Questions tagged [exponential-function]

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

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3answers
25 views

Taylor series at $a=1$

I have to find Taylor series at $a=1$ for $ f(x)=\begin{cases} \frac{e^{x}-e}{x-1},\quad &\text{if } x\ne1\\ e,\quad &\text{if } x=1\\ \end{cases} $ I ...
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1answer
42 views

An integral involving trigonometric and exponential function

Prove that $$ \int_{0}^{\infty}x^{2019}\sin(\sqrt{3}x)e^{-3x}=\dfrac{2019!\sqrt{3}}{2^{2021}\cdot 3^{1010}} $$ Hence generalize the integral for any other value than $2019$.I know it can be done by ...
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0answers
44 views

Solution to $\int_0^{\infty} \frac{1}{\sqrt{t^3(t+\tau)^3}}\exp\left[- \frac{2t+\tau}{t\left(t+\tau\right)}\right] \, dt$ [closed]

I am trying to solve the following integral: $$ \int_0^{\infty} \frac{1}{\sqrt{t^3\left(t+\tau\right)^3}}\exp\left[- \frac{2t+\tau}{t\left(t+\tau\right)}\right] \, dt $$ Is there an analytical ...
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1answer
16 views

On the existence of a holomorphic function [duplicate]

I have just encountered this exercise which has me stumped: We are asked to prove that if $ f $ is holomorphic on the unit disk $ D $ and if $ f(z) \neq 0 $ on $ D $, that there is a holomorphic ...
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0answers
27 views

optimize sum of log sigmoid with sum of sigmoid constraint

I have a non-convex constraint optimization problem to find its approximate solutions: First define a parameterized function $g(s_i) = g(s_i;\mathbf{x})= \operatorname{sigmoid}(x_0 + x_1 \cdot s_i)$, ...
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1answer
83 views

Why can $e^x$ be defined as the unique function $f(x)$ such that $f(x)=f'(x)$ and $f(0)=1$?

The definition that $e^x$ is the unique function $f(x)$ such that $f(x)=f'(x)$ and $f(0)=1$ has two problems for me: How is $e^x$ the unique function that satisfies this property? $ke^x$ also has ...
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3answers
31 views

$f(x) = x^2 - 6x + 4$. What is the maximum value of $\frac{1}{f(x)}$?

The solution to this in my book says $\frac{1}{5}$. But greater values can be achieved by using $f(x)$ values approaching zero, right ?
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1answer
56 views

Help isolating “t” in this equation: $1.1 ^ t + 1.2 ^ t + 1.5 ^ t = 1,000,000,000$

My math skills are a bit rusty :( I'd like to isolate the exponent "t" in this equation. $$ 1.1 ^ t + 1.2 ^ t + 1.5 ^ t = 1,000,000,000 $$ So if I apply log on both sides, I'd have this: $$ \log(1....
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0answers
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Request for help inverting difficult function (or avoiding inverting said function)

[Edit - input values are expected to be constrained so that no complex intermediate nor final results are involved] First time poster. So I have been working at this for a few weeks and am now ...
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Mathematical analysis question [closed]

Could someone help solve this problem please Solve $f(x,p) < g(y,p)$ where $$ \begin{split} f(x,p) &= 2 \exp\left(-\frac{xp}{2}\right)\\ g(y,p) &=\left[\frac{yp +2}{2}\right] \cdot \exp(...
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1answer
35 views

How to argue $\sum_{j=1}^\infty\sum_{k=j}^\infty\frac{1}{k!}=e$? [duplicate]

In an answer to an unrelated question, the following result was used without any argument. $$\sum_{j=1}^\infty\sum_{k=j}^\infty\frac{1}{k!}=\sum_{n=1}^\infty\frac{n}{n!}=e$$ Its true (checked ...
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2answers
41 views

