Questions tagged [exponential-function]

For question involving exponential functions and questions on exponential growth or decay.

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exponential function for pythagoras tree?

the pythagoras tree is a fractal generated by squares. for each square, two new smaller squares are constructed and connected by their corners to the original square. how do i create an exponential ...
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1 answer
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Distance between exponential and first degree polynomial function

Consider the functions $$ \begin{split} f(x) &= x^{2x}\\ g(x) &= 10x+2 \end{split} $$ they are defined on the interval $0 \le x \le 1.5$ I'm trying to find the maximum vertical distance ...
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Is my proof wrong or what could improved about $e^x\geq x+1$

Is the following proof true and new ? The proof : We suppose for $a,x>0$ fixed such that $a\geq 1 $ : $$e^{ax}\leq 1 +ax$$ using Bernoulli's inequality we have : $$1 +ax\leq (1+x)^a$$ We have : $$e^...
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A bound on a function involving exponentials [closed]

Does there exist constants $c,x_0>0$ such that $$\sqrt{e^{1/x}-1}\left(1-\sqrt{e^{-1/x}}\right)\leqslant \frac{c}{x^2}$$ for all $x>x_0$? So far I have only been able to show that this is $O(x^{...
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Approximation of PDF with summation to infinity, cos(x) and exp(x)

I would like to implement this probability density function in C++. However, on this current form, the algorithm takes a lot of time to return a result (especially because it include a summation). Do ...
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2 votes
3 answers
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Why does the matrix exponential $e^A$ always exist?

Why does $e^A$ always exist for any given $n \times n$ matrix $A$? I can't find anything discussing this question, which is quite suprising, since it is such a general question.
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How many integer values of $y$ are there such that there are no more than $2021$ integer values of $x$ wherein $\log_2(x+y^2+1)-3^{y^2+y-3x}<0$?

How many integer values of $y$ are there such that for each value of $y$, there are no more than $2021$ integer values of $x$ satisfying the condition that $\log_2(x + y^2 + 1) - 3^{y^2 + y - 3x} < ...
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1 vote
1 answer
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Assistance with mathematics for fiction project

I am working on a project to develop a rigorously mathematically-defined magic system. The "magical energy" (which is in units of energy) is quantized in what effectively amounts to ...
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How do I find the partial sum of the Maclaurin series for $e^x$?

In one of the problems I am trying to solve, it basically narrowed down to finding the sum $$\sum^{n=c}_{n=0}\frac{x^n}{n!}$$ which is the partial sum of the Maclaurin series for $e^x$. Wolfram | ...
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$\lim_{x \to 0+}{{x^\epsilon}\ln x} = \lim_{x \to \infty}{\ln x/{x^\epsilon}} =0$ for every $\epsilon>0$.

I want to show that $\lim_{x \to 0+}{{x^\epsilon}\ln x} = \lim_{x \to \infty}{\ln x/{x^\epsilon}} =0$ for every $\epsilon>0$. I know it can be easily proved when using L'Hôpital's rule but I'm not ...
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Find all values of $m$ such that the equation $3^{x^2 + 2mx + 4m - 3} - 2 = \left|\dfrac{m - 2}{x + m}\right|$ has two distinct roots on $[-4; 0]$.

Consider the equation $3^{x^2 + 2mx + 4m - 3} - 2 = \left|\dfrac{m - 2}{x + m}\right|$. All values of $m$ such that the above equation has two distinct roots on $[-4; 0]$ are $$\begin{aligned} &&...
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If $0<x,0<\alpha$, then $0<x^\alpha$

If $0<x,0<\alpha$, then $0<x^\alpha$ I know that If $x<y$ and $a>1$, then $a^x<a^y$ And I know the exponential laws. How can I prove it without differentation?
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2 answers
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If $x,y,\alpha>0$ and $x>y$, then $x^\alpha>y^\alpha$

If $x,y,\alpha>0$ and $x>y$, then $x^\alpha>y^\alpha$ I know it's obvious when we use differnetation rule for exponential function, but I'm not allowed to. Is there any way I can show it ...
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3 votes
1 answer
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How to create synthetic data for a decaying curve in order to extrapolate it beyond some point? [closed]

