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

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3
votes
3answers
50 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 ...
1
vote
4answers
32 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 ...
0
votes
1answer
33 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 ...
0
votes
0answers
33 views

Integral of an exponential of rational function

I have an integral of the form $\int_{a}^{b} \text{exp}\left(\frac{\lambda}{\rho^2 m + \sigma^2_u}\right) \frac{1}{m^2}\text{exp}\left(-\frac{\lambda}{m}\right)$. Can this be found analytically?
2
votes
2answers
48 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 ...
0
votes
4answers
79 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 ...
0
votes
2answers
37 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.
0
votes
1answer
30 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 ...
3
votes
2answers
139 views

Exponential function to logarithmic function

i'm stuck on completing this equations. Is this correct? $$z=a e^{-bt}$$ $$\ln(z)=\ln(a)+\ln(e^{-bt})$$ $$\ln(z)=\ln(a)+(1)(-bt)$$ $$\ln(z)=\ln(a)-bt$$
1
vote
1answer
105 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 ...
1
vote
1answer
17 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 ...
-4
votes
1answer
32 views

Find the percent increase or decrease

For each function, find the percent increase or decrease that the function models $y=1298 \cdot 1.63^x$ $f(x)=2 \cdot 0.65^x$
1
vote
1answer
50 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 - ...
0
votes
0answers
87 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 ...
0
votes
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 ...
1
vote
2answers
28 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 ...
1
vote
3answers
50 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.
1
vote
3answers
61 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 ...
0
votes
1answer
28 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: ...
0
votes
0answers
7 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 ...
0
votes
0answers
51 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, ...
2
votes
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 ...
2
votes
0answers
36 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( ...
2
votes
0answers
26 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
votes
1answer
38 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 ...
1
vote
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 - ...
1
vote
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 ...
0
votes
0answers
17 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 ...
0
votes
1answer
27 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 ...
2
votes
0answers
39 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 $\forall s \in \mathbb C$, ...
-1
votes
1answer
32 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 ...
1
vote
2answers
88 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 ...
0
votes
2answers
35 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 ...
0
votes
2answers
79 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
vote
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
vote
1answer
45 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 ...
4
votes
5answers
230 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
votes
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 $C_{t}$.
2
votes
1answer
71 views

Checking derivation of y = a^x

Can you tell me if there are any flaws with this derivation of $y = a^x$... The assumptions are that the derivative $$\frac{d}{dx}e^x = e^x$$ and that the derivative $$\frac{d}{dx}\ln x = ...
0
votes
1answer
43 views

Something that grows faster than NP class of problem does

I have a theoretical question. F.ex. we have a NP-class of problem, i.e. which do need exponential time on deterministic Turing machine. Is there anything that is growing faster than exponent does. ...
1
vote
0answers
27 views

Can this be solved analytically?

I have a sum of two Gaussian type functions, $g_1(x) = C_1 Exp[-\alpha (X_1-x)^2]$ and $g_2(x) = C_2 Exp[-\beta (X_2-x)^2]$ and have found that the derivative w.r.t. $x$ is $f(x) = 2 C_1 (X_1 - x) ...
19
votes
2answers
308 views

How to prove $ \lim_{n \to \infty} e^n \cdot \left( \sum_{k=0}^{n-1} ({k-n \over e})^k/k! \right)- 2 \cdot n = \frac 23$?

I observed for the function $$ f(n)= e^n \sum_{k=0}^{n-1}\left(\dfrac{k - n}{e}\right)^k \cdot \dfrac{1}{k!} \tag 1$$ with small $n$ that ...
1
vote
2answers
79 views

Contour integral $\int^\pi_{-\pi}(a-\cos\theta)^b\exp(c\cos\theta)d\theta$ assuming $a>1$, $b>0$, $c>0$

Under the condition $a>1$, $b>0$, $c>0$, is there any good function to express the following integral? $$ \int^\pi_{-\pi}\left(a-\cos\theta\right)^b\exp\left(c\cos\theta\right)d\theta $$ I ...
6
votes
1answer
170 views

$\exp(\ln(x))=x$ and $\ln(\exp(y))=y$.

Let $(A,1_A,|\cdot{}|)$ be a unital Banach algebra, for instance $A=M_n(\Bbb R)$ or $M_n(\Bbb C)$. What is the union of all open unit balls $B_{\|\cdot{}\|}$ where $\|\cdot{}\|$ ranges over all ...
0
votes
1answer
20 views

Question about calculating exponent of polynomial

$V=R_{3}[X] $ and $T:V->V$ is a linear transformation : $T(p(x)) = p(x) + xp'(x)$ I need to find $e^{T(1+x+x^{2}-x^{3})}$ I don't understand how to do it? what does it mean to calculate exponent ...
1
vote
1answer
71 views

How can I solve this equation: $e^{2x^3 - 6x^2 + 3} = 0$ [closed]

I don't remember what I supposed to do in this situation...I know that it's necessary transform both sides of the equation in the same base. However, what I need to do when i have a 0? My equation: ...
5
votes
6answers
88 views

Need explanation for simple differential equation

I can't figure out this really simple linear equation: $$x'=x$$ I know that the result should be an exponential function with $t$ in the exponent, but I can't really say why. I tried integrating ...
0
votes
0answers
16 views

Exponential Integral Function representation by sum

I have an expression for the Exponential Integral Function as the followings: $$ E_{L+1}(x) $$ where L is a positive integer larger than zero; and x is real number larger than zero. Now I have this ...
3
votes
0answers
54 views

Roots of derivative of q-expontial function

Let the q-deformation of the exponential function be defined by $$ e_q(z)=\sum_{n=0}^\infty{\frac{z^n}{[n]_q!}}. $$ Eq. (1.8) of this paper provides the product representation $$ ...
0
votes
1answer
14 views

find exponential line going through 3 points

I have 3 points:(0,0);(55,64);(137,200) How could i get the formula going through those 3 points? They line up in an exponential line like this one: