Mathematical intuition is the instinctive impression regarding mathematical ideas which originate naturally without regard to formal mathematical proofs. It may or may not stem from a cognitive rational process.

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Is it possible to always get the optimal formula regardless of the derivation method?

Today I've tried to solve a geometric problem (collision point between two circles in a specific situation). I found a working solution but I'm not sure if it was optimal (maybe my solution took more ...
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How to explain the significance of $\pi$ to a child? [closed]

In honor of $\pi$ Day, I thought I would pose this question. How would you explain the significance of $\pi$ to a child of, say, 9 years of age? While that's certainly an age that is old enough to ...
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28 views

Determine weaker hypotheses for Evaluation Fundamental Theorem of Calculus (Abbott p 202 T7.5.3)

(p 200 T7.5.1) If $f:[a,b] \to R$ is integrable and $F:[a,b] \to R$ satisfies $f(x)= f'(x) \; \forall x \in [a,b]$, then If $g$ is integrable on $[a,b]$, then $\int_a^b f = F(b) - F(a)$. ...
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What does Riemann-Stieltjes integral calculate when $\alpha(x) \neq x$?

When we get Riemann-Stieltjes integral becomes standard Riemann integral which calculates area under the curve. We have that $$ s(f,\alpha,P)=\sum_{k=1}^nm_k\Delta\alpha_k \ \text{ and }\ ...
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Population average age decreases with births AND deaths (kind of)?

When a baby is born, it's easy to see the average age of the population decreases. Intuitively, therefore, when a person dies, the average age of the population must increase to compensate. However, ...
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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 ...
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Modus operandi for proving Evaluation Fundamental Theorem of Calculus (Abbott p 200, Spivak p 272 T14.2)

1. How can we presage to use Mean Value Theorem to start the proof? 2. Mean Value Theorem engenders a point in an open interval. Shouldn't this be $x_i \in (t_{i - 1}, t_i) $? After ...
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Choices of epsilons in proof : $(b_n) \to b$ implies $\left\{\frac{1}{b_n}\right\} \to \frac{1}{b}$ (Abbott pp 47 T2.3.3.iv) [closed]

Original became long, ergo I ask anew. The trick is to look far enough out into the sequence $(b_n)$ so that the terms are closer to b than they are to 0. Consider the particular value $e_0 = |b|/2$. ...
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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, ...
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Relation of common divisors leading to integer results

When dividing an integer $a$ by 3 and 7 both results in an integer answer, I intuitively feel that $a/A$ with $A=21$ would also be integer, which seems related to the fact that $3\times7=21$. ...
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73 views

What's the intuitive explanation that the volume of a solid is $\frac{1}{3} A_{base} h$?

I can see why the area of a triangle is $A = \frac{1}{2} bh$ because it's half of a rectangle with sides $b$ and $h$, but I fail to see the intuitive explanation for this general volume formula. (Yes, ...
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230 views

Why do statisticians like “$n-1$” instead of “$n$”?

Does anyone have an intuitive explanation (no formulas, just words! :D) about the "$n-1$" instead of "$n$" in the unbiased variance estimator $$S_n^2 = \dfrac{\sum\limits_{i = 1}^n ...
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62 views

Derivative of exponential functions

Can anyone present an intuitive reason for why the derivatives of exponential functions, lets say, as apposed to polynomials, grow more rapidly than the functions themselves? i.e. $$ y = e^{x^2}\\ ...
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When is a series expansion related to its derivative by a polynomial equation?

Is there some common theory behind the following two examples? Example 1. Let $p(t) = \sum_{n \geq 0} (-1)^k t^{2k}/(2k)!$, and $x = p(t), y = p'(t)$. Then $x^2 + y^2 = 1$ identically. Example 2. ...
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608 views

Why are we interested in irreducible representation but not faithful representation?

I am reading some materials of representation theory (of a group). The motivation of representation theory is to represent (by a homomorphism $h: G \to GL(V)$, from the group $G$ to a vector space ...
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53 views

How to presage Prove by Contrapositive, for Sequential Characterizations of Limit and Continuity? (Abbott pp 106 t4.2.3, 110 t4.3.2)

Dafinguzman answered consummately this question initially but it became too long. I want to question for different beliefs. 1. $(ii) \implies (i)$ in both Theorems 4.12 and 4.19 posit sequences ...
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62 views

Intuition and counterexamples for higher-order derivative test

In the higher-order test we keep differentiating a function till we find the n'th derivative (n being even) to be greater than or less than zero thereby identifying it as a minimum or maximum. My two ...
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52 views

Geometry of $k$-forms and $k$-vectors

In this question I was trying to see why $k$-forms are selected as the way to generalize vector calculus rather than $k$-vectors and a comment providing links to other questions made me end up with ...
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How would one arrive at the formulas for divergence and curl?

