For questions on elementary functions, functions of one variable built from a finite number of exponentials, logarithms, constants, and $n$th roots through composition and combinations using the four elementary operations $(+, –, ×, ÷)$.

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0
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1answer
24 views

What are equivalent parametric equations?

What are equivalent parametric equations? Is there a fast method to prove that 2 parametric equations are non-equivalent?
-1
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1answer
42 views

fourth grade math [on hold]

The science projects are set on tables.There are $99$ long tables used.Each long table holds $7$ projects. The rest of the projects are set up on short tables. Each short table can hold $4$ ...
0
votes
2answers
32 views

Minimum of a+b+c given an exponential equation

I am reviewing for my exam next week and in my notes we have this problem: Given x,y,z as real numbers and $4^{\sqrt{5x+9y+4z}}-68\cdot 2^{\sqrt{5x+9y+4z}}+256=0$. Find the product of the minimum and ...
0
votes
3answers
41 views

Find the values of $\sin z=0$

Hello can someone check my work and tell me what to do next. Find the values of $\sin z = 0$ we know that; $$\sin z = {{{e^{zi}} - {e^{ - zi}}} \over {2i}}$$ So; $${{{e^{zi}} - {e^{ - zi}}} ...
3
votes
2answers
79 views

How do I find the value of $\tan\bigl(\frac{\pi}{4}-3i\bigr)$ in terms of elementary functions?

The question is to compute $t=\tan \left( {{\pi \over 4} - 3i} \right)$. So I change it into $$t={{\sin ({\pi \over 4} - 3i)} \over {\cos ({\pi \over 4} - 3i)}}$$ Then $$\eqalign{t= & ...
1
vote
1answer
39 views

Finding the inverse of a non linear function

Let $F:\mathbb{R}^2\to \mathbb{R}^2$ be the diffeomorphism given by $$F(x,y)=(y+\sin x, x) $$ Find $F^{-1}$. I know that the answer is $F^{-1}(x,y)=(y,x-\sin y$), this can be shown to be true by ...
27
votes
11answers
1k views

Farewell 2014 welcome 2015 - “Math Golf” [closed]

In programming, Code Golf is a competition in which the participants are trying to implement an algorithm with code which is as short as possible. Also, Stack Exchange has a site dedicated to Code ...
1
vote
1answer
23 views

Examples of functions with properties

I'm looking for example of functions defined on $[0,+\infty)$ with the following properties: 1) continuous, twice differentiable 2) $f(0)=0$, $\lim_{x\to+\infty} f(x)=1/3$ 3) $f^{\prime}>0$, ...
2
votes
0answers
20 views

Number of terms in $f'$ as a polynomial of the number of terms in $f$

Take an elementary function $f$. The derivative $f'$ is an elementary function as well. Let $G(f)$ be the minimum number of terms required to express $f$ as a formula comprising a finite number of ...
0
votes
0answers
69 views

Nice approximations of sums by integrals.

Let $f(x):\Bbb Z^+\rightarrow \Bbb R^+$ be a non-monotone function. If for every $m\in\Bbb N$, $$S(m) =\sum_{n=1}^N\frac{1}{(1+f(n))^m}$$ be sum of interest, then is there a way to approximate this ...
3
votes
2answers
50 views

Proving that a function is absolutely monotonic on a given interval

According to the following Wolfram Alpha article: http://mathworld.wolfram.com/AbsolutelyMonotonicFunction.html, the function $f(x) = -\ln(-x)$ is absolutely monotonic on the interval $[-1,0)$. My ...
3
votes
1answer
48 views

Asymptotic elementary expression for the antiderivative of $x^x$

It is well known that there exists no elementary function $f$ with $$\int x^x\,dx \quad = \quad f$$ Is there an elementary function $g$ such that $$\int x^x\,dx \quad \tilde{} \quad g$$ in the ...
0
votes
4answers
63 views

How do I figure out if a 'function is odd or even'?

I am currently doing an highschool math problem and I do not know what the question is asking for when it asks 'state which functions are odd and which are even for the below'. $f(x)= x^2+1 $ ...
0
votes
1answer
27 views

Is there an intersection point for $f_1 = 100n²$ and $f_2 = 2ⁿ$?

