# Questions tagged [exponential-function]

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

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### 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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### 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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### 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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### 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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### 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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### 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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### $f(x) = x^2 - 6x + 4$. 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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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 $Y=\min(S,T)$. $\rightarrow Y\sim ... 2answers 45 views ### Derivative of vector functions [closed] i am struggling the derivation of the function on f). Does anyone know how to solve that? 2answers 84 views ### how would you prove that polynomial functions are not exponential? here is one proof that I know but I am not totally sure if it is acceptable- exponential functions are exponential: no matter how many times you differentiate them e.g- f(x)=e^x first derivative f`(... 0answers 27 views ### Exercise with density of$(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^{-...
The following excerpt is from Stewart Calculus: Early Transcendentals, 7th Edition. My question is how does the author arrive at: $$\lim_{h \to 0}\frac{a^h-1}{h}=f'(0)$$