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

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1answer
27 views

Evaluate if series with exponential diverges or converges

The task is to evaluate for what values of $a \in \Bbb R_+$ does the series $$\sum_{n=1}^\infty \frac{a^n \times n!}{n^n}$$ converge. I've already checked with the ratio test that it converges for $ a ...
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2answers
69 views

How to solve exponential equations?

How to solve the equation: $$2\cdot 3^x +2^{2x}+5^{2x-1}-13^x+10=0$$ Well the answer can be found by trial & error to be $x=2$. But I am not able to proceed in a systematic way. I cannot see ...
4
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1answer
73 views

If $\theta$ is a rational number, is $e^{i\pi\theta}$ algebraic?

I want to know if $\theta$ is a rational number, is $e^{i\pi\theta}$ an algebraic number or not? For the first step I tried to write it $(e^{i\pi})^\theta$, that equals $(-1)^\theta$, but I think ...
9
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1answer
65 views

if $f(x + y) = f(x)f(y)$ is continuous, then it has to be injective.

Let $f$: $\Bbb R$ $\rightarrow$ $\Bbb R$ be a non-constant function such that $f(a + b) = f(a)f(b)$ for all real numbers $a$ and $b$. Prove that if $f(x + y) = f(x)f(y)$ is continuous, then it has ...
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2answers
26 views

Simple e equation

$$e^{-x}-x+1=0$$ $$\frac{1}{e^x}=x-1$$ $$e^x(x-1) = 1$$ $$\therefore e^x = 1, x-1 = 1$$ Where $$x=0, x=2$$ Or, $$e^x = -1, x-1 = -1$$ Where $$x=nil,x=0 $$ Therefore, there is no solution to ...
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1answer
13 views

How to Find Variables of Exponential Function Based on Other Information

Given the exercise in the screenshot below, I don't understand why, in order to find the value of the constant 'r', we need to equate r2 to 0.55 (as they did in the screenshot), when we actually need ...
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0answers
14 views

How can I “map” a parameter with range $[0,∞]$ to a “ratio parameter” such as probability?

Newbie in the house! On one hand, I have this sense that there exists one non-arbitrary, a priori or 'natural' function to map $[0,∞]$ into ratio parameter, natural in the sense $e^x$ is natural. On ...
0
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1answer
18 views

Population decline.

I'm looking at a question here and I'm a bit confused on how I'm supposed to solve it. A population of 460 decreases at 5% monthly. How many years will it take for there to be 100 left on the island? ...
1
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1answer
64 views

A basis for the algebra $\mathbb{C}\{z^{\alpha}(1-z)^{\beta}\}$?

Let us consider the domain $$ \Omega=\mathbb{C}\setminus (]-\infty, 0]\,\cup\,[1,+\infty[) $$ (the doubly cleft plane). On it, we have the functions, $z^{\alpha}(1-z)^{\beta}$ for $\alpha,\beta\in ...
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2answers
37 views

How can I get Maclaurin series for $\frac{x^2 + 3e^x}{e^{2x}}$?

The answer for it is $$3 + \sum_{k=1}^n (3+k(k-1)2^{k-2})\frac{(-1)^k}{k!} x^k + o(x^n)$$ Well, I've tried to change every $e^x$ to $1 + x + \frac{x}{2!} + ... + o(x^n)$ and got nothing useful. I know ...
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0answers
27 views

Simplifying a probability distribution function using an exponential function

I have a pdf for a variable $r$ given two other variables $m, \kappa$ defined as follows: \begin{align} p(r|m,\kappa)=\frac{I_0(\kappa r)}{I_0(\kappa)^m}r\psi_m(r), \end{align} where $\psi_m(r)$ is ...
1
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1answer
18 views

Exponential of a non terminating matric

So I understand how to calculate the exponential of matrices that eventually terminate; however, how to approach the cases in which the matrix does not seem to truncate? For example with the matrix ...
0
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2answers
19 views

What's the exact value of $y$?

Given that $\frac{dy}{dx}=e^{x-y}$ and $y=1$ when $x=0$ find the exact value of $y$ when $x=1$. After my attempts. I stuck in $$y=e^{1-y}+1-e^{-1}$$ How to proceed?
2
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1answer
39 views

Calculating the time for 8% to double an investment when compounded quarterly

I'm trying to use the following formula to calculate the amount of time it will take an investment at 8% interest to double. I'm using the following formula: ...
1
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1answer
86 views

Summation of square root sine

Is it possible to solve this? $$y = \sum_{N=1}^{x}\sqrt{\sin N}$$ I know it is possible to solve $$y = \sum_{N=1}^{x}{\sin N}$$ by expressing sinN as its exponential before doing geometric ...
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0answers
51 views

What is different between $\frac{1}{1+\lambda x}$ and $\exp{(-\lambda x)}$

I want to choose a function $f(x)$ which has properties: $f(x)$ closes to $0$ when $x$ goes to $+\infty$ . I have two option for that $f(x)=\frac{1}{1+\lambda x}$, where $\lambda$ is tuning ...
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3answers
35 views

Simplifying a $\lim_{x\to\infty}$ problem.

