.
Answer:

Chords each have a role that they normally play in typical Western music, which is called their chord function. There are three possible chord functions: tonic, dominant, and subdominant. Tonic chords feel stable; they do not lead to anywhere in particular. $$I$$ is the most common tonic chord, but $$iii$$ and $$vi$$ are also considered to be functionally tonic. Dominant chords feel unstable and want to resolve to a tonic chord (though resolutions to $$iii$$ are exceedingly rare in most music genres). $$V$$ and $$vii^{o}$$ are dominant chords. Subdominant chords ($$ii$$ and $$IV$$) are intermediate chords between tonic and dominant chords that enrich the transition between them, giving a harmonic progression of Tonic $$\implies$$ (Subdominant) $$\implies$$ Dominant $$\implies$$ Tonic $$\textbf{Okay, but what makes a chord have a particular function?}$$ This is something that music theorists have a variety of opinions on. One school of thought is that it basically comes down to how individual tones in a chord want to move, and where they resolve to (i.e. voice leading). I especially like this explanation for dominant chords, because the pull of the leading tone to the key center is – by far – the most important part of the $$V$$ and $$vii^o$$ chords’ resolution to tonic chords. The explanations I’ve seen for how voice leading creates tonic and predominant function ring a little less true to me, but one of the cool things about music theory is that there really isn’t a single “right” answer, only useful explanations! There are no fundamental rules to make music sound “good” – music theory just gives some places to start.

Subject: Calculus

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Question:

What is a delta-epsilon proof? Why are they useful, and can you walk through an example?

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Benjamin C.
Answer:

A delta-epsilon proof is a technique to verify that the value of a given non-infinite limit* is correct. The reason it’s called a delta-epsilon proof is that the formal definition of a limit usually uses these Greek letters. The standard definition of a non-infinite limit (using these letters) is as follows: Let $$f(x)$$ be a function that is defined on an interval which contains $$x=a$$, but is not necessarily defined at $$x=a$$. We say that $$\lim_{x \to a} = L$$ if for every $$\epsilon > 0$$ there exists a number $$\delta > 0$$ such that $$|f(x)-L| < \epsilon$$ whenever $$0<|x-a|<\delta$$. Okay, so what concept is this definition getting at? Basically, it's saying that $$\lim_{x\to a}f(x)=L$$ if and only if you can get $$f(x)$$ to stay as close as you you want (within $$\epsilon$$) to $$L$$. by restricting $$x$$ to be really close to $$a$$ -- specifically, making $$|x-a| <\delta$$. The point of a delta-epsilon proof is to say: "You want me to get $$f(x)$$ within $$\epsilon$$ of $$L$$? Here's a $$\delta$$ that I can use to make that happen. Since I can always do this for any $$\epsilon$$ you give me, the limit is $$L$$." These proofs are really the only way to actually prove the values of non-infinite limits, and are thus very useful. Here's an example that illustrates the basic process you can use for simple delta-epsilon proofs. While more complex delta-epsilon proofs (like those using quadratic functions) use new insights or techniques, they all use the same process and basic logic. $$\textbf{Example: Prove} \lim_{x \to 4} 5x-7=13$$ Note that delta-epsilon proofs all start with a bunch of scratchwork, which we then use to write up a formal proof. The goal of this scratchwork is to manipulate the expression $$|f(x)-L|<\epsilon$$ to look like $$|x-a|<\delta$$. This is the reverse of the process in the proof, and by writing down all of our steps here we can just copy backwards for the formal proof. The reason to do it this way (and not just start with the way we use in the proof) is that it’s much easier to simplify $$|f(x)-L|$$ than it is to do a ton of complicated construction to get from $$|x-a|$$ to $$|f(x)-L|$$. Hopefully this will be clear after the example. $$\underline{Scratchwork}$$ In this problem, $$f(x) = 5x-7$$, $$L=13$$, and $$a=4$$. We want to find a $$\delta$$ that allows us to transform $$|(5x-7)-13| < \epsilon$$ into $$|x-4| < \delta$$. Simplifying the first expression, we get $$|(5x-7)-13| < \epsilon$$ $$|5x-20| < \epsilon$$ $$5|x-4| <\epsilon$$ $$|x-4| < \epsilon/5$$ This means that our proof will work if we choose $$\delta = \epsilon/5$$. We can now write up the formal proof: $$\underline{Proof}$$ Let $$\epsilon > 0$$ be an arbitrary positive number. Define $$\delta = \epsilon/5$$, so $$\delta >0$$ as well. Then if $$|x-4| < \delta$$, we have $$|x-4| < \epsilon/5$$ $$5|x-4| <\epsilon$$ $$|5x-20| < \epsilon$$ $$|(5x-7)-13| < \epsilon$$. Therefore, for any $$\epsilon > 0$$ we can choose $$\delta = \epsilon/5$$ to make the expression $$|x-4| < \delta \implies |(5x-7)-13|<\epsilon$$ true. Thus, by the definition of a limit, $$\lim_{x \to 4}5x-7 = 13$$. *by “non-infinite”, I mean a limit which has a finite value at a finite point

