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

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1answer
20 views

When using exponential growth why is the form $a \cdot e^{ct}$ used instead of $a\cdot b^t$?

My maths book is not very forthcoming. My guess would be that it is because when you use the form $a \cdot e^{ct}$ is easier to differentiate to see the rate of growth.
1
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1answer
34 views

Bound for the exponential distribution using chebychev's inequality

Let X be a exponential distribution with rate lambda. I'm supposed to find the exact value of P(|X-u x>= k * sigma ) for any k>1 then compare it to the bound I get from chevychevs. How the heck do I ...
9
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3answers
155 views

Proof of the inequality $e^x\le e^{x^2} + x$ [duplicate]

The question is to prove the inequality $e^x\le e^{x^2} + x$. I tried the Taylor expansion like ${e^x} = 1 + x + \frac{{{x^2}}}{{2!}} + \frac{{{x^3}}}{{3!}} + ...$ and $x + {e^{{x^2}}} = 1 + x + ...
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1answer
46 views

Inverse of a function

How can we find the inverse of $f(x) = 3x + e^{2x} $? I am not able to separate the $x$ even taking the logarithm on both sides.
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1answer
18 views

Calculating the Lyapunov exponent of the times-m map, $E_{m}$.

I'm trying to compute the Lyapunov exponents for $E_{m}$, where $E_{m}:S^{1}\to S^{1}$, $x\mapsto mx\mod 1$. The Lyapunov exponent is given by ...
0
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2answers
31 views

Bounds involving exponential

I know that $(1+(1/n))^n \to e$ and $(1-(1/n))^n \to 1/e$ as $n \to \infty$. Both these sequences are increasing. I saw this by plotting; is there an analytical way to see this? Taking the derivative ...
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1answer
33 views

Property of Exponential.

Can anyone please explain how $$\exp[y/\alpha]=\exp[y-\log\alpha]$$ ? I tried as : $$\exp[y/\alpha]=\exp(y/\exp[\log\alpha])=?$$ I think $$\exp(y)\exp[-\log\alpha],$$ can be written as ...
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1answer
21 views

How to find function equation from a given asymptote?

I want to build a $i$ function with the following 4 parameters: $a$ is the left inflexion point $b$ is the right inflexion point $s$ is the maximum of the function $k$ is the asymptote value ($y=k$) ...
2
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1answer
48 views

Evaluate $\int_0^i e^z dz$

Evaluate $\int_0^i e^z dz$. Let $z(t) = it$, where $ 0\leq t \leq 1$. Then, $\int_0^i e^z \, dz = \int_0^1 ie^{it} dt = e^i - e^0 = e^i - 1 = \cos{1} + i\sin{1}$. This looks fairly simple, but I ...
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0answers
9 views

What to look for in a distribution to conclude that it is exponential?

I have the following problem: There are two types of oranges, say A and B. $50\%$ of all type A oranges are edible for more than two weeks, while only $30\%$ of all type B oranges are edible for more ...
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0answers
29 views

How to prove an exponential function preserve the positive semi-definite property?

If $f(x) \in \Re$ has the positive definite property, $\sum_i\sum_j a_i a_j f(x) \geq 0$ for $a_i,a_j \in \Re$, then $e^{f(x)}$ has the positive definite property. How can i prove it? And, i guess a ...
3
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1answer
68 views

Matrix exponential for Jordan canonical form

Let $X$ be a real $n \times n$ matrix, then there is a Jordan decomposition such that $X = D+N$ where $D$ is diagonalisable and $N$ is nilpotent. Then, I was wondering whether the following is ...
12
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2answers
697 views

Show that $e^{x+y}=e^xe^y$ using $e^x=\lim_{n\to\infty }\left(1+\frac{x}{n}\right)^n$.

I was looking for a proof of $e^{x+y}=e^xe^y$ using the fact that $$e^x=\lim_{n\to\infty }\left(1+\frac{x}{n}\right)^n.$$ So I have that ...
2
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1answer
33 views

Convergence to zero of exponential map

Let $A$ be some matrix and for any $x \in \mathbb{R}^n$ we have for $$x(t):=e^{At}x$$ that $\frac{d}{dt}||x(t)||<0$ if $x$ was not zero. Then I was wondering if we can conclude that $||x(t)|| ...
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1answer
20 views

Order of operations in polynomial with exponent

I have a simple question about whether or not my approach is correct in simplifying a polynomial, here it is, $(n(n+1)/2)^2 = ((n^2+n)/2)^2 = 1/4(n^4+2n^3+n^2)$ I apologize if you find that hard to ...
2
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2answers
58 views

justifying a definition of $e^x$

I'm studying the different definitions of $e^x$ and showing their equivalences. If we define $e^x$ as the limit of the sequences of functions $f_n=(1+x/n)^n$, for all real $x$, how do we go about ...
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1answer
10 views

Regions, Secants and Edges

In a circle, I have N points then I would make secants and edges then I have regions, which has sides as edges. My questions is when N increases, the regions double but it will not last long, so why ...
2
votes
2answers
57 views

Taylor expansion of logarithm function.