Show that this function is decreasing

I have to show that the function $f(x)=(\ln x)^2(1-\ e^{-\frac{t}{x}}), t>3$ is decreasing on $[\max(e^4,2t),\infty[$ and deduce that $f(x)\leq \max(16,(\log2t)^2)$, $x\geq 1$. The exercise ...
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0answers
7 views

Understanding an exponential backoff equation and why I don't see the results I expect

I want to emulate the results of http://exponentialbackoffcalculator.com/. However, if my interval is 5, my rate is 3, and my "attempt" is 2 and I use their proposed equation of ...
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1answer
34 views

Modeling the problem

I'm having a problem while solving this problem, when i implement the solution it happens a loop and it doesn't go to the answer Consider a model for the long-term eating behavior of students at a ...
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1answer
42 views

How to prove inequality $(2n-1)x^{-1} + (2n+1)x^{2n-1} > \frac{(2n+1)^2}{2n}, \quad n > 2$ and $ 0 < x < 1$

How to prove the following inequality for $n > 2$ and $ 0 < x < 1$: $$(2n-1)x^{-1} + (2n+1)x^{2n-1} > \frac{(2n+1)^2}{2n}, \quad n \in \mathbb{N}, x \in \mathbb{R}$$ I tried using ...
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0answers
36 views

Prove that $\exp(x+y) = \exp(x)\exp(y)$. [duplicate]

For every real number $x$, we define the exponential function $\exp(x)$ to be the real number \begin{align*} \exp(x) = \sum_{n=0}^{\infty}\frac{x^{n}}{n!} \end{align*} Prove that $\exp(x+y) = \exp(x)\...
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1answer
36 views

Explicit solutions to $x^y=y^x$ using Lambert $W$

A Flammable Maths video gives the solutions to the title equation by $y=-\frac{x}{\ln x}W(-\frac{\ln x}{x})$. This makes a lot of sense, given that Wikipedia gives $W_0(-\frac{\ln x}{x})=-\ln(x)$ for $...
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1answer
57 views

Searching for real solutions of $2^x = \frac{3}{4}x + \frac{5}{4}$

I'm looking for real solutions to the following equation: $$2^x = \frac{3}{4}x + \frac{5}{4}$$ By plotting the functions I've found that the solutions should be $\pm 1$, however I this approach is not ...
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1answer
15 views

Exponential decay of temperature of tea

A cup of tea is poured at 98 degrees. After two minutes it has reached 94 degrees. The difference between the temperature of the tea and the room temperature (22 degrees) falls exponentially. Find the ...
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2answers
63 views

Integration that looks like fourier transform [closed]

There is a second order integral which looks like a fourier transform as shown below. $$ \int_{-a/2}^{a/2} \int_{-a}^{a} \left(1-\frac{|x|}{a}\right) e^{jkx \sin\theta \cos\phi} e^{jky \sin\...
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0answers
38 views

Product representation of the exponential series.

Let ${a_n} = \prod\limits_{j = 1}^n {\gcd (j,n)} .$ Comparing OEIS A067911 and A170911 suggests that there are integers $b_n$ such that the $n-$th partial sum of the product $\prod\limits_{k = 0}^\...
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0answers
53 views

Can anyone help me to understand how this upper and lower bound for the binomial coefficient are derived?

I found this very interesting upper and lower bound for the binomial coefficient that is taken from a book that I have not heard of. Could someone help me understand how it is derived? Here is the ...
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5answers
92 views

${\log}_{a}{x}\neq {\int}^{x}_{1}{\frac{1}{t}}dt$

In most calculus textbooks, $\ln{x}$ is defined to be ${\int}^{x}_{1}{\frac{1}{t}}dt$. Some textbooks validate this definition by demonstrating that this function $\int^{x}_{1}{\frac{1}{t}}dt$ has all ...
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4answers
48 views

Limits to infinity of exponential functions

I am completly blocked trying to prove the solution of the limits below 1) $$ \lim _{m\to\infty}\left(\cos\left(\frac{x}{m}\right)\right)^m\\1 \quad \text{ for } a\to +\infty;\quad 0 \quad \text{ ...
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0answers
14 views

