Derivative of a derivative of a graph

[BioEngine507]

So I have a 2 page assignment and I want to make sure I'm writing it correctly so say you have a graph on the y axis is the cars sold and on the x axis you have the given year. so the f(x) would be the cars sold that year and correct me if I'm wrong.. f'(x) would be showing the rate of increase in the sales during the time period..? And what would f''(x) be?

 

[pka]

So I have a 2 page assignment and I want to make sure I'm writing it correctly so say you have a graph on the y axis is the cars sold and on the x axis you have the given year. so the f(x) would be the cars sold that year and correct me if I'm wrong.. f'(x) would be showing the rate of increase in the sales during the time period..? And what would f''(x) be?

The better description is to say that the derivative \(f'(x)\), is the rate of change in the function. If the rate is positive the function is increasing; a negative tells us the function is decreasing. Now \(f''(x)\) is the rate of change of the derivative.

 

[Dr.Peterson]

So I have a 2 page assignment and I want to make sure I'm writing it correctly so say you have a graph on the y axis is the cars sold and on the x axis you have the given year. so the f(x) would be the cars sold that year and correct me if I'm wrong.. f'(x) would be showing the rate of increase in the sales during the time period..? And what would f''(x) be?

First, since you asked about writing correctly, I'll reword your question a little (because cars sold are not numbers, they are cars):

Say you have a graph, such that the y axis represents the number of cars sold, and the x axis represents years. So f(x) would be the number of cars sold in a given year, and f'(x) would give the rate of increase in the sales during the time period. And what would f''(x) be?​

There's a major problem here. Your independent variable is discrete -- there is one point for each individual year; there is no such thing as the number of cars sold in year 2019.3! With a discrete domain, you don't have a function that can be differentiated in the first place, because you can't make a small change in x.

Perhaps you need to tell us what the assignment was, as given to you (at least the part that matters to us at the moment). Were you given this function (in which case your instructor is to blame), or is this an example you made up in response to a question asking you to do so? In the latter case, I'd want to see what conditions your function is supposed to meet.

 

[BioEngine507]

This is the exact graph I was given and told that the percentage [y axis] could be what I interpreted it as. Something to do with sherlock holmes books. I have to write two pages about the graph and what the f'(x) and f''(x) would be representing

 

[Dr.Peterson]

Do you notice all the points on the graph where it makes sharp bends? Are you aware that this means it is not differentiable at those points?

Then, do you see that those points are the only points of the graph where the value is meaningful (as I already pointed out)?

Possibly if you quoted what is said in introducing the graph (such as their explanation of what it means - beyond "something to do with"), and the first question that is asked, we could interpret what they want.

But as it stands, the only way I see to make sense of it is to replace f' with "slope from one year to the next", and f" with "change in slope from one year to the next"; or to talk about average slope. Or, they may want you just to write two pages of "Does not apply".

By the way, you said it was a "2 page assignment", so I assumed the question was that long, so you wouldn't want to quote it entirely. If you meant, as now appears, that the answer has to be that long, but the question is not, please do quote it all.

 

[BioEngine507]

The actual assignment says
Concider the following graph f(t)
Type up a narrative response that presents and analysis of the given graph. In your write up, you should attend to the following questions, though analyzing the graph beyond the given questions will be looked apon favorably. Make sure that all claims are thoroughly supported.

- what information is the graph giving you?

-why would using the first dirivative f'(t), as a construct be intresting to use within the given context?

-what is the estimated first dirivative at each of say 5 different points in time?

-where is the value of f'(t) the greatest? Where is the value of f'(t) the lowest?

-what information would the second derivative of f(t) give you?

That is the exact assignment word for word

 

[Dr.Peterson]

I guess you need to ask your teacher!

In addition, you imply that you were given this graph with no explanation beyond what the image itself shows. That is problematic in terms of the first question, as you don't know what information it gives you. We can guess; it would appear to give the percentage of ... what? Of books published in each year that contain the phrase "Sherlock Holmes" somewhere? Maybe ... but maybe not (in fact, not).

Math is not about guessing what something means; it requires definitions! If someone gives you a graph, it is their responsibility to tell you what it means. What is the base of the percentage? What is being counted (words, books, ...)? What is the corpus (the set of data called "English (2012)")? What does "smoothing of 3" mean? If I were given this assignment, I would at least try to find the site for "ngrams" to learn what I could about it (see here, for example); and I might, if I were bold, turn in a grade of F for the teacher who has failed to provide required information. Just as an answer can be wrong, a question can be wrong. This one is.

I'm guessing that perhaps your teacher gave each student in the class a different graph, and most of them may have made sense for the questions being asked; perhaps this graph is an anomaly, and you'll just be given a more suitable graph if you ask.

 

[BioEngine507]

This is what I was handed in class. Idk what im supposed to do I have a two page paper to write and they only info I have is that my teacher is stupid and I maybe should just write her two pages on how stupid she is for giving us this assignment. The first thing I did was go to ngram and look to see if there were lables on the graph which there is not.

 

[Dr.Peterson]

Go to your teacher and point out some of the things I've said, such as that the function is not differentiable, and that insufficient data is provided. Ask what you are expected to do.

Beyond that, I have nothing more to say. I think the assignment is a mistake; but maybe that's because I'm taking it literally (as mathematics), and your teacher may expect you to make some sort of approximation, such as looking at the overall shape rather than actual derivatives at specific points. Possibly you have been taught something relevant that you will be reminded of.

The important thing is, when an assignment doesn't make sense to you, you are allowed to ask questions about it! In fact, part of the point of the assignment may be to connect what you are learning to the real world, including learning to ask the same sort of (polite) questions that you would ask of an employer who gives you a poorly explained assignment, as well as recognizing that math concepts often don't apply directly to something, but still have some relevance.

By the way, please write back and tell us the answer you get. I'm hoping to learn that the teacher is not stupid, and to find out what an assignment like this is given for, for my future reference.

 

[Cubist]

Perhaps you are meant to imagine that the function has been smoothed in a such a way that it has become differentiable while retaining the same general shape (just more curved rather than having the joined flat sections). But I'm guessing. So I also recommend that you talk to your teacher. I'd also try to get an idea about how much you're expected to write for each question. And is the main focus derivatives, interpreting graphs, or ngrams? (Or maybe a roughly equal split?).

Regarding ngrams I found the following article. It gives a quick overview of ngrams and their pitfalls. Be warned there is some profanity on the page. https://www.wired.com/2015/10/pitfalls-of-studying-language-with-google-ngram/

FYI There is a field of study called "discrete time systems", in which a series of numeric samples (for example audio wav files) can be analyzed. There is the notion of a numeric 1st, 2nd, etc derivative which can be calculated from sets of adjacent samples.