Expand $f(x) = \log(1 + x)$ around $x = 0$ to all orders. More precisely, find $a_n$ such that for any positive integer $N$, we have$$f(x) = \left(\sum_{n=0}^{N-1} a_nx^n\right) + E_N(x) \text{ for ...
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5answers
114 views

Where is my thinking wrong? [duplicate]

I have always known that $a^n=a*a*a*.....$(n times) Then what exactly is the meaning if $a^0$ and why will it be equal to $1$? I have checked it in the internet but everywhere the solution is based ...
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1answer
95 views

Calculating limit of sequence by Euler $e$

I'd like to calculate the following by the only sequence concept, not to involve the other concepts like limit of function, etc. problem $\begin{align} \tag{i} &\lim_{n\rightarrow \infty} ...
2
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3answers
70 views

Proof for which exponent is greater

Is there a way to prove which one of these is bigger? $e^{(a+b)}$ or $e^a + e^b$? Thanks
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2answers
151 views

Newton's law of cooling, soup

Newton's law of cooling states that the temperature $T(t)$ of an object at time $t > 0$ changes at a rate proportional to the difference between the temperature of the object and the temperature ...
0
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1answer
53 views

What's a real world example of double exponential function and factorial function? [closed]

As the title asks. I'm looking to create very fast growing numbers. If there's a better solution than these two please let me know as well.
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3answers
69 views

Does $ \sum_{k=0}^n k {n \choose k}$ have a convenient exponential equivalent? [duplicate]

I know that: $${n \choose 0} + {n \choose 1} + ... + {n \choose n} = 2^n.$$ Does $$0 {n \choose 0} + 1 {n \choose 1} + ... + n {n \choose n} = ??$$ have some convenient simplification?
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0answers
20 views

Exponent Laws and the Ceiling Function

Suppose I have $f(x) = 5^{\lceil \frac x 3 \rceil}$, where $x \in \Bbb N$. If I were to simplify $f(x+4)$, can I do the following: $f(x+4) = 5 ^{\lceil \frac {x+4} 3 \rceil} = 5^{\lceil {\frac x 3} ...
6
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4answers
434 views

Instance of Mean Value Theorem

What's an elementary proof that there exists $c\in (a, b)$ with $$e^b-e^a=(b-a)e^c$$ without calculus, just using the standard properties of the exponential function? *By calculus I mean ...
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1answer
40 views

Prove that $1+x \le e^x \le 1+x+x^2$ for every $|x| \le 1$ [closed]

I get the following inequality formula. $$1+x \le e^x \le 1+x+x^2\quad\text{if}\quad |x| \le 1$$ I know $\displaystyle e^x = 1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+...$ but I cannot prove the ...
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3answers
126 views

How to prove that n! have a higher order of growth than n^(log n)?

I am aware that n^n have a higher order of growth than n!, but how about n^(log n)? Is there a way to get an alternative form of n^(log n) such that when taking the lim n to infinity [alternative ...
2
votes
1answer
25 views

Do these functions have the same order of growth?

I have a list of functions and was confused whether they have the same order of growth. $$f(n) = \Theta(g(n))$$ Given functions: $\log^2 n, \log(n^2) $ My method: I took the logs of both functions ...
0
votes
2answers
40 views

Constructing an exponential function

I need to create a number picker via slider where you can pick numbers from 1 to a million, but the lower numbers should have a better resolution so you could choose 1, 13,43,50 easily but when it ...
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1answer
10 views

Exponential distribution: Fuel pumps

There are 2 fuel pumps (one active and one reserve). If the first one fails, the reserve pump is brought online. The life of fuel pumps follows an exponential distribution with an expected life of 2 ...
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2answers
64 views

Find function which satisfies the initial value problem: $\displaystyle 6 \sec x \,\frac{dy}{dx} = e^{y + \sin x} $

Find the function which satisfies the initial value problem: $\displaystyle 6 \sec x \,\frac{dy}{dx} = e^{y + \sin x} $ $\displaystyle \;\ y(0) = -9 $ So as far as I understand it, I should move ...
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0answers
8 views

Reconciling aggregate and component growth rates

I'm working with a very basic basic forecast model using Compound Annual Growth Rate and I need to reconcile the forecasts at different levels of detail. Suppose I have two business lines with ...
0
votes
1answer
70 views

Is the power of a complex exponential signal always zero?

Is the power of a complex exponential signal always zero? For example say I have the function $ f(t) = Ae^{i\omega t}$ Then, I think power is defined as: $P=\int_{-T/2}^{T/2} f^2(t) dt$ So is it ...
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0answers
72 views

Properties of $\sum\limits_{n\ge2}\frac{n(n-1)x^n}{n!}$ for $x\in \mathbb{R}$

This is from an MCQ Contest. for all integer $n$ greater than or equal to $2$ Let $$\forall n\geq 2\qquad u_n=\dfrac{n(n-1)x^n}{n!}, \qquad x\in \mathbb{R}$$ for all $x$ in ...
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1answer
59 views

What is the integral of $e^{x+e^x}$?

