0
votes
1answer
79 views

how $2x=x$ , related to differential calculus [duplicate]

can anybody please tell me what's happening here ? $$1^2=1$$ $$2^2=2+2$$ $$3^2=3+3+3$$ $$x^2 = x+x+\cdots+x \mbox{ ($x$ times)}$$ differentiating both the sides $$2x = 1 + 1 + \cdots+1 \mbox{ ...
0
votes
1answer
33 views

Integer solutions of an equation that is set to a number

How many integer solutions for $a$ and $b$ in $(ab)/(a+b)=3600$? My attempt: $(ab)/(a+b)=3600$ = $ab=3600(a+b)$ = $ab=3600a+3600b$ =$ab=3600a=3600b$ Dividing $3600b$ on both sides ...
1
vote
2answers
113 views

What is the probability of going bankrupt in roulette?

Imagine that the bank has the money $M_1$ and the player has the money $M_2$. The rules are the following: You win with a chance of $\frac{17}{36}$ and lose with $\frac{19}{36}$ each round. Now you ...
0
votes
0answers
41 views

Card Shuffling and Convergence in Probability

There are $4n$ cards, and we denote the set of cards with number $4k,k \in \{1,2,\ldots,n\}$ as $S$. The we shuffle the whole cards randomly, which means that each permutation will happen with the ...
1
vote
1answer
51 views

The rate change of the radius of a coil.

Suppose I have a tube of radius $r_0$ that I want to wrap a sheet of length $l$ and thickness $\Delta x$. Assuming the radius changes only when the paper overlaps the where the previous section ...
0
votes
1answer
54 views

A Bug Crawls Along a Square

A wire of length $4$ is bent into a square. At time $t = 0$, a bug starts crawling from the corner of the square to an adjacent corner, and continues traveling along the rest of the square until it ...
8
votes
2answers
356 views

Euler's identity in matrix form

I assume everyone is familiar with the famous mathematical identity due to L. Euler: $$ e^{i \, \pi} + 1 = 0,$$ where $i^2 = -1$ and $e$ is the base of natural logarithms. I was wondering if this ...
19
votes
12answers
727 views

hand evaluate $\sqrt{e}$

I have seen this question many times as a example of provoking creativity. I wonder how many ways are there to evaluate $\sqrt{e}$ as accurately as possible. The obvious way I can think of is to use ...
7
votes
2answers
137 views

What is the integral $\int x^t/\Gamma(1+t) \, dt$? (In general: relation between series and integrals)

(The question arises from playing with translating series into integrals) I wanted to see, what it means to have a "continuous" relative for powerseries and other series; the most simple one ...
1
vote
1answer
46 views

$[x-\frac{1}{n}, (n-1)x+\frac{1}{n}]$ contains an integer $\forall x\in \mathbb{R}$ and $\forall n\in \mathbb{N}$

For any real number x: Prove that among the numbers x,2x,...,(n-1)x ,there is one that differs from an integer by at most $\frac{1}{n}$. any hints for a pigeon solution. Non-pigeon solution ...
1
vote
1answer
55 views

shorter proof of generalized mediant inequality?

Show $\frac{a_{1}+...+a_{n}}{b_{1}+...+b_{n}}$ is between the smallest and largest fraction $\frac{a_{i}}{b_{i}}$, where $b_{i}>0$. Attempt Assume the largest is $\frac{a_{n}}{b_{n}}\Rightarrow$ ...
0
votes
1answer
68 views

Proving using squeeze principle

This problem sounds very confusing. Please help me solve this problem.
1
vote
1answer
57 views

Prove the derivative

Let $f(x) = (x^2-1)^{\frac{1}{2}}, x>1$. How do I prove that the $n$th derivative of $f(x) > 0$ for odd $n$, and the $n$th derivative of $f(x) < 0$ for even $n$?
1
vote
2answers
79 views

A question about indeterminate forms

Are there any set of numbers into which any of the indeterminate forms we see in a calculus course, like 00, n/0, 1infinity, etc has an answer? I'm asking that because, thanks to the Net, I took ...
14
votes
4answers
510 views

Find all bijections $\,\,f:[0,1]\rightarrow[0,1],\,$ which satisfy $\,\,f\big(2x-f(x)\big)=x$.

A friend of mine gave me the following problem: Find all functions $f:[0,1]\to[0,1]$, which are one-to-one and onto and satisfy the following functional relation: $$ f\big(2x-f(x)\big)=x, \tag{1} $$ ...
4
votes
0answers
164 views

Measuring how change in input variables contributes to output in non linear equation.

How do we measure how a variable contributes to an output as its value increases, and how it relates to other input variables? Let's say we're playing a video game, where you can buy items to augment ...
5
votes
2answers
177 views

Compute the integral $\int_{0}^{2\pi}|\cos^n t|\ dt $ for $n \in \mathbb{Z}$

I need some help on computing this integral. I thought of this when solving another question on S.E. Evaluate$$I=\int_{0}^{2\pi} |\cos^n t|\ dt $$ for $n \in \mathbb{Z}$. Observation: This ...
2
votes
2answers
384 views

So close yet so far Finding $\int \frac {\sec x \tan x}{3x+5} dx$

Cruising the old questions I came across juantheron asking for $\int \frac {\sec x\tan x}{3x+5}\,dx$ He tried using $(3x+5)^{-1}$ for $U$ and $\sec x \tan x$ for $dv$while integrating by parts. below ...
7
votes
1answer
180 views

A functional relation which is satisfied by $\cos x$ and $\sin x$

Assume that the functions $f,g : \mathbb R\to \mathbb R$ satisfy the relations \begin{align} \left\{ \begin{array}{ll} f(x+y) &=& f(x)f(y)-g(x)g(y), \\ g(x+y) &=& f(x)g(y)+f(y)g(x), ...
28
votes
5answers
1k views

Can this ant find its way back to the nest?

