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

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1answer
15 views

finding sequence for e converging at some speed

I want to find an infinite sequence that conerges to e so that the kth term of the sequence is less than 10^-k away from e. Obviously, I've considered the Taylor series, but asymptotic bounds on the ...
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1answer
35 views

differentiate of an inverse function of mixed exponential and algebraic form

Let $f(x)= e^{2x} + x^5 + 1$ Find $(f^{-1})'(2)$ Find $(f^{-1})''(2)$ There is a missing link in my brain with regards to dealing with a function containing exponential and algebra. :/ I'm ...
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2answers
310 views

Increasing/Decreasing Test with Exponential Function

The goal is to find the intervals by which the function $f(x) = e^{x} - e^{2x}$ is increasing and decreasing, as well as any local maxima/minima, intervals of concavity, inflection points, asymptotes, ...
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2answers
33 views

Upper bound for product of exponents

From here we have the bound $$\left(1-\frac1N\right)^N\leq e^{-1}$$ where $N$ is a positive integer. Written another way, it is ...
5
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2answers
124 views

Upper bound for $(1-1/x)^x$

I remember the bound $$\left(1-\frac1x\right)^x\leq e^{-1}$$ but I can't recall under which condition it holds, or how to prove it. Does it hold for all $x>0$?
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4answers
127 views

Why is $\frac{d}{dx}\exp(x) = \exp(x)$?

What is the explanation for $$(e^x)'=e^x$$ I searched the SE, 'cause this can't be the first time this has been asked. But the question seems hard to formulate and search for here and on Google. Any ...
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1answer
34 views

Growth of exponential functions with different base and/or exponent

Assuming that the following is trivial: $\lim_{x \to +\infty} \frac{2^x}{2.1^x}=0$ $\lim_{x \to +\infty} \frac{2^x}{2^{x^2}}=0$ What is the most simple, intuitive way to show that: $\lim_{x \to ...
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0answers
92 views

Integral involving Modified Bessel function, exponential and power

I am trying to evaluate the following integral: $$ \int_{b y}^\infty \frac{e^{-x}(-1+I_0[2\sqrt{bx}]-\sqrt{bx}I_1[2\sqrt{bx}])}{x^2}dx $$ Even though there are a lot of integrals involving the ...
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1answer
49 views

You can write ${\left( {\frac{1}{2}} \right)^x}$ as ${2^{ - x}}$ , can the same be done with ${\left( {\frac{2}{3}} \right)^x}$?

You can write ${\left( {\frac{1}{2}} \right)^x}$ as ${2^{ - x}}$ as: ${\left( {\frac{1}{2}} \right)^x} = {({2^{ - 1}})^x} = {2^{ - x}}$ But what about ${\left( {\frac{2}{3}} \right)^x}$? Can it be ...
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0answers
65 views

Development of imaginary exponent without appealing to “ambiguity” between $i$ and $-i$

Is there a way to develop the definition of the imaginary exponent, $z^i$, for complex $z$, that does not appeal to the notion that $i$ and $-i$ are "qualitatively indistinct" and that does not rely ...
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0answers
34 views

How many number of ways are there for getting a special prime?

Definition of special prime : Any integer (+ve, -ve or 0) that is divisible by at least one of the single digit primes (2, 3, 5, 7) is a special prime. Thus -21, -30, 0, 5, 14 etc are special ...
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1answer
114 views

Error analysis of exponential function

By definition: $$ e^x = \lim_{n \rightarrow \infty} ( 1 + \frac{x}{n} ) ^ n$$ I am interesting in calculating the error $$\left | e^x - \left( 1 + \frac{x}{n} \right) ^ n \right|$$ for some fixed $n ...
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1answer
46 views

Calculate $10,000e^{-\int_2^{10}\left(0.05+0.01/(t+1)\right)\,dt}$

This equation is used as an example in a text book with a given answer of $\approx$ 6,617 I cannot get to this solution as somewhere along the way I must be making an error. If it is a problem with ...
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1answer
42 views

How to best solve a system of equations like this one

I need some help trying to solve a system of two equations with two unknowns. Background: This is not homework, just hobby. I have some LEDs that I needed to identify, but the manufacturer datasheet ...
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3answers
140 views

How to solve $e^{ax}+e^{bx}+e^{cx}+d=0$

How to solve an equation like $e^{ax}+e^{bx}+e^{cx}+d=0$ (i.e. to write $x=...$) where $a,b,c,d$ are fixed non-zero real numbers. I have tried assuming that $x=ln(y)$ for $y>0$ but it goes ...
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1answer
67 views
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1answer
247 views