Approximating integrals with a sharply peaked integrand

I am working through a textbook on laser trapping and cooling (by Metcalf and van der Straten), but I have purely mathematical question. During a derivation they arrive at the following differential ...
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1answer
33 views

Doubt on conditional expected value

Let $S$ and $T$ two random indipendent variables with exponential distribution, and let $\mathbb{E}(S)=\alpha,\mathbb{E}(T)=\beta$ . 1) Find the distribution of $Y=\min(S,T)$. $\rightarrow Y\sim ...
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2answers
45 views

Derivative of vector functions [closed]

i am struggling the derivation of the function on f). Does anyone know how to solve that?
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2answers
84 views

how would you prove that polynomial functions are not exponential?

here is one proof that I know but I am not totally sure if it is acceptable- exponential functions are exponential: no matter how many times you differentiate them e.g- f(x)=e^x first derivative f`(...
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0answers
27 views

Exercise with density of $(X,Y)$ [closed]

Let $(X,Y)$ a random variable with density $f(x,y)=ke^{-(2x+3y)}$, $x>0,y>0$. a) Find the value of $k$. $\rightarrow k=6$ b) Find the CDF of $(X,Y)$. $\rightarrow F(x,y)=(1-e^{-2x})(1-e^{-...
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1answer
28 views

Understanding the proof of derivative of exponential function.

The following excerpt is from Stewart Calculus: Early Transcendentals, 7th Edition. My question is how does the author arrive at: $$\lim_{h \to 0}\frac{a^h-1}{h}=f'(0)$$
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1answer
35 views

Integrating the product of an exponential and a derivative

I have the following problem that I'm unsure how to tackle: $\frac{dm}{dt} = \frac{dn}{dt} - \lambda m$ I tried using the integrating factor method with IF = $e^{\lambda t}$ so I end up with: $me^{\...
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0answers
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Need and equation to calculate mortality based on concentration

I've gone around many circles with this. I suspect the solution will be something involving exponential decay but not sure how to fit it. I have a mechanistic microsimulation model of mosquito ...
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0answers
22 views

How to derive this inequality for the integral of $exp(x)^2$?

Im looking at the solution of some exercise problem. It uses the following inequality $$\frac{1}{\sigma} \int_0^{\theta-\theta_0} \left(\exp(x/\sigma) \cdot A - \frac{1}{2\sigma}\right)^2 \,dx \leq \...
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2answers
53 views

Solve nonlinear equation

How to solve nonlinear equation: $$x+2.1*\frac{100}{1+e^{(10-q(x))/3}}-2=0,\\ here \quad q(x)=\frac{100}{1+e^{(10-x)/3}}$$ Are here any numerical method suitable to solve or any package? I tried ...
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2answers
28 views

Convergence of a series involving $e^{in\phi}$

Consider the following series: $\displaystyle\sum_{n=1}^{\infty} \frac{(n+1)e^{in\phi}}{n^2}, \phi \in \mathbb{R}, \phi ≠ 2\pi k$ for $k \in \mathbb{Z}$ Then: $\displaystyle\sum_{n=1}^{\infty} \...
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0answers
25 views

Alternative Laurent expansion of $1-e^{-x}$

I wonder if there is Laurent expansion of $f(x) = 1 - e^{-x}$ where $x>0$, is real number. It is known that the expansion : $$ g(x) = 1- e^{-x} = x - \frac{x^2}{2!} + \frac{x^3}{3!} - ... $$ for $ ...
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0answers
31 views

Why I am getting similar $\beta$ for minimization $\sum_i (\log(y_i)-X_i\beta)^2 $ and $\sum_i (y_iX_i\beta-e^{X_i\beta})$?

