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

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1answer
55 views

independent Exponential distribution P(X > Y + 1)

$X$ and $Y$ are independent exponentially distributed random variables with parameters $a$ and $b$. Calculate $P(X > Y + 1)$. I have let $X-Y=Z$ and Then $P(Z>z)=1-P(Z\leq z)$ $1 - P(X-Y\leq ...
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4answers
52 views

Exponent calculation

How to calculate the decimal powers of any number? (without using log ) Example: $$10^{0.3010} \approx 2$$ I have asked to my maths teacher and many such persons and no one knows the answer. The ...
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2answers
53 views

Simple equation $2^x = 16$ [closed]

Solve the following equation: $$2^x = 16$$ What is $x$? For $x = 4$, how do the $16$ and $2$ relate?
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1answer
86 views

Since $2^n = O(2^{n-1})$, does the transitivity of $O$ imply $2^n=O(1)$?

Let us assume that $f(n)=2^{n+1}$, $g(n)=2^n$ be two functions. Now, use limit to find $O(f(n))$: $\lim_{n\to\infty} \dfrac{2^{n+1}}{2^n}=2$. This is not equal to infinity, so the limit exists, hence ...
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0answers
32 views

Approximations for finite n in limit-based definition of the exponential function

The exponential function can be defined via: $$ e^x = \lim_{n \rightarrow \infty} \left( 1 + \frac{x}{n} \right)^{n} = \lim_{n \rightarrow \infty} g(x; n) $$ In my problem, I actually have the right ...
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1answer
79 views

Searching two matrix A and B, such that exp(A+B)=exp(A)exp(B) but AB is not equal to BA.

We know that if two matrix $A$ and $B$ commutes then $\exp(A+B)=\exp(A)\exp(B)$. I am trying to find two matrix that does not commute but $\exp(A+B)=\exp(A)\exp(B)$ is true for them. Can anybody give ...
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2answers
70 views

Identifying the exponential function $f(x)=e^x$ from its functional equation

Prove that if $f(x+y)=f(x)f(y)$ for all $x,y$ and $f(x)=1+xg(x)$ where $\lim_{x\to 0}g(x)=1$, then: a) $\exists f'(x)$ $\forall x$ b) $f(x)=e^x$ I would really appreciate your help.
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1answer
83 views

Is this a valid proof of $\lim _{n\rightarrow \infty }(1+\frac{z}{n})^n=e^z$?

Define the function $g_n(z)=(1+\frac{z}{n})^n$ for $\:n\in \mathbb{R^+}$. Then $\frac{d}{dz}g_n(z)=n(1+\frac{z}{n})^{n-1}\cdot\frac{1}{n}=(1+\frac{z}{n})^{n-1}$ Define $g_{\infty}(z)=\lim ...
2
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1answer
21 views

normal equations of $ y(t) = \gamma e^{\lambda t} $ for minimizing the error

Let $ y(t) = \gamma e^{\lambda t} $ and we have the points $(0,2)\ (1,0.7)\ (3, 0.3)$. The task is to get the parameter so that error is minimal. So we need to get the matrix for the normal ...
2
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3answers
38 views

inverse of quadratic log functions

Can a Log function with a quadratic have an inverse function? The specific question is to find the inverse of $$f(x) = \log_2(x^2-3x-4)$$ The function already fails the horizontal line test, but ...
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1answer
55 views

Attitude toward risk taking and the exponential utility function [closed]

I want to know some reference/book on the following topic: "Attitude toward risk taking and the exponential utility function". Thanks in advance.
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2answers
113 views

Integral of the exponential function

I am searching the indefinite integral of this function: $\dfrac{\exp(x)}{(1+x)^{5/3}}$. Thank you alot.
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3answers
65 views

$x^y < y^x$ for $y\ll x$?

Sorry if this is a naive question; I am not very good at mathematics. It seems obvious that for many $x$ and $y$, $x^y < y^x$ if $y \ll x$, e.g. $2^{10} > 10^2$. If $x$ and $y$ are very close ...
3
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3answers
287 views

Evaluation of the integral of $e^{-(x^2+y^2)}$ over a disk

Show that $$\renewcommand{\intd}{\,\mathrm{d}} \iint_{D(R)} e^{-(x^2+y^2)} \intd x \intd y = \pi \left(1 - e^{-R^2}\right)$$ where $D(R)$ is the disc of radius $R$ with center $(0,0).$ I ...
2
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4answers
41 views

Number of distinct real roots with $e^{-x}$ in the equation

How to find the number of distinct real roots of the equation $$13x^{13}-e^{-x}-1=0$$ I know that we generally find number of real roots by observing number of sign changes in $f(x)$ and $f(-x)$ but ...
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1answer
34 views

