Talk:Critical Points

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Contents

Response to checklist

Original response to checklist, in black, completed by Kate 21:06, 14 July 2011 (UTC) It looks great. I only have one note, but it's non-essential, so I'm going ahead and moving this to the "finished" list. AnnaP 16:22, 15 July 2011 (UTC)

References and footnotes

  • All images cited on click-through
  • All mathematical content is general knowledge and therefore uncited

Good writing

Quality of prose and page structuring

  • Section headings and first sentences make the purpose and relevance of each section clear
  • Simpler content is placed at the top - single variable over multivariable
  • Have you thought about explaining what to do when the derivative isn't factorable?
Kate 18:00, 15 July 2011 (UTC): I did think about it, but my (somewhat uneasy) conclusion was that that's beyond the scope of this article. In the single variable section, at least, I'm deliberately trying to give a mostly picture-based explanation, something that will make some sense even if you don't know how to differentiate or aren't comfortable with harder algebra. My thought was that "We need to solve for when this equation is zero" is an understandable goal no matter whether or not you know how to solve the equation, and that if you do know how to solve most polynomials, then you don't need me to explain it; while if you don't know how to solve them, a full explanation of that is a whole textbook, and would only distract from the point at hand. I'm not 100% satisfied with this conclusion, because it does seem to leave some holes, so I'd be very glad to hear your opinion.

Integration of images and text

  • Images are referred to and explained explicitly

Links to other pages

  • The HelperPage template links to the page that uses this one
  • The body of the page links to the related pages on differentiability and on gradients

Examples, calculations, applications, proofs

  • Numeric examples are provided for each main idea
  • Effort has been made to integrate the calculations well into the text

Mathematical accuracy and precision of language

  • All statements are accurate to the best of my knowledge.
  • The definition of critical point was checked in several calculus textbooks (3 online resources, 3 physical textbooks) and appears to be generally accepted
  • Terms are defined (except for in the case of differentiable, which is such a big topic that I just linked to the helper page on it)

Layout

  • Paragraphs are short
  • Hide/Shows are used so that the page is shorter
  • Page has been viewed in windows of different sizes, spacing and pictures look good.

Old comments

General Comments

Critical Points in Single Variable Calculus Section

I'd suggest talking about the derivative as telling us about how one variable changes with respect to another variable once you mention the idea of it being the slope. This can just make it a bit less abstract--you can find an example such as the number of prey animals as a function of the number of predators. In a situation like that, a critical point will represent an equilibrium point. A couple of paragraphs on that would help readers who will struggle with abstraction and demonstrate another situation where critical points are relevant. AnnaP 7/5

Kate 19:17, 5 July 2011 (UTC): I tried to do this. It doesn't really seem like a helpful example to me, and I don't know whether that's because I explained it poorly or because I like abstraction better.
I actually like what you've written. I'll come back later when I have some more time to think about specific improvements


For your example function in the first derivative test section, writing it out in its unfactored form will make things a bit more clear. AnnaP 7/5

Kate 20:39, 5 July 2011 (UTC): But I did write it out in unfactored form…?
All I see is this:
"Let's work with the function f(x) = \frac{x^4}{4} + \frac{x^3}{3} - 3x^2 + 2. The derivative of this function in factored form is f'(x) = (x+3)(x)(x-2)."
What I meant in my original comment is that it's helpful to see the derivative unfactored, so people can see where the factored form comes from. 7/7AnnaP
Kate 17:40, 11 July 2011 (UTC): Oh, I didn't realize you were talking about the derivative! Changed.


This sentence: "In this case, the first derivative test tells us that the point is neither a maximum nor a minimum, and we must turn to the second derivative test to fully understand what's happening at the point." makes it seem like you plan on explaining the second derivative test. If that's your plan, then I'd leave it, but if that's not your plan you might want to rework it somehow. AnnaP 7/5

Kate 19:17, 5 July 2011 (UTC): I changed this sentence so that hopefully it sounds more like an ending. I don't really want to get into the second derivative test here, I'm trying to keep the page as short as possible.
It's much better now AnnaP 7/9

Critical Points in Multivariable Calculus Section

When you take the gradient, it'll look a bit nicer if you make your parentheses big, so instead of \nabla f = ( \frac{\partial f}{\partial x} , \frac{\partial f}{\partial y}), you'll have \nabla f = \left( \frac{\partial f}{\partial x} , \frac{\partial f}{\partial y}\right) AnnaP 7/5

Kate 18:23, 5 July 2011 (UTC): I didn't know how to do that and for some reason it didn't occur to me to look it up, so thanks! :)
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