It has been some years since I've taken multivariable calculus now, but there's something I really never understood: how people would discover the expressions for divergence and curl. I mean, the ...
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61 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) = ...
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So we don't need to choose delta, epsilon, or $N \in \mathbb{N}$ in delta-epsilon or sequence convergence proofs?

(http://math.stackexchange.com/a/700667/85079) I would write the proof with all my bounds $\eta$ and then choose $\eta$ to make the conclusion match the arbitrary $\epsilon$. ...
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74 views

If $g \ge 0$ is continuous on $[a,b]$ and $g(x_0) > 0$ then $\int^{b}_a g > 0$ (Abbott pp 199 q7.4.4c)

True or False. If $g \ge 0$ is continuous on $[a,b]$ and $g(x_0) > 0$ for $\ge 1$ point $x_0 \in [a,b]$, then $\int^{b}_a g > 0.$ 1. Need determine if true or false. Ergo do we need ...
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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. ...
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Differential forms turn infinitesimal stuff rigorous?

First of all, I know that infinitesimals are not well defined in standard analysis and they have rigorously nothing to do with differential forms. My doubt is on the intuition between one relationship ...
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Proof. sup{ f(x) } - inf{ f(x) } $\ge$ sup{ |f(x)| } - inf{ |f(x)| } (Abbott pp 198 q7.4.1)

Let f be a bounded function on a set A, and set $S = \sup\{f(x) : x ∈ A\}, I = \inf \{f(x) : x ∈ A\},$ $S' = \sup\{|f(x)| : x ∈ A\}, I' = \inf \{|f(x)| : x ∈ A\}.$ Show that $S - I ≥ S' - I'$. ...
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When does one proof of one direction of an If and Only If proof suffice?

Would someone please explain when this is admissible (please expound on $\color{darkred}{sometimes}$)? In advance of starting an Iff proof, how would one divine/previse if this convenience (of a ...
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Within If and Only If Proofs, why can the proof for one direction be easier than the other?

For $ P \iff Q$, my initial sentiment is that because P and Q are equivalent, the total of two proofs (one for each direction) should entail the equivalent level of "difficulty" or "exertion", as well ...
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Intuition. 3 Equivalences of Riemann integrability (Abbott pp 191 q7.2.4)

Not questioning about proofs. For this entire question, posit $f$ is a bounded function on $[a,b]$. ♪ f is integrable signifies $\inf \{ \, U(f, P) \, \} = \sup \{ \, L(f, P) \, \}$ where $P$ is any ...
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Intuition. Cauchy criteron for Riemann integrability (Spivak pp 239, S. Abbott pp 189 thm 7.2.8)

1. Why $\inf U(f,P') \le U(f, P)$ and $\sup L(f, P') \ge L(f,P) $? I tried to research but I can't find where Spivak defined it $P'$? 2. Why are there two partitions P', P''? Not the same? ...
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109 views

Easiest proof $\sup A + \sup B ≤ \sup(A + B).$ No epsilons, sequences. (S.A. pp 18 q1.3.9d)

(question 2. http://webcache.googleusercontent.com/search?q=cache:DohoRC3-bU8J:www.maths.usyd.edu.au/u/UG/IM/MATH2962/r/PDF/tut01s.pdf) Essay By definition of A + B and sup(A + B), for all a ∈ A and ...
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If sup A < sup B, there exists an element b ∈ B that's an upper bound for A. (S.A. pp 18 q1.3.8)

My Figure: By definition of $\sup B$, $\sup B$ is an upper bound for $B$. Set $e = \sup B − \sup A > 0$. By Lemma 1.3.7, there exists an element $b ∈ B$ satisfying $\begin{align} & \sup B − ...
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42 views

Intuition behind failure rate.

The failure rate of the exponential distribution is a constant, $\lambda$, as the exponential distribution is memoryless. So say we have that $\lambda = \frac{1}{10}$. What is that telling us? The ...
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96 views

Deep Understanding of Independence of Probabilities

I really want to have a deep understanding of the independent probabilities of two events. That means to me that I just do not want to use and know the definition. I want to fully understand the why. ...
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Can we find the $n$ that minimizes $\log{(c/n)}\log{(v)}(k^n v^2)^{\log{(c/n)\log{(v)}}}$?