So the puzzle is like this: An ant is out from its nest searching for food. It travels in a straight line from its nest. After this ant gets 40 ft away from the nest, suddenly a rain starts to ...
3
votes
1answer
56 views

If a function is smooth is 1 over the function also smooth

If $f(x):\mathbb{R}\rightarrow\mathbb{C}$ is $C^\infty$-smooth. Is $1/f(x)$ also $C^\infty$-smooth? $f(x)\neq0$
7
votes
2answers
222 views

Come up with some fun “equation Limericks”

We were discussing "Limericks" in my Calculus class. Specifically, "equation Limericks". A Limerick is a poem with five lines. The first, second, and fifth lines should have nine syllables each and ...
-1
votes
1answer
63 views

Computation of integral $\int_{0}^{1}\ln(p)\ln(1-p)p^{2}\,dp$

I want to compute this integral: \begin{equation*} J=\int_{0}^{1}\ln(p)\ln(1-p)p^{2}dp \end{equation*} It will be great if you can detail the proof. I tried to do change of variable it does not ...
0
votes
0answers
77 views

What is the maximal distance between two points $A, B \in [\Delta x, \Delta y]$ as they evolve with time?

Let us place two monkeys in a box with dimensions $\Delta x \times \Delta y$, Alice and Bob. The room is fairly large and there are no obstacles within. Alice would like to be within the minimal ...
4
votes
1answer
350 views

Hilarious Comic … DiffyQ and infinity ensue…

I ran across this comic, and it's gold. It is orginially published here If I am correct, the first panel alone defines a self-referential loop if not a differential Equation: $X$: Amount of Black ...
1
vote
1answer
411 views

How do you calculate work (KJ) and Power (W) when jogging on a treadmill?

The following is known... The Weight of the Person, The Angle/grade of the Treadmill, The Time that they are on the treadmill, The velocity of the treadmill. How do I calculate Work in Kilojules and ...
0
votes
1answer
42 views

Adjust a range of given values. [duplicate]

If I have a number anywhere on the range 140 - 350 and I want to match it to the correlated range "0 - 360" what function can I run it through? i.e.:140 would go through the function and return 0.350 ...
2
votes
0answers
54 views

Symmetry between differentiation and integration [duplicate]

I want to make clear, that I am interested in the question: Why does integration need a bigger spectrum of functions than differentiation and not why integration is harder!!! as experience told me, ...
2
votes
2answers
131 views

How to make a box which has the largest possible volume?

I have sheet metal in form of an equilateral triangle and I want to fold it to make a container for the screws. How should I cut and fold to make the a box with largest volume? Basically I cut the ...
2
votes
1answer
40 views

least number of planes intersecting a finite number of points in space, but not intersecting origin.

Let $$\mathbb{R}^*=\mathbb{R}-\{0\}$$ and $$N=\{0,...,n\}$$ and $$\mathcal{M}=\{ A\subseteq \mathbb{R}^3\times\mathbb{R}^* \mid (\forall\mathbb{x}\in N^3:\mathbb{x}\ne 0)(\exists(\mathbb{a},d)\in ...
3
votes
2answers
218 views

Rotating the graph of a function

I think I have answered the following puzzle: Rotations Under what conditions can you rotate the graph of a function about the origin, and still have the resulting graph being the graph of a ...
1
vote
2answers
108 views

Differing by $2$ “puzzle”

I've been thinking about the following: For (1) I drew $y= 0$, $y=2x$ and $y = 2x$, $y = 4x$. The expression I wrote down if $y = mx + b$ is one of the lines then $y_2 = (m \pm 2)x + b$ is the ...
8
votes
1answer
112 views

Navigating a Field of Sprinklers

Everyday I walk to class and I have to walk through a sequence of sprinklers. I usually watch them for a second and try to plan a path in which I never have to stop or back track and will not get wet. ...
3
votes
3answers
316 views

Convergence of $x_n = \cos (x_{n-1})$

I define the sequence $x_n = \cos (x_{n-1}), \forall n > 0$. For which starting value of $x_0 \in \mathbb{R}$ does the sequence converge?
3
votes
2answers
125 views

Different Representations of Numbers in Subsets of $\mathbb R$

I think I've mentioned sometime before about logarithmic number system. In this system, a real number $r$ is represented by $(\text{sgn}(r), \log|r|) \in \{-1, 0, 1\} \times \mathbb R$. If $\mathbb R$ ...
1
vote
3answers
219 views

Deconstructing $0^0$ [duplicate]

Possible Duplicate: Zero to zero power It is well known that $0^0$ is an indeterminate form. One way to see that is noticing that $$\lim_{x\to0^+}\;0^x = 0\quad,$$ yet, ...
28
votes
6answers
3k views

Where is the flaw in this argument of a proof that 1=2? (Derivative of repeated addition)

Consider the following: $1 = 1^2$ $2 + 2 = 2^2$ $3 + 3 + 3 = 3^2$ Therefore, $\underbrace{x + x + x + \ldots + x}_{x \textrm{ times}}= x^2$ Take the derivative of lhs and rhs and we get: ...