In the following curve , I would like to extend the measurements beyond $x$=1 in order to have a better estimate of the green curve compared to red line. Note: I do not have the analytical form of the ...
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Other approaches to evaluate $\lim_{h\to 0} \frac{4^{x+h}+4^{x-h}-4^{x+\frac12}}{h^2}$

$$\lim_{h\to 0} \frac{4^{x+h}+4^{x-h}-4^{x+\frac12}}{h^2}=?$$ I evaluated the limit by using the Hopital rule,$$\lim_{h\to 0} \frac{4^{x+h}+4^{x-h}-4^{x+\frac12}}{h^2}=4^x\lim_{h\to0}\frac{4^h+4^{-h}-...
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How do I solve this exponential equation without using graphical/visual methods? [closed]

I want to solve this equation for t: $3 = e^{0.1t} + e^{-0.2t}$. Can this be done without plotting a graph and iteratively finding the value of t?
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2 votes
2 answers
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Multiclass Classification: Why do we exponentiate the softmax function?

In the context of neural networks, we use the softmax output in multiclassification models. Firstly, let $P(y) = \sigma (z(2y-1))$, which comes from the definition of sigmoid units. We define $\bf z=\...
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$Ω≕\{(x,y;b):b^x=y\}$ ⇛ Howmany unique values does each∘ $⟦b,x,y⟧_Ω$ take (for fixed values of other two parameters), determined respectively by what? [closed]

Given an exponential relation $(b,x)⧟y$ defined by objects ⧼$x,y,b;Ω$⧽ ≘⧼$\text{power,potence,base;structure}$⧽, in which multiple values are allowed (i.e, corresponding to an exponential map that is ...
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Does $e^X=\lim_{n \to \infty}\left(\text{Id}+\frac{X}{n}\right)^n$ hold for matrices?

Let $X$ be a $d \times d$ real matrix, $d>1$. Is it true that $$ e^X=\lim_{n \to \infty}\left(\text{Id}+\frac{X}{n}\right)^n\,\,\,? $$ Edit: It seems that this question is a duplicate. To make it ...
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How to solve for $a$ in the equation $a^{10} = 100$?

Solve for the unknown $a$ in the equation: $$a^{10} = 100.$$ I know the following is allowed: $$ \log 100 = \log\bigl( a^{10} \bigr). $$ But after that, I get lost. I know the answer is $1.5847$. ...
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2 votes
1 answer
59 views

Proof of an elementary infinite sum identity [duplicate]

How to prove the identity $$\sum_{k=1}^{\infty}\frac{k^{k-1}}{k!}e^{-k}=1\,?$$ It seems innocent enough to have a short proof, but I can't find one after quite an amount of work.
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1 vote
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Fundamental Theorem of Calculus and Chain Rule for differentiating integral with multiple functions (exponential-weighted moving average in ODE form)

I am trying to derive the ODE version of the continuous exponentially-weighted moving average on Integral form given as such: $m(t) = \frac{1}{w} \int_{-\infty}^{t} x(\tau)e^{-(t-\tau)/w} \,d\tau$ I ...
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1 vote
3 answers
46 views

Prove that $e^{-(1-z)\mu} \le z + (1-z)e^{-\mu}$ for $\mu > 0$ and $0 \le z < 1$.

I have tried performing the power series expansion on the RHS and get $$ z + (1-z)e^{-\mu} = 1 + (1-z)\left[-\mu + \frac{\mu^2}{2} + ...\right] $$ And I cannot see why it is necessarily larger than $e^...
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Three equations with 3 variables and one variable is an exponent

Can anyone help me in solving this ? Can you show me the steps I need to make. $$\frac{P}{T} = \frac{c}{T^a+f}$$ where, if T=6, P = 94; T=12, P =109; T=24, P=116.5 find a, c ,f Thanks and Regards,
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Mertens theorem in an arbitrary banach algebra

I'm analyzing the Mertens theorem that states that when at least one of two convergent series $\sum_{i=0}^\infty a_i$ and $\sum_{j=0}^\infty b_j$ is absolutely convergent than their Cauchy product $...
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-3 votes
1 answer
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How to find derivative when x is in the exponent? [closed]

Hi I was practicing for my final math IB exam and stumbled upon this calculus question where I have to find the derivative of a function whose x is written as an exponent. I'm not too familiar with ...
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1 vote
2 answers
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Can $x^{\mathrm{cm}}$ ($x$ to the power $\mathrm{cm}$) be defined?