As far as i know, there is, and it can be answered by resolving the equation $100n^2 = 2^n$. But how can it be resolved? I already tried to transform the equation using $\log 2$, but i just don't get ...
0
votes
4answers
66 views

How to produce 100 with only four 3s and mathematical symbols

An approach like this doesn't count!        $\underset{k \in \mathbb N, k \leq (3+3/3)} {\sum k^3}$ Thanks for your answers but I need something more strong... You are ...
0
votes
2answers
38 views

Help with parametric quadratic equation

We have the equation $x^2+ax+2a+1=0$ which has real roots $x_1$ and $x_2$ and a is a parameter. I need to answer to the following questions: Find all values of a for which $x_1=(a-1)x_2$. For which ...
1
vote
4answers
38 views

Simplfiying $x(5xy+2x-1)=y(5xy+2y-1)$

I want to simplify: $x(5xy+2x-1)=y(5xy+2y-1)$ to $(x-y)(5xy+something-1)=0$ but I can't figure out what to do with the $2x$ and $2y$ on both sides.
0
votes
2answers
38 views

Inverses / Bijections

Let $f:A\to B$, and $g:B\to A$ such that $$ g(f(a))=a \ ,\ \forall\ a \in A, $$ and $$ f(g(b))=b\ ,\forall\ b \in B. $$ Does this mean that $f,g$ are inverses and bijections? Bests
0
votes
0answers
29 views

Solve this Simple quadratic equation $cU^2-2(a+ b)U+2(a-b)V- cV^2=0$

I need help solving this symmetrical quadratic equation where $\gcd(a,b,c)=1$$cU^2-2(a+ b)U+2(a-b)V- cV^2=0$ Is there an easier method than the quadratic formula? Any hint?
2
votes
2answers
28 views

If $\sin \theta = \frac{3}{5}$, what are the possible values of $\cos\theta$?

If $\sin \theta = \frac{3}{5}$, what are the possible values of $\cos\theta$ ? I do know that .. $\sin^2\theta + \cos^2\theta = 1$ .. and you can solve that equation, resulting in $\cos \theta = ...
2
votes
1answer
28 views

Finding an upper bound for this expression

Given two non-negative numbers $a$ and $b$, I'm trying to found an upper bound for $-\sqrt{a} + \sqrt{a+b}$ But I'd like the bound not to be dependable on any square roots and have only one term. ...
4
votes
5answers
46 views

Tool for expressing $x=f^{-1}(y)$ if $y=f(x)$ is given

I have many equations of nature - $y=ax^{12}+bx^5+5x^4+1$ For all these equations, I need to express x in terms of y. What tool or software would you recommend for this? I would much prefer to ...
3
votes
0answers
38 views

Prove that an equation has no elementary solution

There are methods proving that a polynomial isn't solvable in radical extensions (see Abel–Ruffini theorem) or proving that an integral or a differential equation has no solutions expressible through ...
0
votes
2answers
46 views

Asymptotic approximation of the arctangent?

That is, I am looking for an algebraic function $f(x)$ that approximates $\arctan x$ for large values of $x$. The approximation could be reasonably modest -- perhaps something like $$\tan (f(x)) = ...
1
vote
0answers
39 views

Are there any functions which were proposed as elementary by mathematicians but not considered elementary now?

Are there any functions which were proposed by various mathematicians to be included in the set of elementary functions because of their properties but not considered elementary as of now?
0
votes
0answers
47 views

Monotonically decreasing function for multiplication product?

I have a set of numbers $S = [100,999]$ for which I want the maximum product $p$ such that $p = a \times b$ for all $a,b \in S$ also fulfilling some condition $C$. I would like $p$ to be monotonically ...
2
votes
1answer
504 views

Need to prove that the equation has only 1 solution.

I have been trying to solve the following equation: $5^x+7^x=12^x.$ Obviously, x=1 is a solution but how do I prove that there are no other solutions.
3
votes
2answers
79 views

prove a sequence without knowing its convergence

Let $(X_n)$ be the sequence with $X_1=2$ and $X_n=\sqrt{5X_{n-1} + 6}$ for all $n\ge 2$. How can you prove that it is convergent? IF given convergence, I know that its limit is $6$, but the question ...
1
vote
2answers
39 views

Inequality Expression

what happens when an inequality expression is divided by the same number. Did the sign stay the same each time or change directions? If the sign changed, explain what made the direction of the sign ...
0
votes
0answers
56 views

Autocorrelation of Raised Cosine Function

Let us define the raised cosine function as follows: $f \left( x \right) = \dfrac{\left( 1 + \cos \left( x \right) \right)}{2}$, for $- \pi < x < \pi$. $f \left( x \right) = 0$, elsewhere. I ...
0
votes
2answers
36 views

Evaluate $\lim_{x\to 0}\frac{1}{x^4}\int_{-x}^{0}\sin(t^3)\,dt$

Evaluate $$\lim_{x\to 0}\frac{1}{x^4}\int_{-x}^{0}\sin(t^3)\,dt.$$ I use L'Hopital's Rule and get -1/4. The solution says 1/4. Any ideas?
5
votes
1answer
138 views

Is there a theory of integration in elementary terms for definite integrals?

