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

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TAMS TOURNEMENT: exponential question (very hard) [on hold]

What is the sum of the roots of $(2−x)^{2012} −x^{2012} = 0$ Any tips or solutions to this question would be greatly appreciated!
1
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0answers
29 views

Differentiation of $\exp(A)$

Let's say we have $${\sigma(\exp(a\cdot X^{-1} \cdot a^\mathrm{T}))}/{\sigma X}$$ when I know that the term inside the exponent is essentially a scalar. Should I differentiate according to ...
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0answers
10 views

Continuously Compound Interest [on hold]

If $5,000 per year flows uniformly over an 8 year period and earns 3% interest, compounded continuously, then the present value, to the nearest dollar, is...?
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3answers
30 views

prove that this is a cauchy sequence

Prove that $\{x_n\}$ = $e^{-n}$ is a Cauchy sequence. I tried to prove this by proving that, For all $ϵ>0$, there is a positive $N$ s.t. for all $n>N$, $|e^{-n}|< ϵ$ For all $ϵ>0$, ...
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1answer
24 views

Given the density function: $\frac{1}{2}\exp\left(-\frac{x}{2}\right), \space x > 0$ find $P\left(\sum_{i=1}^{81}X_i > 170\right)$

Suppose that $X_1,X_2...X_{81}$ are independent random variable with the same probability density function $$\frac{1}{2}\exp\left(-\frac{x}{2}\right), \space x > 0$$ Find ...
7
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1answer
73 views

Does $\exp(2ir\pi)$ equal $1$? What's wrong?

Since $e^{ix}=\cos x+i\sin x$, thus $e^{2\pi i}=\cos2\pi+i\sin2\pi=1$ Now I take arbitrary real number $r$ then $e^{i2\pi r}=(e^{i2\pi})^r=1^r=1?$ But this cannot be true since $\cos2\pi ...
2
votes
2answers
83 views

Is the function $y = a^x + b$ exponential?

What exactly is an exponential function? Some of the sources at which I looked said that it's a function where the rate of change at $x$ $(f'(x))$ is proportional to the value at that point $(f(x))$, ...
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2answers
79 views

Solving $\sinh(ax) = bx$

I need an equation that expresses $x$ in function of $a$ and $b$. $$\sinh(ax) = bx$$ I'm a newbie in mathematics, and i don't know where to get help. I need to get this problem solved in order to ...
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3answers
33 views

Finding a closed-form formula for a sequence that is defined recursively

$$a_0 = 0, a_1 = 1 \quad \text{ and } \quad a_n = a_{n-1} + 2a_{n-2}\quad \text{ for }n\geq 2$$ a) Find $a_2,a_3,a_4,a_5$ b) Find a closed form-formula for $a_n$ I found the value to be ...
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1answer
56 views

How find steps to Question: $x = 10^{x-1}$. Answer: $x = 1$

I created an equation a bit ago where I knew the answer, but not how to solve it. Equation: $$x = 10^{x-1}$$ Answer: $x = 1$ I can not see to find any documentation related to this problem. I know ...
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0answers
15 views

Decisions on the order of integration with double integrals (when Deriving PDF via CDF) (Bank Problem)

Consider the following problem: Gandalf, Saruman and Radagast go to a bank together. There are two open counters which Gandalf and Saruman immediately go to get their service. Radagast goes to the ...
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votes
3answers
390 views

Solution to differential equations $y(0)=1$ and $y^{(n)}=y+1$

When I was solving some differential equations, I asked myself the following: Is there a function has the following: $$y'=y+1$$ $$y''=y+1$$ $$y'''=y+1$$ $$......$$ $$......$$ If the initial value is ...
0
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1answer
25 views

Integral of Euler's formula

Why is $=\int\limits_{-\infty}^{\infty}\cos(-tx)dF(x)+i\int\limits_{-\infty}^{\infty}\sin(-tx)dF(x)=\int\limits_{-\infty}^{\infty}\cos(tx)dF(x)-i\int\limits_{-\infty}^{\infty}\sin(tx)dF(x)$? I know ...
0
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1answer
19 views

How to model function with unknown exponents?

I know the Cobb-Douglass function which describes the production quantity: $$Q(K, L) = A \cdot K^\alpha \cdot L^\beta$$ Also I do know multiple assignments of K ...
0
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0answers
11 views

compounding signups to a website

I am interested in figuring out how many people will sign up to a website. The figures are: 2000 people sign up per hour, that figure multiplies by 1.5 every day - so: Day 1 - 2000 per hour Day 2 - ...
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0answers
14 views

Can an integral in the exponent of an exponetial function be written as a product?

