1
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0answers
69 views

Geometrical application of generation function for permutation

It is quite well known that the generation function for permutations is represented as $$(1+x)(1+x+x^2)\dots(1+x+x^2+x^3...+x^{n−1})$$ (See, e.g., question The generating function for permutations ...
2
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3answers
90 views

Intuition. Equivalence of Characterization of Limits and Continuity (Abbott p106 t4.2.3, p110 t4.3.2)

What are the intuitions of these equivalences? Not questioning about proofs or any rigour. I question both equivalences jointly because they look similar. And Are there any figures? ...
4
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0answers
95 views

Visualize $f(b) - f(a)$ withOUT Mean Value Theorem (Stewart p 282 figure 4) [closed]

How can we visualize $\color{green}{f(b) - f(a)}$ withOUT the Mean Value Theorem or rewriting it as $\color{dodgerblue}{\dfrac{ f(b) - f(a) }{ b - a }} $ ? I'm trying to understand ...
1
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2answers
82 views

Is this a counterexample to “continuous function…can be drawn without lifting” ? (Abbott P111 exm4.3.6)

I'm au courant with http://math.stackexchange.com/a/288133 and http://math.stackexchange.com/a/422001. They're both Abbott P111 exm 4.3.6 which proves "a continuous function is sometimes described, ...
1
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0answers
60 views

Intuition for $\inf(AB) = \inf(A)\sup(B)$. Difference for sets and functions? (Abbott pp 199 q7.4.5)

1. What's the intuition for $\inf(AB) = \inf(A)\sup(B)$? Figure please? I know I must posit $A,B \subseteq R$ as bounded sets. If they're unbounded, $\sup$ doesn't exist. I believe $\inf(AB) = ...
1
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3answers
96 views

If $\int^{b}_a f > 0$ then there is some interval and $\delta > 0$ on which $f(x) \ge \delta$ (Abbott pp 199 q7.4.4d)

True or False. If $\int^{b}_a f > 0$, then $\exists \; [c,d] \subseteq [a,b]$ and $\delta > 0$ such that $f(x) \ge \delta$ for all $x \in [c,d]$. 1. We need to determine if true or false. ...
1
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1answer
99 views

Cauchy's Generalized Mean Value Theorem. Required function. (S.A. pp 140 t5.3.5)

Cohen, Henle. Calculus pp 827, (http://www.vias.org/calculus/09_infinite_series_10_06.html) I revised the footnote in pp 14 http://www.math.uga.edu/~pete/2400calc2.pdf. This theorem can be ...
1
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1answer
191 views

If f' exists and f'(c) > 0 then f'(x) > 0 for all |x - c| < d for some d. (S.A. pp 137 question 5.2.8b)

If $f'$ exists on an open interval, and there is some point $c$ where $f'(c) > 0$, then there exists a d-neighborhood $\{x \in \mathbb{R} : |x - c| < d\} = V_d(c)$ around c in which $f'(x) > ...
3
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0answers
34 views

Intution, Figure. Negation of Continuity and Uniform Continuity (S.A. pp 117 T4.4.6)

Every time I need negation, I have to write out all the logical symbols to negate manually. I know how to determine these negations myself. But I want to compehend intuition or figure like ...
3
votes
0answers
68 views

Intuition on Axiom of Completeness

♪ (J. Stewart. Calculus 6th ed. pp 682) Axiom of Completeness = AoC = A nonempty set of real numbers that has an upper bound has a least upper bound. AoC is an expression of the fact that there ...
2
votes
1answer
239 views

Direct proof. Square root function uniformly continuous on $[0, \infty)$ (S.A. pp 119 4.4.8)

(http://math.stanford.edu/~ksound/Math171S10/Hw8Sol_171.pdf) Prove for all $e > 0,$ there exists $d > 0$ : for all $x, y \ge 0$, $|x - y| < d \implies |\sqrt{x} - \sqrt{y}| < e$. (a) ...
1
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0answers
58 views

Visualising the derivative/slope $f'(x_0)$ of $f:\mathbb{R} \rightarrow \mathbb{R}$ as a line segment

A function $f:I\subseteq \mathbb{R} \rightarrow \mathbb{R}$ that is differentiable at $x_0 \in I$ obeys the following equality for all $h\in (I-x_0)$ (i.e. for all $h\in \mathbb{R}$ such that $x_0+h ...
6
votes
1answer
302 views

Ways to visualize the real numbers?

I was just wondering if there are any diagrams for visualizing subsets of the real numbers, or totally 'radically' different ways of looking at them as a real line? The model of the line relies on ...
4
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1answer
145 views

Intuitive understanding a theorem in analysis

Is there a way to intuitively understand/visualize the following theorem in analysis? Let $(f_n)$ be a sequence of real functions differentiable in a finite/infinite open interval $(a,b)$. Suppose ...
8
votes
3answers
862 views

Is there a geometric interpretation of the exponential function of real numbers?

I can visualize the exponential function with the graph $y = e^x$, but I can do that for almost any function. In addition to its graph, the function $f(x) = x^n$ can be visualized as the volume of a ...
0
votes
0answers
221 views

Visualization of 2-dimensional function spaces

As a follow-up question to what is the norm measuring in function spaces I just had an idea: How about visualizing function spaces as normal planes. What I have in mind is to have an orthogonal ...