Recent content by cra18

This is EXACTLY my source of confusion. It seems so silly, when you put it like that though. I find it confusing to read about a variable x taking the value x or the value x + h . When I think of a variable x taking a value a , I interpret that as x=a , so variable x taking value...

Actually, I am still confused on a fundamental thing: I am used to plotting values of the independent variable on one axis and the dependent variable on a different axis. But when looking at a functional relationship like y=f(x) , what enables us to plot both x and x + h , for some...

Thanks for the replies, they were very helpful.

I am currently reading Calculus Made Easy by S. P. Thompson, and the author's idea of what it means for a variable to "vary" seems fundamentally different from my own, so I was hoping someone could help me correct my understanding. Here is the excerpt I'm having trouble with: Those...

I never said that a variable is a placeholder for a set. I said that a variable is a placeholder for any of a set of numbers --- for any element of some set of numbers. This is a definition I decided upon because of its simplicity, and because the alternative concept of a "general number"...

(I hope this question is in the proper place.) I am confused about what effect the universal quantifier has on a variable. My understanding of variables is very simplistic: I view a variable as simply a placeholder for any of a set of possible values, where that set is the universe of...

Thanks for all of the answers. And that was incredibly helpful lurflurf.

You're right. It would have to be something like \frac{\mathrm{d}}{\mathrm{d}x}f : a \mapsto \left(\frac{\mathrm{d}}{\mathrm{d}x}f\right)(a) = \frac{\mathrm{d}}{\mathrm{d}x}f(x)\vert_{x=a}. This is indeed extremely cumbersome, which is probably why I haven't been able to find an example doing...

Thanks for your answer. But what do people mean generally? Are they referring to the rule, or the variable value of the output of the rule?

I have seen over and over statements like: \begin{aligned} &f(x)~\text{is a function of}\dots \\ &\text{Let}~f(x)~\text{be a function that}\dots. \end{aligned} This is probably a dumb question, but am I justified in feeling annoyed at these statements? The annoyance stems from my...

If we have the function f : x \mapsto f(x) = 3x^2, I am used to Lagrange's prime notation for the derivative: f' : x \mapsto f'(x) = 6x. I'm fond of this notation. But it has been mostly abandoned in my engineering courses in favor of Leibniz's notation, using differential operators such...