Activity for TextKit
| Type | On... | Excerpt | Status | Date |
|---|---|---|---|---|
| Question | — |
Which other Real Analysis textbooks unusually recommend ending delta-epsilon proofs with a cluttered, bedecked $\epsilon$? 1. Most textbooks conclude $\delta-\epsilon$ proofs tidily with $\epsilon > 0$ alone, as in red beneath. But what's the official term for this alternative $\delta-\epsilon$ proof, as in green beneath? 2. I forgot the particulars of another textbook that I read, not the one quoted below. It advise... (more) |
— | 12 months ago |
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A: Expanding the Integration problem. This is just middle school algebra. $\frac{1}{b-a} \left[ -\dfrac{1}{2}(a + b){\color{red}{(b^2 - a^2)}} + \dfrac{1}{4}(b+ a)^2(b-a)\right] \equiv \frac{1}{b-a} \left[ -\dfrac{1}{2}(a + b){\color{red}{(a + b)(b - a)}} + \dfrac{1}{4}(b+ a)^2(b-a)\right] \equiv \frac{1}{b-a} \left[ -\dfrac{1}{2}{\c... (more) |
— | over 1 year ago |
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What did James Stewart mean by "the line integral reduces to an ordinary single integral in this case" ? 1. Please see the question in the title, in reference to the paragraph beside my two green question marks in the image below. 1. How do you symbolize "the line integral reduces to an ordinary single integral in this case"? $\int^ba f(x {\color{goldenrod}{, 0)}} \, dx = \int^ba f(x) \, dx $? 1. F... (more) |
— | over 1 year ago |
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A: $g(x)\xrightarrow{x\to\infty}\infty$ Implies $g'(x)\leq g^{1+\varepsilon}(x)$ My questions 1. Can you cite or scan the question from the source? I am leery, because the claim appears false. 2. Is g(x) supposed to be convex? Game plan I shall construct, on top of $f(x)=x$, a function which occasionally jumps up a constant amount over littler and littler intervals... (more) |
— | almost 2 years ago |
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Why can an easily graphable definite integral, be labyrinthine to evaluate? How can I explain to 16-year-olds, who just started calculus, why it's so nettlesome and tricky to symbolically integrate definite integrals, when their graph looks so unremarkable and straightforward? I used Desmo to graph $\int0^{\pi/2}\frac{x^2\log^2{(\sin{x})}}{\sin^2x} \, dx$ below. u/camel... (more) |
— | almost 3 years ago |
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De-mystifying tricks – If $\{x_n\}$ converges, then Cesaro Mean converges. >Exercise 2.3.11 (Cesaro Means). (a) Show if $\{xn\}$ is a convergent sequence, then the sequences given by the averages $\{\dfrac{x1 + x2 + ... + xn}{n}\}$ converges to the same limit. I rewrote and colored the official solution. >Let $\epsilon>0$ be arbitrary. Then we need to find an $N \... (more) |
— | about 3 years ago |
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Why aren't $z_1=f(xy)$ and $z_2=f(x/y)$ functions of 2 variables? Hagen von Eitzen answered that $z1, z2$ >depend on only one variable - there's no comma between the parentheses. John Doe commented >the function $f(xy)=e^{xy}sin(xy)+(xy)^3$ may look like a multivariable function in x and y, but it can be written more simply as a univariate function, $f(t... (more) |
— | about 3 years ago |
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Don't downvote just because someone posted screenshot. I've seen comments of admitting to downvoting a post just because that poster posted screenshots. If someone is writing math, then they ought use MathJax, not screenshots. But if you're quoting someone else, especially if you are alleging someone else made a typo or mistake, then screenshots a... (more) |
— | about 3 years ago |
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If $n = xm$ and $n \rightarrow \infty$, then $m \rightarrow \infty$? Did Stewart prove the titular result, also underlined in green in the image below, either by himself or as an exercise, in Calculus Early Transcendentals or in the normal version Calculus? I attempted the proof, but I got nonplussed. $\infty = \lim\limits{n \rightarrow \infty} n = \lim\limits{n \r... (more) |
— | about 3 years ago |
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How can I generalize a picture for the Mean Value Theorem to the Generalized MVT? How can I transmogrify this figure for the Generalized MVT? If $f$ and $g$ are continuous on the closed interval $[a, b]$ and differentiable on the open interval $(a, b)$, then $\exists$ $c ∈ (a, b) \ni \cfrac{f'(c)}{g'(c)} = \cfrac{f(b)-f(a)}{g(b)-g(a)}$. Please answer with a picture. Please... (more) |
— | about 3 years ago |
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How can I visualize $\lim\limits_{x \rightarrow \pm \infty} f(x) = \lim\limits_{t \rightarrow 0^{\pm}} f(1/t)$? I'm not asking about the proof that I already understand. I'm longing to understand this graphically. As you can see, I added $1/t$ to Stewart's graphs. Then what? ![enter image description here][1] I can ask this as a separate question, but I feel that the underlying difficulty is the same. ... (more) |
— | about 3 years ago |
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Why doesn't `\;` work? Kindly see https://math.codidact.com/posts/280629. `\;` isn't rendering? And I don't know why some of the MathJax is miffed? (more) |
— | about 3 years ago |
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Isn't it wrong to write that Indefinite Integral = Definite Integral with a variable in its Upper Limit? >$\int f(t) \\; dt = \int{t0}^t f(s) \\; ds \quad \text{ where $t0$ is some convenient lower limit of integration.}$ Isn't this wrong? Because LHS $\neq$ RHS in general! Rather, LHS $\ni$ RHS, because LHS = RHS only if $C = -g(t0)$. By the Fundamental Theorem of Calculus (FTC), LHS = $\int... (more) |
— | about 3 years ago |