Formula to link two exponential values together - doesn't quite work

Basically, I've done a script, and I'm stuck on a formula for it. After I run the code on a cube, based on two different inputs (detail level and vertex average iterations), the resulting size will be ...
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0answers
47 views

Integral of an exponential of rational function

I have an integral of the form $\int_{a}^{b} \text{exp}\left(\frac{\lambda}{\rho^2 m + \sigma^2_u}\right) \frac{1}{m^2}\text{exp}\left(-\frac{\lambda}{m}\right) dm$. Can this integral be found ...
2
votes
2answers
60 views

Different Definitions Of The Sine Function

I was hoping someone could give me a flow chart or high-level map connecting all of the definitions of the sine function, with some of the reasons why we care next to each. I've tried this but I'm not ...
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4answers
84 views

Proving $\log(b^a) = a \log(b)$ using calculus

Sorry, this is a really simple question, but I'm trying to teach myself calculus and can't figure it out. If we define $\log(b) = \frac{db^x}{dx}(0)$ how does one prove $\log(b^a) = a\log(b)$? I ...
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2answers
41 views

help in finding number of solutions of the equation

I wanted to find the number of solutions of the equation: $$3^{(x-1)} + 5^{(x-1)} = 34$$ I can of course find one solution , but how to be sure that there is just one solution.
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1answer
31 views

Solve for coefficients of $y = A(1 - e^{-x/B})$ given two points

I have the equation $y = A(1 - e^{-x/B})$, and two $(x,y)$ pairs. How can I solve for $A$ and $B$? This should be simple, but I've been banging my head against the algebra for a while to no avail. I ...
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2answers
145 views

Exponential function to logarithmic function

i'm stuck on completing this equations. Is this correct? $$z=a e^{-bt}$$ $$\ln(z)=\ln(a)+\ln(e^{-bt})$$ $$\ln(z)=\ln(a)+(1)(-bt)$$ $$\ln(z)=\ln(a)-bt$$
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1answer
110 views

Inverted Circle?

The equation I have is $$\Large x^{\frac23} + y^{\frac23} = 3^{\frac23} $$ I know what the graph looks like, but I don't know how I would find points other than the intercepts mathematically. How ...
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1answer
23 views

Looking for a approximation/solution to my mortgage calculator function

I'm working on a little function, $t(A,y,r)$ that calculates the monthly payment of a fixed-rate mortgage, where $A$ is the amount borrowed, $y$ is the number of years over which the loan will be ...
1
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1answer
56 views

Rewriting a double integral with complex exponential function

Why can we write $$ \begin{align} I_T &= \int_\mathbb{R}\int_{-T}^{T}\frac{e^{-ita}-e^{-itb}}{it}e^{itx}dtdF(x)\\ &= \int_\mathbb{R}\left[\int_{-T}^{T}\frac{\sin(t(x-a))}{t}dt - ...
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0answers
92 views

Please give me an example of the algorithm where $\Theta$ will be equal to $e^n$

Please give me an example of the algorithm where $\Theta$ or $O$ will be equal exactly to $e^n$ . The algorithm should not be simple counting from 0 till $e^n$ . It should be a clear relation of two ...
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1answer
20 views

Graph exponential function

I am having problems understanding why $xe^x + 10e^x$ has two $(x,y)$ intercepts. I understand why there is one $(0,10)$, but am unclear on how to return $(-10,0)$. Any help would be much ...
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2answers
31 views

How Do I set up this problem? continuous compounding

I have no idea how to set up this problem. I am aware of the formula $$A = Pe^{rt}$$ Assume the cost of a gallon of milk is $2.90. With continuous compounding, find the time it would take the cost to ...
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3answers
56 views

Inverse Laplace Transformation of an exponential function

How one could find the inverse Laplace transformation of $\exp(-(b/(b+s))^k)$? Where both $b$ and $k$ are positive.
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3answers
66 views

Proving that $ 1-u = e^{-u - \,u^2/2 - \,u^3/3 -…}$

How can one see that for $-1 < u < 1$ we have the following equality $$ 1-u = e^{-u - \,u^2/2 - \,u^3/3 -...} \,\,\,\,?$$ It's probably easy to prove, however I've tried a couple of things so ...
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1answer
32 views

Normalizing a probability density function

I need to find a normalization term $N(\alpha,\beta)$ for the probability density function: $$PDF(\alpha,\beta)=(x-x_1)^{\alpha}e^{-\beta(x-x_1)}$$ In other words, solve the following equation: ...
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0answers
10 views

Poisson distributed graphs

I am currently reading a paper about poisson distributed graphs and came across the following formula. Apparently the degrees of the graph are distributed binomially through the following ...
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0answers
53 views

How do you solve this differential equation? $\tfrac{dx}{dz} = i (M x)$

How do you solve this differential equation : $\tfrac{dx}{ dz} = i (M x)$ where $M$ is a tridiagonal matrix with elements $100$. That is, $M$ is an array with $100$ elements in triagonal form, ...
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2answers
54 views

