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1d
answered Is this sum differentiable?
1d
comment Is $f(x) =x^{-1}$ an analytic function?
The condition of applicability should read "$|x - a| < a$" instead of "$|x| < a$", of course....
1d
answered Numerical evaluation of first and second derivative
1d
comment Numerical evaluation of first and second derivative
Do you just need a numerically stable expression for $f(x)$ itself, or do you really need the Taylor series at $x = 1$?
1d
answered Is $f(x) =x^{-1}$ an analytic function?
1d
comment How does the continuity of a composite function of floor change when it is integrated, but with floor treated like a constant?
My final claim isn't the most general possible, just a convenient framework into which your $e^{2x + \lfloor x\rfloor}$ fits. If $f\bigl(g(x, \lfloor x\rfloor)\bigr)$ is integrable, then every definite integral $F$ is continuous everywhere, and is an antiderivative at each point where $f\bigl(g(x, \lfloor x\rfloor)\bigr)$ is continuous. As in runaround's answer, however, dividing the domain into intervals $[n, n+1]$ is a good way to see how working with floor differs from using constants, precisely because it allows you to focus on $x = n$ where "the slope jumps (but the height doesn't)".
1d
comment Is this sum differentiable?
Forget differentiability; can you give an example for which the stated series converges...?
2d
comment The role of visualization and intuition in graduate and postgraduate math and developing it
Just a tangential note: You might be interested in the remembrance for Bill Thurston in the AMS Notices.
2d
answered In which cases are $(f\circ g)(x) = (g\circ f)(x)$?
2d
comment Differentiable, Parametrized Curve for Trace $y = |x|$
If you pick a suitable $f$, you can get a $C^{\infty}$ parametrization. :) That's not necessary from your problem statement, but is worth knowing: Smoothness of a mapping doesn't of itself imply "local smoothness" of the image; you need something like regularity (injectivity of the derivative at each point). Examples such as the cycloid or astroid curves show that a real-analytic mapping can have an image with cusps.
2d
comment In which cases are $(f\circ g)(x) = (g\circ f)(x)$?
Related: If $f \circ g = g \circ f$ does that mean that both functions are to and from the same set and both are bijections? Does it tell us anything else?
2d
comment Differentiable, Parametrized Curve for Trace $y = |x|$
Suggestion: Pick your favorite smooth, increasing function $f$ that satisfies $f'(0) = 0$, and put $x = f(t)$, $y = |f(t)|$. (For extra fun, see how smooth you can make the parametrization.)
2d
comment Prove that multiplication is well defined
In this situation, it probably helps to "cheat": Multiply out $(a - b)(c - d)$ as if you know about integer arithmetic, then interpret the result as an ordered pair of natural numbers. Then use the fact that $(a - b) = (a' - b')$, etc. As long as the end result refers only to pairs of natural numbers, you haven't technically done anything illogical.
2d
answered When is a manifold also a vector space?
2d
comment What simple topological properties of conic sections can be explored?
Could you please add details about which metric spaces you're considering, and how you define a "conic section" in these spaces? (I didn't vote to close, but tend to agree the current wording is difficult to address substantively.)
Feb
7
comment Why don't infinite sums make any sense?
@Rob: Welcome to Math.SE. You might consider re-titling your question to something like "Seeming discrepancy with geometric sum formula." :)
Feb
6
answered Is velocity a function of displacemnt?
Feb
6
comment locus of a variable straight line
"taking the equation of a line in parametric form and substitute the given line equations. we get 6 constants, solve" looks like a sketch of a strategy. Could you please clarify where exactly you are unable to make progress in carrying out the details?
Feb
6
comment How did the rule of addition come to be and why does it give the correct answer when compared empirically?
@bzal Here are some Wikipedia links: base-$b$ positional notation, and three examples common in computer science: binary (base $2$), octal (base $8$), and base $16$ (hexadecimal). Wikipedia also has a more extensive list of examples.
Feb
5
answered How does the continuity of a composite function of floor change when it is integrated, but with floor treated like a constant?