We all know that $x \cdot \mathrm{cm}$ is "$x$ times $\mathrm{cm}$" but I was wondering about what is $x^{\mathrm{cm}}$ ($x$ to the power $\mathrm{cm}$) can be define this $x^{\mathrm{cm}}$ ...
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-2 votes
1 answer
25 views

How to find the interval that satisfies this inequality $\max\{1 - x,1\} < \exp(x)$? [closed]

How to find the interval that satisfies this inequality? $$\max\{1 - x,1\} < \exp(x)$$
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1 vote
1 answer
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How do I rigorously compute $\lim_{x\rightarrow0} a^x$ for $a \in \mathbb{R}$?

How do I rigorously compute $$\lim_{x\rightarrow0} a^x$$ for $a \in \mathbb{R}$? I can intuitively and graphically get the answer of $\delta_{a\neq0}$ (Kroenecker delta), and I think also by using the ...
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How to solve system of equations with complex exponentials (in medical imaging)

In the context of medical imaging, I need to solve the following equations for $\phi_1$ and $\phi_2$: \begin{alignat}{2} I_1(\boldsymbol{X}) &= \left\{A(\boldsymbol{X})+B(\boldsymbol{X})\cos(k_e ...
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Error in Exponential Sum Formula

I have noticed that, for the exponential sum formula: $$\sum_{n=0}^{N-1} r^n = \frac{1-r^N}{1-r}.$$ the formula is not correct (left side $\neq$ right side) when N has a factional part (not just ...
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1 vote
0 answers
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Proof of exponential inequality [duplicate]

Claim: For any $c \in \mathbb{R}$ and any constant $\lambda > 0$ we have that $$\frac{e^{-\lambda c} + e^{\lambda c}}{2} \leq e^{(\lambda c)^2/2}$$ The place I observed the inequality proves it by ...
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1 answer
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Proof of the inequality $e^x \leq x+e^{\frac{9}{16}x^2}$

How can we prove the inequality $e^x \leq x+e^{\frac{9}{16}x^2}$? Furthermore, if $e^x \leq x+e^{Cx^2}$ holds, what is the smallest $C$? It is similar to the question proof of inequality $e^x\le x+e^{...
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How to find the expected value of an exponential curve?

I have a function of the form $$ y = \left(\frac{a}{b}\right)^{1/T} - 1$$ where $T$ is a future point in time, such that $$0 < T \leq \infty.$$ Question 1: Given constant values for $a$ and $b$, ex:...
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2 answers
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solving for exact solution of $x^{x^x} = 17$

Im trying to solve for the exact solution of $$ x^{x^x} = 17 $$ I understand that the previous solution, $x^x=17$, does not have an exact closed form solution and requires use of the Lambert W ...
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Way to describe an inverse exponential proportionality.

My understanding is that when describing any function of the form $y = x^a$, where $x$ and $a$ are any positive integers, then it can be summarised that $a$ is exponentially proportional to $y$. Given ...
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1 answer
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How do I draw the image of the following complex region under the power map?

Let me denote $R=\left\{0<Arg(z)<\frac{\pi}{6}\right\}$. I want to understand how $f(R)$ looks like if $f(z)=z^i$. My idea was that I first compute the image for an arbitrary $z\in R$ and then ...
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-1 votes
2 answers
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Show that $1+x+x^2/2+\cdots+\frac{x^n}{n!}\leq e^x$ [closed]

$e^x$ is defined as the inverse of Log. I want to show that $1+x+\frac{x^{2}}{2!}+\cdots+\frac{x^n}{n!}\leq e^x$ for $x\geq 0$ but I don't know how to start. I was thinking to use induction over $n$ ...
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0 answers
26 views

Solution of given complex functional equation:

Consider the following functional equation: $F(x+iy)-F(x-iy)= (e^{ay}-1)f(x,y)$ $f(x,y)$could be any function depending on $F$ What is the general solution to this type of functional equation? Any ...
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2 votes
4 answers
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Number of solutions to $3xe^x+1=0$