I asked a question here and learned that minimizing $\Vert \! \log(y)-X\beta \Vert_2^2$ and $\Vert y-e^{X\beta} \Vert_2^2$ are different. But I still have difficult times to understand why ...
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0answers
10 views

Population recovery in an exponential process

Suppose that we have a population $P_{n}$ that grows exponentially with a growth rate $g$, so $P_{n+1} = P_n (1+g)$. We consider $P'_n$, a counterfactual population that grows as $P_n$ and satisfies ...
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1answer
42 views

Is minimizing $\Vert \! \log(y)-X\beta \Vert_2^2$ and $\Vert y-e^{X\beta} \Vert_2^2$ the same?

I am trying to fit some exponential data ($y$ is the regression target vector, and $X$ is the data matrix, $\beta$ is the coefficients that we want to optimize). However I am getting different ...
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1answer
19 views

Sigmoid Function

Typically sigmoid function is calculated as 1/(1 + exp(-x)) I see sometimes it is calculated as 1 - 1/(1 + exp(x)) ...
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0answers
10 views

Best way to do exponential regression with ranged x values

Given the following data What is the best way to exponential regression to function $f(x) = Ce^{kx}$. I thought of replacing the ranges with the value in the middle so that for example $0 \leq x <...
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0answers
32 views

Possible analytic evaluation of integral

I'm wondering if the following integral can be done analytically? $$I(x) = \int_{x/4}^1 \text{d}y\,y^{3/2+a} \,\,(1-y)^{1/2}\,\, \exp \left(c\sqrt{\ln \left(\frac{4y}{x} \right)} \right),$$ where $a$ ...
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0answers
26 views

How to model a quasi-exponential function?

This is the graphic of official daily deaths caused by Covid-19 in Brazil. And this is the same graphic with an exponential model in red. We see the model don't fit the data. The misadjustment is ...
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4answers
65 views

Find all numbers $a$ for which the equation $a3^x+3^{-x}=3$ has a unique solution

Find all numbers $a$ for which the equation $a3^x+3^{-x}=3$ has a unique solution $x$. How should I approach this? I don't really see any good way to start the problem, I've seen a similar problem ...
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2answers
38 views

I resulted a contradiction solving an inequality

The problem is "Find the range of $k$ when $x<1$ for the following inequality" $$9k-6 \leq (3k-1)3^x$$ . Isn't this equation always true because $9k-6 \leq (3k-1)3^x < (3k-1)3 = 9k-3$ $9k-...
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2answers
35 views

exponential function and inequality

I have a hard time solving this one. I'm sure there is trick that should be used but if so, I can't spot it. $$(3\cdot4^{-x+2}-48)\cdot(2^x-16)\leqslant0$$ Here is what I get but I'm anything but ...
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2answers
32 views

Rational or Irrational?

Here is my question: can an Irrational number; like $e$, be equal to a second irrational number, $\pi$, times an integer, then divided by a second rational number? Such as: $e = \frac{\pi a }{b} $ , ...
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2answers
40 views

What is the $\lim_{z\rightarrow{\infty}}e^{-(z+i)^2}$?

I'm struggling to find the following $complex$ limit (or show that it DNE): $\lim_{z\rightarrow{\infty}}e^{-(z+i)^2}$. I know that we are actually looking at $\lim_{|z|\rightarrow{\infty}}e^{-(z+i)^2}...
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2answers
53 views

Definite integral of exponentials and trig functions

Wikipedia has $$ \int_{-\infty}^{\infty} e^{-ax^2} dx = \sqrt{\frac{\pi}{a}} $$ and $$ \int_{-\infty}^{\infty} e^{-ax^2} e^{-2bx} dx= \sqrt{\frac{\pi}{a}} e^{\frac{b^2}{a}} $$ https://en....
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3answers
80 views

Does it follow that for $ x\ge 785, (2x+3)^{\frac{1.25506}{\ln x}} < 4$

Does it follow that for $ x\ge 785$, $$(2x+3)^{\frac{1.25506}{\ln x}} < 4$$ Note: $1.25506$ is taken from this inequality regarding the prime counting function $\pi(x)$: Here's my thinking: (...

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