Can we find the $n$ that minimizes: $$\log{(c/n)}\log{(v)}(k^n v^2)^{\log{(c/n)\log{(v)}}}$$ ...Here $c$, $n$, $k$ and $v$ are all naturals. I tried taking the derivative, and then setting it equal ...
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145 views

$\frac{x}{10!} = \frac{1}{8!} + \frac{1}{9!}$

I have a pretty simple straightforward question. Q) Find the value of $x$ in the following: $$\frac{x}{10!} = \frac{1}{8!} + \frac{1}{9!}$$ Instinctively, I do the quickest thing I know how to ...
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Prove limit of modulus of quotient of two functions is infinity (S.A. pp 144 question 5.3.9)

We need to prove for all $M > 0$, there exists d such that $0 < |x−c| < d \implies |\frac{f(x) }{ g(x) }| ≥ M.$ Choose $d_1$ so that $0 < |x−c| < d_1 \implies \color{brown}{|f(x) − ...
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If $g'(c) \neq 0$, show $g(x) \neq g(c)$ for all $x \in V_d(c)$ (S.A. pp 144 q5.3.8)

Let $g : (a, b) \to R$ be differentiable at a point $c \in (a, b)$. $V_d(c) := \{ x \in R : |x - c| < d \}$. First Case $g'(c) > 0$: We cannot use the mean value theorem since we only know ...
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57 views

Length of a curve by integration: why won't flat segments do?

Maybe my question is a duplicate, but I guess I don't know the right terminology to find it elsewhere. I would be happy to delete it if someone can point out a duplicate. From elementary calculus, ...
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101 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 ...
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Intuition. If f differentiable and $\lim_{x\to0} f'(x) = L$, then $f'(0) = L$. (S.A. pp 137 q5.2.8c,d) [duplicate]

True/False. (c) If $f$ is differentiable on an interval containing zero and if $\lim_{x\to0} f(x) = L$, then $f(0) = L$. (d) Drop the assumption that $f'(0)$ necessarily exists. Not a duplicate of ...
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197 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) > ...
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Function on $\mathbb{R}$ differentiable at a single point (S.A. pp 136 question 5.2.3)

By imitating the Dirichlet constructions in Section 4.1, construct a function on R that is differentiable at a single point. Tried and assayed. 1. How can you calculate this construction? Where ...
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68 views

Show that “$\Gamma \models S \Rightarrow \Gamma \vdash S$” entails “if $\Gamma \nvdash P \And \sim P$ then $\Gamma$ is satisfiable”

Show that "$\Gamma \models S \Rightarrow \Gamma \vdash S$" entails "if $\Gamma \nvdash P \And \sim P$ then $\Gamma$ is satisfiable" I'm primarily confused with the notation being used here. In ...
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59 views

Why there is this relation between $k$-vectors and $k$-forms?

I've been trying to understand the geometrical meaning of $k$-vectors and $k$-forms on some vector space $V$ of finite dimension $n$ over a field $\Bbb K$. Indeed, as I understood, a $k$-form $\omega ...
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48 views

Why always $\delta = 1/n$? Negation of Continuity and Uniform Continuity? (S.A. pp 117 T4.4.6)

These proofs about negation of continuity and uniform continuity proofs always invoke $d = 1/n$. Where did this emanate from? I know $\lim _{n\rightarrow \infty }\frac {1} {n}=0$. Why not something ...
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Action of a group on itself by conjugation is faithful $\iff$ trivial center

p. 5: A group action of G on X is called faithful (or effective) if different elements of G act on X in different ways: when $g_1 \neq g_2$ in G, there is an $x \in X$ such that $g_1 \cdot x \neq ...
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How small can we make a modulus and still perform linear algebra on these pairs?

We can work with numbers of the form $(a^n + a^m)$, where $a$, $n$, and $m$ are all naturals, and $-v \le m \le v$ and $-v \le n \le v$. There is one more possibility: $a^n$ could be replaced by $0$, ...
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139 views

A Nonabelian group of order of product of primes G has a trivial center - Fraleigh p. 153 15.18

Using Exercise 37, show: A nonabelian group G of order pq where p and q are primes has a trivial center. Reference: http://users.humboldt.edu/pgoetz/Homework%20Solutions/Math%20343/hw...
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Proof blueprint - If $G/Z(G)$ cyclic then $G$ Abelian - Fraleigh p. 153 15.37

(1.) Why didn't Fraleigh state the result in the direct form like in my title? Why state it with the negations and then prove the contrapositive? Isn't this extra unnecessary work? (2.) How do you ...
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190 views

What insight is supposed to be gained from this complex analysis exercise?

Let $C_0$ denote the circle centered around some point $z_0\in\mathbb{C}$ with radius $R$. We can parametrize this circle like this: $$\begin{array}{cc} z(\theta)=z_0+Re^{i\theta}, & \theta \in ...