What is the integral of $e^{x+e^x}$ ? I really just need to know how to get the answer rather than the answer. Thanks for any help.
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3answers
81 views

How to prove that $4^n-3n-1$ is divisible by 9?

How can I prove that $4^n-3n-1$ is divisible by $9$? I tried dividing the expression by $9$ and seeing if the terms cancelled in any predictable way but I still cannot prove it. Maybe there is a ...
7
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1answer
95 views

History of the power series for $e^x$ and compound interest

As discussed in How did Bernoulli approximate $e$?, Bernoulli showed that $2\frac{1}{2} < e < 3$ in this paper: ...
0
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1answer
46 views

Exponential Functions $e^{-2x}$

sketch the graph of the following function $f(x) = e^{-2x}$ for $x \in \mathbb R$ this what i got, y-intercept $x=0$ implies $y=\cfrac{1}{e^{2\times 0}}$ therefore $y=1$ and I have used the ...
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2answers
57 views

Find the derivative of the following function: $\displaystyle y(x) = (\ln{(2 x)})^{5 x}, \quad x> \frac{1}{2}$

Find the derivative of the following function. $$\displaystyle y(x) = (\ln{(2 x)})^{5 x}, \quad x> \frac{1}{2}$$ So the answer given by wolfram seems to be correct: $$5 \ln^{(5 x-1)}(2 x) (\ln(2 ...
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2answers
69 views

How to solve this Limit Algebraically?

$\lim_{x\to\text{-}\infty} xe^x$. This is limit can be easily be seen that it approaches to 0 using graphs. But how to solve it algebraically?
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0answers
21 views

Determine Var[Y(3)/Y(1)]? Fin Math

Let Y (t) = 2e^(0.07t+0.5z˜(t)) Determine Var(Y(3)/Y(1)) under the risk free measure. So I found E*(Y(3)), E*(Y(1)), E*(y^2(3)), and E*(y^2(1)) and divided and subtracted, ect, for variance and got ...
2
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1answer
37 views

How to understand the rate in a Continuous Time Markov Chain.

Suppose there is a CTMC with three status $\{0,1,2\}$, the rate of transition $1\to2$ is $p$, and the rate of $1\to0$ is $q$. I know that it means the time of the status stay in $1$ before transfer ...
0
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1answer
22 views

$\frac{dy}{dx} \;$ of $\; \ln\!\left(y\right)=e^{3y}\sin\!\left(x\right)$

Find $dy/dx \;\;$ if $\;\; \ln\!\left(y\right)=e^{3y}\sin\!\left(x\right)$ So I did: $\frac{dy}{dx} \frac{1}{y} = 3e^{3y} \frac{dy}{dx} \cdot \cos(x) $ moving $\frac{dy}{dx}$ to the right hand ...
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0answers
18 views

Aggregating exponential growth rates

I'm working on a simple forecast model that uses Cumulative Annual Growth Rate (CAGR) to project future growth, and I've run into an apparent paradox. The model includes multiple lines of business ...
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6answers
156 views

Prove that $\lim_\limits{x\to 0}{\frac{e^x-1}{x}}=1$ without derivatives

Prove that $\lim_\limits{x\to 0}{\frac{e^x-1}{x}}=1$. I currently know only one approach (using L'Hopital 's Rule and derivatives) as follows: $$\lim_\limits{x\to ...
0
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2answers
67 views

The radioactive gas Radon-222 has a half life of $3.8$ days. Initially, there are $2 · 10^9$ atoms in a sample. How many atoms remain after $10$ days?

I already calculated the number of half lives past: $10/3.8= 2.6$ I don't know if I'm on the right path. I don't know what I'm doing. Please help or at least give me a hint.
0
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1answer
32 views

Sigmoid function increasing for large values of variable x

I am looking for a function that involves the sigmoid function but for large values of variable $x$ increases. Maybe sth like this, but there is a part missing: $f(x)= A +B\ \frac{1}{(1+e^{-x})}\ +\ ...
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3answers
68 views

Why does $n^{\ln {\ln n}} = ({\ln n})^{\ln n}$?

Note: this is (part of my solution to) a homework question. Please DO NOT tell me the answer! I am trying to compare the following functions: $$n^{\ln {\ln n}} \qquad\qquad ({\ln n})^{\ln n}$$ ...
1
vote
1answer
98 views

Understanding orthogonality of functions in the context of Fourier series

Show that $$\int_{x_0}^{x_0+L}\mathrm{e}^{-2\pi i r x/L}\mathrm{e}^{2\pi i p x/L}\mathrm{d}x =\begin{cases}L & \space \mathrm{for} \space r=p,\\0&\ \mathrm{for}\space r\ne p.\end{cases} ...