Let's call a real number explicit if it can be expressed starting from integers by using arithmetic operations, radicals, exponents, logarithms, trigonometric and inverse trigonometric functions. For ...
3
votes
2answers
43 views

Solution of $(1-x)^p= x$ in $(0,1/n)$

Let $n,p$ be positive integers. The equation $$ (1-x)^p = x $$ has a unique solution $x_p$ in the interval $(0,1)$. This follows by the monotonicity properties of $(1-x)^p$ and $x$. My question is: ...
2
votes
1answer
70 views

Injective map from real projective plane to $\Bbb{R}^4$

Consider the mapping $\Bbb R^3\rightarrow\Bbb R^4$ given by $$f(x,y,z)=(x^2-y^2,xy,xz,yz)$$ which passes to the quotient and can therefore be viewed as a map from the projective plane $\Bbb ...
9
votes
11answers
432 views

How to prove $(1-\frac1{36})^{25}\lt\frac12$?

How to prove the inequality? $(1-\frac1{36})^{25}\lt\frac12$ I'm in trouble. Thank you very much for your help
3
votes
7answers
119 views

Find the derivative of $y=x\sqrt{9-x}$

"Find the derivative of $y=x\sqrt{9-x}$." So this is what I have and now I'm stuck. \begin{align} y' &= x \frac{d}{dx}\left[(9-x)^{1/2}\right] + (9-x)^{1/2} \frac{d}{dx}(x)\\ &= x ...
0
votes
0answers
83 views

On a differential equation problem of international mathematical competition for university students

I am trying to solve problem 2 of this competition: http://www.imc-math.org.uk/imc2009/imc2009-day2-solutions.pdf I have other thought but i couldn't fill in the detail. Consider the initial value ...
1
vote
0answers
53 views

Power series solution for a DE with Frobenius method

The given DE is $(x²-3)y"+2xy'=0$ Since there is a singular point ($x=\pm\sqrt{3}$) I used the Frobenius method. I found two indicial relationships: $-3r(r+1)=0$ and $-3(r+1)(r+2)=0$ because I have ...
1
vote
1answer
26 views

Composition of mappings on finite sets

If I'm working in the realm of finite sets on the form $\underline{n} = \{1, \ldots, n\}, n \in \mathbb{N} $. Consider any two transformations $f :\underline{n} \to \underline{m}$ and $g ...
0
votes
1answer
38 views

Find a polynomial with certain conditions.

Suppose that: $$f(x) = 3\frac{x^4+x^3+x^2+1}{x^2+x-2}.$$ Find a polynomial $h(x)$ such that $f(x) + h(x)$ has horizontal asymptote of 0 as $x$ approaches positive infinity.
1
vote
0answers
36 views

Is it possible to find set of functions so that their antiderivatives are also in this set?

Antiderivatives of elementary functions sometimes are not elementary functions. Is it possible to add finite number of non-elementary functions to elementary function set so that their ...
1
vote
4answers
56 views

Equation , powers of two

I want to find the sum of the roots of the equation $$4(4^x + 4^{-x}) - 23(2^x + 2^{-x}) + 40 = 0 $$ in real numbers. I tried the substitution $ 2^x = t $ but then it turns into a quartic equation ...
28
votes
0answers
628 views

Generalization of Liouville's theorem

As proposed in this answer, I wonder if the answer to following question is known. Let $E = E_0$ be the set of elementary functions. For each $i > 0$, inductively define $E_i$ to be the closure ...
37
votes
4answers
3k views

Why can't we define more elementary functions?

$\newcommand{\lax}{\operatorname{lax}}$ Liouville's theorem is well known and it asserts that: The antiderivatives of certain elementary functions cannot themselves be expressed as elementary ...
1
vote
0answers
27 views

Definition of standard functions [duplicate]

In many texts and books about calculus we see There are functions $f$ for which the anti-derivative cannot be expressed in terms of standard functions or there are many integrals that cannot ...
3
votes
3answers
134 views

$4 \sin 72^\circ \sin 36^\circ = \sqrt 5$

How do I establish this and similar values of trigonometric functions? $$ 4 \sin 72^\circ \sin 36^\circ = \sqrt 5 $$
1
vote
0answers
64 views

Logarithm and “basic” functions.

To express the antiderivatives of $\frac{1}{x}$, we cannot apply the formula $\int x^n dx=\frac{x^{n+1}}{n+1}+C$ and we need to introduce a new function, the logarithm. But how can we prove that ...
1
vote
0answers
50 views

Repository of functions which do not have elementary integrals [duplicate]

If there is some function and I suspect that the primitive function cannot be expressed using elementary functions, I would like to have some argument that there indeed is no such expression. One ...
4
votes
2answers
78 views

value of $a+b$ of the following function

$f(x)=x^3-3x^2+5x\;$ and $\;f(a)=1,f(b)=5.\;$ Find $a+b$. I know only one real root exist for each equation as derivative of the function is always positive .I do not intend to use the formula of ...
1
vote
0answers
14 views

efficiently solve for values of a coefficient in a function, so for those values, the function intersects another function a specific number of times.

This is my summer assignment for my freshman "Intro to Numerical Methods with Matlab: Unit 2" course. The task: "Write an efficient Matlab code, which will take any closed $f(x)$ and $g(x)$ and ...