I am asked to simplify/calculate the following integral: $$\frac{\int d^3q \exp \left( -\beta C \int d^3q [u(q) u(-q)|q|^2] \right)|u(q)|^2}{\int d^3q \exp \left( -\beta C \int d^3q [u(q) u(-q)|q|^2] ...
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1answer
20 views

Fourier transform of a function with sine [duplicate]

I don't know how to compute the Fourier tranform of this function: $f(x) = \frac{\sin \pi a x}{\pi x}$ I know that $\frac{\sin \pi a x}{\pi x} = \frac{e^{i \pi a x} - e^{- i \pi a x}}{2i \pi x}$ ...
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2answers
48 views

How to get last digit of $7^{7^7}$

I want to find the last digit of $7^{7^7}$. I found out already that $7^7$ (mod 10) last digit is 3. But how do I use that to get the last digit of the whole thing? Thanks
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2answers
65 views

Transforming the exponential function of a sum into the sum of functions

Is there a way to transform the function $$\exp(A+B+C),$$ where $\exp(\cdot)$ is the exponential function, into a sum $$f(A)+f(B)+f(C)?$$
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2answers
93 views

$[1-x_n]^{n}\to 1$ implies that $n x_n\to 0$ as $n\to\infty$?

As the title says $[1-x_n]^{n}\to 1$ implies that $n x_n\to 0$ as $n\to\infty$? We know that $\left(1+\frac{x}{n} \right)^n \to e^x$ as $n\to\infty$. This implies (not sure why) that ...
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2answers
63 views

Differentiability proof of exponential function $\sum_{n=0}^ \infty \frac{x^n}{n!}$

$$f(x)=\sum_{n=0}^ \infty \frac{x^n}{n!}$$ I want to prove that $f$ differentiable on $x$ in $[0,1]$. I am not clear with using the definition of differentiability. I can prove it is ...
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1answer
35 views

Express recurrence in closed form

I am having trouble understanding the process of expressing the following recurrence in its' closed form. First of all, I do not really understand what "closed form" means. If someone could elaborate ...
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1answer
34 views

Finding nth derivative of an exponential function and its value at the origin.

I have a function defined as $f(x) = e^{-\frac{1}{x^2}}, $if $ x\ne0$; $0$ if $x =0$. where $f:[0,\infty) \to \mathbb{R}$ I am asked to prove the following: (a) that the nth derivative is of the ...
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1answer
11 views

Application of Borel-Cantelli for sequence of two parameters

Let $(A_{m,\ell})_{\ell \geq 0, m \geq 0}$ be a sequence of events in some probability space. How to show by using Borel Cantelli that, if $$\sum_{\ell \geq 0, m \geq 0} P(A_{m,\ell}) < \infty,$$ ...
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2answers
39 views

Relation on $\int_1^x\exp{t^2}dt$

Could you give me some leads to show the following relation : $$\forall x>0, \int_1^x\exp{t^2}dt = \frac{1}{2x}\exp{x^2} + \frac{1}{4x^3}\exp{x^2} - \frac{3}{4}\mathbb{e}+ \frac{3}{4} \int_1^x ...
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0answers
41 views

If $x^a + x^b = x^c + x^d$ how do $a ,b , c , d$ relationship are?

I used to solved these equation style and it's accidentally found an answer from matching $a, b, c,$ and $d$ relationship when $x^a + x^b = x^c + x^d $ (I assume that $ab = cd$) and found that's ...
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2answers
46 views

Find all real $a$ such that $6a^2+3=9^a$

Find all real $a$ such that $6a^2+3=9^a$ The problem seems to be very easy, but now i can't see an easy way to find if there are other roots than $1$. Tried using the derivative but that didn't help ...
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1answer
50 views

Power series $e^{-x^2}$

How would I create a power series of $f(x)=e^{-x^2}$ around $x_0=1$ without using a Taylor series? I need to know this for my upcoming exam so I would be really grateful to anyone who could show me ...
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1answer
33 views

exponential term evaluation doesn't make sense in this example

I am studying for my final and doing some practice questions, but I am confused by something: Here the solution says k at 0 we get N/2, but there is no way that answer is correct. If k is at 0 the ...
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1answer
19 views

Prove that the function is of exponential order and proving in mathematics

I'm currently learning about the Laplace transform and in my textbook in college and I have this definiton: Function $f$ is of exponential order if there exist constants $M>0$ and $a$ such that ...
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5answers
145 views

Why does the sum of the reciprocals of factorials converge to $e$?

I've been asked by some schoolmates why we have $$ \sum_{n=0}^\infty \frac{1}{n!}=e.$$ I couldn't say much besides that the $\Gamma$ function, analytic continuation of the factorial, is defined with ...
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3answers
72 views

The limit of $e^{\frac{1}{x^2 + y^2}}$ as $(x, y) \to (0, 0)$

$$\lim_{(x, y) \to (0, 0)} e^{\frac{1}{x^2 + y^2}}$$ This really should be a simple limit question, I've done similar things many times before, but I'm very out of practice with limits and cannot for ...
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1answer
14 views

The time to fix a TV,is an exponential random variable with parameter $\lambda=\frac{1}{2}$.What is the probability that a fix take more than 2 hours?