It is required to analytically(without any online tool) find the number solutions to the equation $3xe^x+1=0$. As there is no solution at 0, the problem is $$e^x=-\frac {1}{3x}$$ at $x=-1$,$e^x$ is ...
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2 votes
1 answer
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Compute $I(x,y,z)=\int_{0}^{T} \frac{1}{(T-t)^{3/2}t^{1/2}} \exp \left(-\frac{\left(z-x-\frac{t}{T}(y-x)\right)^{2}}{2 \frac{t(T-t)}{T}}\right)$

I am looking at the integral $$I(x,y,z)=\int_{0}^{T} \frac{1}{(T-t)^{3/2}t^{1/2}} \exp \left(-\frac{\left(z-x-\frac{t}{T}(y-x)\right)^{2}}{2 \frac{t(T-t)}{T}}\right) dt$$ where $x,y,z$ are real ...
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  • 2,050
2 votes
1 answer
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$f(n)=$, for even integer’s $n$, $\lim_{n\to\infty} \int_{0}^{\infty} \frac{e^x}{^nx} dx$

$f(n)=$, for even integer’s $n$, $$\lim_{n\to\infty} \int_{0}^{\infty} \frac{e^x}{^nx} dx$$ $^n x$ is the tetrative function or ‘power tower’, which when $n=\infty$ can also be written in terms of y, ...
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2 votes
5 answers
75 views

How to find the next level needed for a player to level up using exponents?

Excuse me I've not been in a math class for over 15 years, I'm new to Mathematics stack exchange so hopefully someone can help me refine my question. I'm writing a program in C# and I have this ...
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0 answers
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Prove that sequence $b_n=a_n-\ln n,n\in\mathbb{N}$ is positive and decreasing if $a_0=1, a_{n+1}=a_n+e^{-a_n}, n\in\mathbb{N_0}$ [duplicate]

You have the following recurrence formula: $$a_0=1,\quad a_{n+1}=a_n+e^{-a_n}, \quad n\in\mathbb{N_0}$$ Now define sequence: $$b_n=a_n-\ln n,\quad n\in\mathbb{N}$$ Prove that: $$0 \lt b_{n+1} \lt b_n, ...
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1 vote
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Find one sided limit to show equivalence

I'm working on a problem that looks at growth models and I'm trying to show that the equation, $\frac{dN}{dt}$ = $aN^{\gamma}$ - $bN^{\gamma}$($\frac{N^{\gamma -1} - 1}{1 - \gamma}$) is equivalent to ...
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Proof of power rule for negative integer exponents involving indeterminate forms

Proof of the power rule for negative integer exponents Theorem statement: $$z \in \Bbb Z^-, \ \ f(x) = cx^z \implies f'(x) = czx^{z-1}$$ Proof: $$f'(x) = \lim_{d \rightarrow 0} \frac{c/(x+d)^{|z|} - ...
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1 answer
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Find parameter to catenary interpolate a specific point

I'm working with the catenary equation and this equation is given by $$ f(x) = a \cdot \cosh\left(\dfrac{x}{a}\right) $$ I know this function pass at the point $(x_0, \ y_0)$ and therefore I want to ...
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1 vote
1 answer
44 views

$f(x)=e^{a+g(x)}$ can be written as $f(x)=e^a \left(1+\sum_{n=1}^\infty c_n e^{inx} \right)$

Let $a\in \mathbb R, \{b_n\}_{n=1}^\infty \subset \mathbb C$, suppose $g(x)=\sum_{n=1}^\infty b_n e^{inx}$ absolutely converges. And let $f:\mathbb R \to \mathbb C, f(x)=e^{a+g(x)}$. Then, prove that $...
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0 answers
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Can one integrate an exponential in a polynomial without expanding the polynomial?

I am trying to solve a problem related to failure rates where multiple failures can occur, assuming a Poisson process for failures following an exponential distribution. I am trying to build a ...
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1 answer
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How to represent 1/log(x) as integral of exponentials on domain [1, infty)? [closed]

Is there a way to represent $\frac{1}{\log(x)}$ as an integral of exponentials on the domain $x \in [1, \infty)$?
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