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

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2answers
29 views

Can't solve exponential equation using logs?

I can't figure out why my method isn't working. I know it is possible to solve this using a substitution but I don't know when to use the substitution. In general when are you supposed to substitute ...
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1answer
25 views

Integral Euler's formula equals integral $\frac{sin(t)}{t}$ dt

For real $c$ we should have that \begin{align} \int_{-T}^{T} \frac{e^{itc}}{2it} dt = \int_{0}^{T} \frac{\text{sin}(tc)} {t}dt. \end{align} However, for me this is not directly clear. I know that $e^{...
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3answers
50 views

How to show that $2 ^\binom{N}{2} \sim \exp(\frac{N^2}{2}\ln(2))$

How can we show that: $2 ^\binom{N}{2} \sim \exp(\frac{N^2}{2}\ln(2))$ for large N.
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2answers
30 views

More than one way to solve a exponential equation?

What techniques can you use to solve the following equation: $$5 \times 2^x = 2 \times 3^x$$ I know we can use logarithms, but I don't have a lot of confidence solving exponential equations in ...
0
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1answer
28 views

equivalent characterization of the complex exponential

I want to prove the following statement: Define the complex exponential function $\exp$ by $\exp(z)= \sum_{n = 0}^\infty \frac{z^n}{n!}$. Then $\exp$ can be characterized by $$\frac{\mathrm df}{\...
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1answer
34 views

A simple question about limits.

This may seem like a simple question, but I feel as if it is wrong but I am unsure why. Is it possible to evaluate a limit in two stages for example: say you know that $x(1- a)\rightarrow b$ as $x\...
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4answers
72 views

Prove that $\lim\limits_{n \to \infty} (1-\frac{1}{2n+1})^{3n} = \frac1{e\sqrt{e}}$

I have this problem that I cannot seem to solve. I tried splitting it into two factors $$\lim\limits_{n \to \infty} (1-\frac{1}{2n+1})^{2n}\times \lim\limits_{n \to \infty} (1-\frac{1}{2n+1})^{n}$$ ...
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2answers
47 views

Sequence solutions of $ax=e^x$

This question comes from my answer to: Solving $4x = e^x$ without graphing and looking for intersection Here I've used a sequence of nested exponentials constructed from $$ x=\frac{1}{a}e^x $$ and a ...
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2answers
93 views

Solving $4x = e^x$ without graphing and looking for intersection

If I want to solve the equation $4x = e^x$, is there a way to solve for $x$ without graphing and looking for intersection?
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1answer
74 views

Determining parameters of $y=ab^x+c$ given 3 points

We can find the parameters for the equation of a parabola through $(x_1,y_1)$, $(x_2,y_2)$, $(x_3,y_3)$ by solving the system \begin{cases} ax_1^2 + bx_1 + c = y_1 \\ ax_2^2 + bx_2 + c = y_2 \\ ax_3^...
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2answers
39 views

Growth of debt: exponential, logarithmic, or linear? [closed]

If I have increasing debt that I don't intent to pay off for a really long time, how would I prefer to have it grow? Exponentially, logarithmically, or linearly?
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1answer
35 views

exponential growth rate

Let's suppose I have $3$ flowers in a field initially and that the number of flowers doubles every month. I can then write that $$N=3(1+0.5/12)^{12t}$$ where $t$ is the time in years. Right? But then ...
1
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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 ...
0
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2answers
74 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
66 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 the ...
0
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1answer
15 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 ...
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1answer
20 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
65 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 ...
0
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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 ...
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1answer
19 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 $M=...
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
42 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: $Q(t)=Q_0\left(1+\frac{.08}{4}\right)^{...
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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
36 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 $\frac{\infty}{\...
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2answers
20 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 cesium-137 ...
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 ...
0
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2answers
69 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
32 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 e^...
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1answer
27 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 $l^{-1}...
0
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2answers
72 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
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1answer
44 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
46 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
35 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 x=...
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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 {(1+1/n)^{hn}-1}{h}=\lim_{n\to\...
3
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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 ...
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1answer
35 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: $$\frac{\bar{X}-\...
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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
34 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?
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1answer
34 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
64 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 $$4^{\frac{x}{y} + \frac{y}{x}} = 32$$ $$2^{2(\...
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 ...
1
vote
2answers
29 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
votes
2answers
116 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: $$\...
5
votes
3answers
74 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 ...