Info, advice, help thinking, or empathy: What do they want?

I've noticed that when people say something about their lives, there can be a huge difference in the kind of response they might be looking for. Eventually, I boiled it down to four categories: Information, advice, help thinking, and empathy.

For example, a few weeks ago, Eric told me that he needed to go to the DMV (technically the RMV because Massachusetts is weird). Here's what my response and the (hypothetical) results could be:

• Information: "You can check what documents you need online."
• Win condition: Eric brings a document he didn't realize he needed.
• Lose condition: Eric was going to bring an extra document just in case, but the website incorrectly told him that he doesn't need it.
• Advice: "I'd go earlier in the day to avoid people coming in during lunch or after school or work."
• Win condition: Eric takes my advice and avoids long lines.
• Alternate win condition: Eric doesn't take my advice, but still makes the best decision for himself. (He doesn't mind lines that much and prefers to sleep in.)
• Help thinking: "What day are you thinking of going?"
• Win condition: By asking leading questions, I help Eric turn a vague idea of needing to go to the DMV into a concrete plan for getting to the DMV and back that fits in his schedule.
• Empathy: "Oof, dealing with bureaucracies is no fun."
• Win condition: Eric feels like I understand how much he doesn't like bureaucracies.
• Bonus points: Having me as an emotional ally makes Eric feel better about his trip.

Unfortunately, people don't tend to communicate which response or responses they want, and sometimes they don't even know themselves until they get frustrated by getting the wrong thing. Information instead of advice: "I don't care, just tell me which one I should pick." Advice instead of information: "I already know what I want, just tell me if this option will do it." Help thinking instead of empathy: "I'm upset; I don't want to think about that right now." Empathy instead of help thinking: "I don't even know if I'm angry about it."

And, a fairly popular one, advice instead of empathy: Person A has a problem. Person B proposes a solution. Person A wants person B to understand their feelings, and so explains why the feelings are still present despite the existence of this solution. Person B understands that the proposed solution was unsatisfactory, and modifies it to deal with the concerns brought up by person A. Person A instead feels that person B shot down their explanation for why their feelings were valid. Person A may attempt to explain why their feelings are valid a second time, at which point person B becomes frustrated that person A keeps shooting down their proposed solution.

Ultimately, as in Cool Hand Luke, what we've got here is failure to communicate. Rather than working together either to understand A's feelings or to improve B's solution, they're talking past each other, often without even realizing why the conversation's "not working."

I've noticed that different people tend to have different types of responses that they default to, and that people seeking a certain kind of response gravitate towards the people who will give it to them. That solution isn't always enough, though. In the above example, person A is already in a conversation with person B, and it's possible that person B would be fantastic at empathizing with person A, but person B is just bad at recognizing that that's what person A wants. It would be great for A to recognize what's happening and say, "I'm not looking for advice right now. I'm looking for empathy," or for B to recognize what's happening and say, "You're not looking for advice, are you? What are you looking for?"

Approximations in physics classes, solved.

If you've taken a physics class, you may have encountered a frustrating situation like this one:

Instructor: Using small angle approximations, approximate $\tan x$ for small $x$.
Student: Okay.
$$\tan x=\frac{\sin x}{\cos x}\approx\frac x1=x.$$
I: Great, now approximate $\tan x-\sin x$.
S: Okay, based on earlier work,
$$\tan x-\sin x\approx x-x=0.$$
I: Nope, that's not a good approximation.
S: What? But you said $\tan x\approx x$ was good!
I: But you get zero, so it's a bad approximation. Instead, you should do something like
\begin{multline*}
\tan x-\sin x=\frac{\sin x(1-\cos x)}{\cos x}=\frac{\sin x(1-\cos^2x)}{\cos x(1+\cos x)}\\
=\frac{\sin^3x}{\cos x(1+\cos x)}
\approx\frac{x^3}{1\cdot(1+1)}=\frac{x^3}2.
\end{multline*}
S: What.

It was never clear to me in physics classes what makes an approximation good enough. It seemed like a hodgepodge of heuristics. At some point, though, I realized that what they are looking for is very simple: The first nonzero term of the Taylor series.

In each problem, there's always some quantity or difference that's "small," and the thing to be estimated (or sometimes its reciprocal) can be written as a Taylor series in terms of that small quantity. The first nonzero term of the Taylor series then provides a good estimate for whatever it is that you're trying to approximate.

The "first nonzero term of the Taylor series" approximations behave nicely. You can multiply quantities together and then approximate the product, or you can approximate them first and then multiply the approximations together, and you'll get the same answer. The same goes for compositions like $\ln(1+\sin x)$. And, the same goes for sums and differences, except when you would get zero. But, of course, the case where you do get zero is often the most interesting one, because then you compute the higher order effects.

In our example, that's exactly what happened. The Taylor series for tangent and sine are
\begin{align*}
\tan x&=x+\frac{x^3}3+\dotsb\\
\sin x&=x-\frac{x^3}6+\dotsb
\end{align*}
The first term of the difference is exactly $\frac{x^3}2$, and the earlier computation becomes an interesting example of how you can compute the third derivative of $\tan x-\sin x$ at zero knowing only the first derivative of $\sin x$ along with the identity $\sin^2x+\cos^2x=1$.