Subject: Economics

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Question:

Why do would anyone want minimum wage laws? Don’t they create a binding price floor, which means that they create deadweight loss?

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Benjamin C.
Answer:

Before we answer this question, we should define the terms we are using. A price floor is a lower limit on the price of a good or service. Price floors can be non-binding or binding. A non-binding price floor is one which has no effect because it is lower than the existing market price, like a law that says cars must be sold for at least $1 each. A binding price floor is one that changes the price of the good or service, like requiring cars to be sold for $1 million each. Minimum wages are price floors – they create a lower limit on the cost that companies can pay for workers’ labor. Since there are millions of workers in the US that work for exactly the minimum wage in their area, it is a pretty reasonable assumption that many (if not all) US minimum wage laws are examples of binding price floors. Note that binding price floors $$increase$$ the equilibrium price; this terminology can trip students up! Deadweight loss is a measure of how much worse a situation is from the ideal situation. For example, if the $1 million price floor for cars was signed into law, most people would not be able to afford cars. This would hurt people who now cannot afford a car, and it would likely hurt businesses who sell cars (since even though they make $1 million per car they sell, they don’t sell very many cars). From this example, it is clear that binding price floors can create deadweight loss. In general, economic theory says that deadweight loss happens if you have a price that’s different from the perfect competition equilibrium price. Thus, if we put a binding price floor on a perfectly competitive market, we will have deadweight loss. However, in a monopsony market (monopsony is just monopoly with the roles reversed – it’s a single buyer rather than a single seller) binding price floors can actually $$reduce$$ deadweight loss. This is because monopsony markets have artificially low prices, just like monopoly markets have artificially high prices. Putting a binding price floor that increases the price to what it would be in a competitive equilibrium eliminates deadweight loss, by the same economic theory that makes binding price floors cause deadweight loss in competitive equilibrium. Given that more and more people are employed by enormous corporations like Amazon, Walmart, McDonald’s, etc., it is reasonable to assume that the US labor market is not perfectly competitive. This gets into what I think is the most important reason for minimum wage laws: the real world is a lot more complicated than the simple theories we learn in ECON 101. The economic literature on the effects of minimum wage laws has mixed (often contentious) findings, but it seems that small increases in minimum wages don’t have significant negative effects on employment (1).Since the goal of minimum wage laws is to protect workers, it seems that minimum wage laws mostly have the intended effect without huge downsides. (1) ROPPONEN, OLLI. “RECONCILING THE EVIDENCE OF CARD AND KRUEGER (1994) AND NEUMARK AND WASCHER (2000).” Journal of Applied Econometrics, vol. 26, no. 6, 2011, pp. 1051–1057. JSTOR, www.jstor.org/stable/23018264. Accessed 10 June 2021.

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