I got the following question to solve: The time to fix a TV in hours, is an exponential random variable with parameter $\lambda = \frac{1}{2}$. What is the probability that a fix take more than 2 ...
2
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1answer
36 views

Rewriting $e^{-a|t|}$

here I have to prove the fourier transform of $e^{-a|t|}$ , the beginning of the proof is to rewrite $e^{-a|t|}$ as: $e^{-at} U(t) + e^{at} U(-t)$, I know how to continue the proof starting from this ...
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3answers
26 views

How did we come to this inequality $|\frac{e^x-1}{x}-1|\leq|\sum_{n=1}^{\infty}{\frac{|x|^n}{(n+1)!}}|$

How did we come to this inequality $|\frac{e^x-1}{x}-1|\leq|\sum_{n=1}^{\infty}{\frac{|x|^n}{(n+1)!}}|$. Using $e^x=\sum_{n=0}^{\infty}{\frac{x^n}{n!}}$. I have this inequality in the proof of ...
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2answers
59 views

Prove or disprove that $p_n > e^{p_n - p_{n-1}}$ for large enough $n$.

Let $p_n$ denote the $n$-th prime. Prove or disprove that for large enough $n$ we have $$p_n > e^{p_n - p_{n-1}}.$$ The inequality trivially holds for all the twin primes larger than $7$. With ...
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1answer
28 views

How to obtain and graph a function that first grows exponentially and then decays exponentially?

I would like to know what the equation of a curve is if it grows exponentially (let's say it doubles each time). This would be: $f(x)=2^x$ But then I would like the line to exponentially decay (let ...
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1answer
17 views

Rate of convergence of the difference of two exponentials

I would like to find the convergence rate of the following function: $$f(x) = |e^{-ax}-e^{-bx}|,$$ with $a,b>0$ and $x\to+\infty$. By finding the convergence rate, I mean finding the largest ...
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1answer
36 views

Is the exponential map for $\text{Sp}(2n,{\mathbb R})$ surjective?

For $\mathfrak{g} := {\mathfrak s}{\mathfrak p}(2n,\mathbb{R})$ and $G = \text{Sp}(2n,{\mathbb R})$, is the exponential map \begin{equation} \text{exp} : \mathfrak{g} \to G \end{equation} surjective? ...
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1answer
30 views

Exponential series is cosh(x), how to show using summation?

I want to show that $$\cosh(x) = \sum_{n=0}^{\infty} ‎\frac{(x)^{2n}‎}{(2n)!}‎ $$ I know that $cosh(x) = \frac{exp(x)+exp(-x)}{2}$ but i cant seem to get there from the original series. I know that ...
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3answers
74 views

Exponentiation of a $2\times 2$ matrix

We know: $$\exp(At)=I+ \sum^{\infty}_{n=1}\frac{A^nt^n}{n!}$$ Here $$A= \begin{pmatrix} 0 & 1 \\ -w^2 & 0\end{pmatrix}$$ is a $2\times 2$ matrix, $I$ is identity matrix. How to show: ...
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1answer
22 views

Inequality between factorial and exponential

Trying to find a nice way to simplify the question: Which is bigger 2000! or 1000^2000? I don't know what kind of reasoning I can apply here.
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0answers
24 views

Functions which can be solutions of exponential algebraic equations.

Can the equation $p(f,e^f)=0$ have a solution in $L^{\infty}([0,1])$? Here $p(x,y)$ is a polynomial with complex coefficients, and $L^{\infty}([0,1])$ is the space of Lebesgue measurable ...
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1answer
25 views

Find the root of C [duplicate]

Can u help me to find a root for C (except c = 0) in below equation. $$ce^{-c}-{10\over5}(1-e^{-c})^2=0$$ by expanding this I got, $$ce^{-c}-2 + 4 e^{-c}-2e^{-2c}=0$$ now grouping, ...
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0answers
73 views

Solving an exponential equation without the quadratic formula

High school math student here. In my homework I was asked to solve $16^x +4^{x+1} - 3= 0$ and I used substitution to get $x=\log_4{(-2+\sqrt7)}$. However, this was in the chapter on ...
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1answer
25 views

Calculating growth rate

Let's say I want to have saved $200 in one year. The first week I afford to save $1. I'm curious to find out how the calculation would look like to understand the following: By how much would I ...
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4answers
48 views

Extending $2^n > n $ from set of natural to set of real numbers

I was given a task to prove that $2^n>n$ for any $n \in N \cup \{0\}$. I am aware that this can be solved by induction and that the solution is pretty easy but instead of meddling with induction ...
0
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1answer
22 views

Solving a binomial when one of the terms is in the form $e^x$

Say I have the function $y=4e^{-2x}-3x$. I can use a graphing calculator to approximately determine the roots, but how do I find an exact answer?
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0answers
20 views

Total demand for two different prices, where market shares are determinened by logit model

The setting is simple, i.e. formula for demand of service/product is linear $$ d = \alpha - \beta p $$ where $ \alpha $ is maximum demand, $ \beta $ is some coefficient, and $ p $ is price. There ...
0
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3answers
39 views

Simplify $\frac {3^{(-3+x)}6^{(3-x)}}{3\cdot4^x}$ [closed]

$$\frac {3^{(-3+x)}6^{(3-x)}}{3\cdot4^x}$$ What is the simplest form?