What do you get when you differentiate a $e^{f(x)}$-like function

I need help with exponential functions. I know that the derivative of $e^x$ is $e^x$, but wolfram alpha shows a different answer to my function below. If you, for example, take the derivative of ...
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0answers
37 views

Functional Equation involving derivatives and time-steps [duplicate]

I am attempting to solve the equation $$f(x + 1) = f'(x)$$ for distributions $C \rightarrow C: f(x)$ My first guess to exploit the fact that this seems similar to identity $$\sin\left( ...
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0answers
27 views

Parameterizing an implicit curve

I have to parameterize this curve: $$F(x,y)=y-x^2+x-e^{-yx^2}=0$$ But I don´t know how to do it. thanks
0
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1answer
39 views

Choosing a branch of the square root

Assume $O$ is the compliment of the non-positive part of the real line to the complex plane. This is an open and connected set. Only one of the values of $\sqrt z$ in $O$ has positive real part. With ...
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0answers
74 views

How to approximate large sum of exponential variables

Is there any way to approximate the following sum: $$ \sum_{i_1=1}^N \sum_{i_2=1}^N \cdots \sum_{i_k=1}^N \cdots \sum_{i_N=1}^N \exp(-r_{i_1} - r_{i_{k+1}} - r_{i_{2k+1}} - r_{i_{3k+1}} \cdots - ...
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2answers
28 views

evaluate exponential using Euler identity

let us consider following exponential $e^{-j*\pi*k/2}$ and $e^{j*\pi*k/2}$ we can decompose it as $cos(\pi*k/2)-j*sin(\pi*k/2)$ and second one same with plus sign ...
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0answers
18 views

Sufficient condition for a indefinite integral to be an elementary function

I would like to find a sufficient condition on two polynomials $P(s)$ and $Q(s)$, such that the function $s \mapsto Q(s)e^{P(s)} $ has a primitive integral of the form $s \mapsto R(s)e^{P(s)} $ (with ...
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1answer
28 views

Need help creating a special function

I'm creating a special function in a game and needed some help with the maths end of it. Essentially, I need a programmable, non-linear function so that $f(100) = 0$, and $f(0) = 100$ (or some other ...
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0answers
42 views

Why does the Riemann Xi function $(\xi(s))$ have order of growth 1

Why does $s(s-1)\xi(s)$, have order of growth 1? In other words, why is it that $\forall \epsilon > 0 $ $\exists A_{\epsilon},B_{\epsilon} \in \mathbb R_+$ so that $\forall s \in \mathbb C$, ...
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1answer
35 views

Don't understand answer from exponential growth question

"A biology student finds a large glass bottle which can be used to grow a bacterial culture. She has a bacterial culture that doubles in size every minute, and with the amount she currently has, she ...
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2answers
89 views

How to Show Polynomial Growth < Exponential Growth (Without L'Hopital!)

Can anyone offer me a way to show that exponential growth trumps polynomial growth, without using L'Hopital's Rule? When I learned function growth speeds in high school, the closest thing to a proof I ...
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2answers
37 views

Probability Random Variable question Need Help Please

You have a set of ten light bulbs - the lifetime of each of them being given by an exponential RV with mean 1000 hrs. Find the probability that.... (a) at least 7 of the bulbs function for 1500 or ...
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2answers
81 views

how to convert 1e+11 into number?

What will be 1e+11 in number? I know e2 means * 10^2 but i am confuse with this above question. what will be its value? I know how to use exponential function when ...
1
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1answer
55 views

How to find the parametric equation of $x^y=y^x$ without Lambert W function?

This is sort of a follow-up to my previous question. I've done basic conversions of parametric to to cartesian and back as part of my A-level, but never anything more advanced than a sin/cos ...
1
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1answer
47 views

$\mathrm{Ei}(x)$, the exponential function, some question.

I have a question involving with $\mathrm{Ei}(x)$, define as $\int_{-x}^{\infty}e^u \cdot u^{-1} \mathrm{d}u$. My question is, when I have a expression say $\exp(x) \cdot \mathrm{Ei}(x)+1$. I want ...
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votes
5answers
243 views

Why $e^x$ never equal $x$?

Je veux savoir pourquoi $x=e^x$ n'a aucune solution dans $\Bbb R$. Lorsque j'ai essayé de tracer le graphe de la fonction $e^x$, j'ai trouvé en fait qu'elle est une fonction strictement croissante ...
0
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1answer
11 views

Consumption change calculation

I want to calculate yearly consumption change according to the following formula: $$C_{t+1}=C_{t}e^{x_{t}}$$ I need to calculate ${x_{t}}$. I have the consumption data $C_{t+1}$ and $C_{t}$.