So I have a problem regarding limits in my calculus class: $$ \lim x\rightarrow\infty \frac {(1+2x^{1/6})^{2016}}{1+(2+(3+4x^6)^7)^8}$$ Basically what I've identified is that it's an ...
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2answers
19 views

Exponential decay involving logarithm [closed]

In 2011 reactor $X$ released $4.2$ times the amount of cesium-137 as was leaked during reactor $Y$ disaster in 1986? Using; A = Pert Half-life = $30.2$ years. a) What year will ...
0
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1answer
62 views

The irrationality of $\pi/e$ is listed as open yet the infinite product formula for it seems to suggest a way to prove it.

And the formula of all rational products seems to suggest that taking some n as n approaches infinity, the formula will have an always increasing amount of uncancelled primes(so provably non ...
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2answers
67 views

How does one compute $I = \int_{0}^{\infty} e^{-2t^{2/3}}dt$?

As stated in the title, I seek an effective way of computing $$I = \int_{0}^{\infty} e^{-2t^{2/3}}dt.$$ My initial impression was to try to make a transformation to spherical coordinates, but my ...
2
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0answers
26 views

Baker Campbell Hausdorff formula and bernoulli numbers

The BCH formula states that the product of two exponentials of non commuting operators can be combined into a single exponential involving commutators of these operators. One may write that $\ln(e^A ...
0
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1answer
25 views

find the proper matrix of exponetial part

I am awkward to calculate the matrix so I would like to get some help $exp(y^{T}V^{T}\Sigma^{-1}S_{X}- \frac{1}{2}y^{T}ly)$ is proportional to $exp(- \frac{1}{2}(y-a)^{T}l(y-a))$ and $a$ is ...
0
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2answers
63 views

Does Lambert W (Product Log) count as an explicit solution?

Say I have an equation that I can solve in $x$ as follows: $$ x = LambertW_{-1}(y)$$ Where LambertW is the product-log function. Can I say I have an explicit solution for $x$? It looks like that, ...
4
votes
1answer
41 views

Finding the intersections between $y = e^x$ and $y = x + 2$ algebraically?

In trying to find the intersections between $y = e^x$ and $y = x + 2$ in terms of $x$, I came up with the equation, $e^x = x + 2$ and subsequently, $x = ln(x+2)$. Beyond that point, I am stumped. ...
2
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1answer
45 views

Explain why $\exp(-7 \log_{10} n)$ approximates $1/n^3$ so well

I was graphing a few functions, and discovered that the graphs of $\exp(-7 \log_{10} n)$ approximates $1/n^3$ are almost the same. Can anyone explain why this is so? Is there a general result for this ...
0
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1answer
33 views

Predict and identify the coefficient of an exponential increasing function

I have done an experiment, by recording the input and the output data, I want to do an extrapolation (meaning predict the output of an input outside the observation region) this is my input vector ...
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0answers
42 views

Confidence interval for exponential - is it the shortest possible?

The confidence interval for an exponential distribution is said to be: $$\frac{2n\bar{x}}{\chi^2_{1-\alpha /2,2n}}<\frac{1}{\lambda}<\frac{2n\bar{x}}{\chi^2_{\alpha /2,2n}}$$ In general we aim ...
4
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2answers
69 views

Swapping limits: $\lim_{h\to 0}\lim_{n\to \infty}\frac {(1+1/n)^{hn}-1}{h}=\lim_{n\to\infty}\lim_{h\to 0}\frac {(1+1/n)^{hn}-1}{h}$

Almost a year ago I asked the question: How to differentiate $e^x$? And in the accepted answer, the following equality appeared: $$\lim_{h\to 0}\lim_{n\to \infty}\frac ...
3
votes
1answer
74 views

Integration Integrate $\int_{-\theta c}^{\theta c} e^{-K/\cos(\theta)} \, d\theta$

I'm trying to integrate $\displaystyle\int_{-\theta c}^{\theta c} e^{-K/\cos(\theta)} \, d\theta$ Numericaly the integrale look like clean, I try various method to have analytic form: Mathematica ...
1
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1answer
34 views

Confidence interval; exponential distribution (normal or student approximation?)

Let's say we have got a sample of size $n$ from an exponential distribution with an unknown mean $\lambda$. We want to construct a confidence interval and so we can compare this: ...
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2answers
32 views

Why does $|\exp{(R^2 i(\cos{2t} +i \sin{2t})}| = \exp{ (-R^2 \sin{2t})} $?

why does |$\exp{(R^2 i(\cos{2t} +i \sin{2t})}$| $= \exp{ (-R^2 \sin{2t})} $ From the question I'm thinking that $i \cos{2t} =0$ but I'm not sure why?
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2answers
33 views

Exponential conjugate equals to reciprocal?

$$\Im[e^{-i x}]=- \sin x $$ Is this true too? $$\frac{1}{\sin x}= \Im[e^{-ix}]$$ If is not true, how can I express the above sine conjugate in terms of exponential?
0
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1answer
33 views

Prove that inverse fourier cosine transform of $\exp(-tkw^2)=\frac{1}{\sqrt{2kt}}\exp(-\frac{x^{2}}{4kt}) $

In the process of solution of a PDE via Fourier cosine transform the author assumes at one step $$F_{c}^{-1}\exp(-tkw^2)=\frac{1}{\sqrt{2kt}}\exp(-\frac{x^{2}}{4kt}) $$ where Fc^{-1} is fourier ...
2
votes
4answers
63 views

Solve for $x$ if $4^{\frac{x}{y} + \frac{y}{x}}$ $=$ $32$ and $\log_3(x+y)+\log_3(x-y)=1$

Question: Solve for $x$ if $4^{\frac{x}{y} + \frac{y}{x}}$ $= 32$ and $\log_3(x+y)+\log_3(x-y)=1$ My attempt: With the first equation $$4^{\frac{x}{y} + \frac{y}{x}} = 32$$ ...
0
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4answers
57 views

help needed to compute derivative of $e^{x\sin x}$

How should I compute the derivative of $e^{x\sin x}$ ? I am a student of class 11, so can you explain me how to do this without high level mathematics ( I know first principles ) I know that ...
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2answers
25 views

Solving a system of equations with an exponential

I've been trying to solve this problem for a while now and can't seem to figure it out. If $3x - y = 12$, what is the value of $\frac{8^x}{2^y}$? The answer should be $2^{12}$ but I'm not sure ...
2
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2answers
115 views

Integral of $\sqrt{ {\rm ln}^2 4 \cdot 4^{2 x} + 1}$

I'm currently taking calculus, and have hit a problem that is causing me confusion. I have the answer to the problem, I just have no idea how to arrive at that answer. The problem is as follows: ...
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3answers
73 views

$f(x)$ is an analytic function in $\mathbb{R}$ such that $f(-x)f(x)=1$. What else can we find out about $f(x)$?

Well, I know that there are some easy things we can say immediately: $f(0)= \pm 1$, follows immediately $f(x)=\pm 1$ is the obvious solution, so let's look for other solutions. Moreover, let's ...
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5answers
137 views

Integrating $\int xe^{-x} dx$ without parts

Can $\int xe^{-x} dx$ ever be solved by integration by substitution without using parts. Or does, as I suspect, substitution fail to yield a solution in this case. Seems that we can't get a ...
0
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2answers
62 views

Problem with the definition of $e$?

I have an issue understanding one of the definitions of $e$ that I found in a textbook I am using. They defined e as the limit of $(1+x)^{1/x}$ as $x\to 0$. But as $x$ approaches $0$ it can come in ...
1
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4answers
79 views

How would one solve the following equation?

This equation is giving me a hard time. $$e^x(x^2+2x+1)=2$$ Can you show me how to solve this problem algebraically or exactly? I managed to solve it using my calculator with one of its graph ...
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1answer
40 views

How can I solve this equation: $xe^{ax}=b$? [duplicate]

How can I solve this type of equation? $xe^{ax}=b$?
9
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1answer
138 views

Nontrivial integral representations for $e$

There are a lot of integral representations for $\pi$ as well as infinite series, limits, etc. For other transcendental constants as well (like $\gamma$ or $\zeta(3)$). However, for every definite ...
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2answers
50 views

Solving an equation with trigonometric and exponential functions

While tutoring about integration and derivatives of functions, we needed to determine what the biggest distance between two functions, one trigonometric $f(x)$, the second exponential $h(x)$ for value ...
2
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2answers
65 views

Has $e^x = ax^2$ a general solution for all $x$?

I was fiddling around with some math and stumbled upon $\exp(x) = a x^2$, finding myself unable to find a solution. Does it even have a general solution $a$ for all $x$? Some googling brought me to ...
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2answers
47 views

About Euler's formula $e^{ix}=\cos x+i\sin x$ [closed]

I think I probably miss something. Can you tell me what it is? In my assumption, that any given 'x' value, $$e^{ix}=\cos x+i\sin x$$ But, why don't I get the same value in the equation when I ...
0
votes
1answer
20 views

What formula represents how to compute a growing stream to reach a fixed total?

Given a growing sequence of values, where each next element in the sequence is the prior value times a constant growth factor, what formula allows the computation of the starting value in order to ...
2
votes
2answers
68 views

Limits problem without L'Hopital

I am prompted to solve the following limit $$\lim_{x\mapsto 0} (\cos(x)^\frac{1}{x^2})$$ I try to approach this problem by doing $$\lim_{x\mapsto 0} (-1+(1+\cos(x))^\frac{\cos(x)}{\cos(x)x^2})$$ ...
3
votes
4answers
81 views

Determine drug concentration over time, given its halflife and dosage

I want to calculate which of two doses is going to have the most active ingredient over the total time of an experiment. So as an example let's say I have a drug which has a halflife of 5 hours, and ...
0
votes
1answer
64 views

exponential function with polynomial exponent

hi guys can anyone help me, I am currently working with some integrals and i am tangled with $$\int_0^1 x^2(1-x)e^{-ax^2+b(1-x)^2}dx$$ and $$\int_0^\infty x^{-1}e^{-ax^2-bx^{-2}}dx$$ I had tried ...