Is a Distribution Exponential if its Mean equals its Standard Deviation

Can someone clarify if it is safe to declare that a distribution is not exponential if the mean and standard deviation are not equal, for example coefficient of variance, c < 1 and that it is ...
0
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1answer
51 views

exponential upper bound on sum of exponentials

For what value of $c_3$ can I guarantee that $(a+b)\exp(-c_3\theta)>a\exp(-c_1\theta)+b\exp(-c_2\theta)$ where $a,b,c_i>0$
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1answer
122 views

Integrating exponential of multiple exponentials

I have a integral term that looks similar to $\int_0^\infty\exp(-u-ae^{-c_1u}-be^{-c_2u})\,du$ where the constants $a,b,c_1,c_2>0$. For the case where $b=0$ I can use the answer from: Integrating ...
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2answers
120 views

What are the properties of the roots of the incomplete/finite exponential series?

Playing around with the incomplete/finite exponential series $$f_N(x) := \sum_{k=0}^N \frac{z^k}{k!} \stackrel{N\to\infty}\longrightarrow e^z$$ for some values on alpha (e.g. ...
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2answers
72 views

Constructing a differential equation for hyperbolic crochet

There is plenty of information about hyperbolic geometry and its melding with crochet, however I have yet to find an exact equation for determining the number of stitches in each row. I will try to ...
3
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1answer
78 views

Prove that $pe^{\alpha q} + qe^{- \alpha p} \le e ^ {\alpha^2/2}$

Prove that, $$pe^{\alpha q} + qe^{- \alpha p} \le e ^ {\alpha^2/2}$$ where $p$ and $q$ are the probabilities of success and failure in a Bernoulli trial ($0 \le p \le 1, 0 \le q \le 1, p + q = 1$) ...
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5answers
81 views

Solve for $x$ in $e^x$

I can't get this to work out right and can't find anything that gives me a rule for how to distribute the $0.5$ here. Any help would be much appreciated. \begin{align*} 0.5 &=\frac{\exp(-3.6 + ...
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0answers
229 views

Exponential integral approximation

I have an equation that contain exponential integral of the form: $$ \begin{equation} E_k\left(\frac{a+b ~x}{c}\right) \end{equation} $$ Where $k\geq 0$ ($k=0,1,2,...$), $a$, $b$, and $c$ are ...
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1answer
184 views

Help find the derivative of $e^{2^x}$ using the definition of the derivative

Let $f(x) = e^{2^x}$, where $e$ is the exponential function. So the $f'(x)$ is: $\begin{align}f'(x) &=& \lim_{h \to 0} \frac{e^{2^{x+h}}-e^{2^x}}{h}\\ &=& ...
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0answers
54 views

How to analyze the asymptotic properties of this function?

Let the function $$f(\mathbf{r})=\int_{\Omega }e^{i\mathbf{k} \cdot \mathbf{r}}d^2\mathbf{k}$$, where $\mathbf{k} ,\mathbf{r}\in\mathbb{R}^2$, and $\Omega \subset \mathbb{R}^2$ is some finite region ...
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0answers
34 views

College number theory problem - need a pointer! [duplicate]

$n$ divides $2^{2^n+1}+1$ $\implies n$ divides $2^{2^{2^n+1}+1}+1$? There are two ways to try to prove this. One is above, the other is its de Morgan counterpart: $n$ doesn't divide ...
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0answers
43 views

Practical explanation of exponential growth

I am presented with the following task: "There are 100 kids in a kindergarden. At one point, a disease breaks out. Let $y(t)$ be the number of infected children after a time $t$ days. In this ...
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2answers
38 views

How do I prove a “double limit”?

Prove $$\lim_{b \to \infty} \lim_{h \to 0} \frac{b^h - 1}{h} = \infty$$ I have never worked with double limits before so I have no idea how to approach the problem. Please don't use "$e$" in your ...
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1answer
87 views

Need your help with the integral $\int_0^\infty\frac{dx}{e^{\,e^{-x}} \cdot e^{\,e^{x}}}$.

Is it possible to evaluate this integral in a closed form? $$\int_0^\infty\frac{dx}{e^{\,e^{-x}} \cdot e^{\,e^{x}}}$$
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2answers
111 views

Show that $\lim_{x\to \infty}\left( 1-\frac{\lambda}{x} \right)^x = e^{-\lambda}$

Based on the definition of $e: = \lim_{x\to\infty} \left(1+\frac1x \right)^x$, how can we show that $$\lim_{x\to \infty}\left( 1-\frac{\lambda}{x} \right)^x = e^{-\lambda}?$$ So far I've tried ...
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2answers
34 views

Finding an equation for a growth formula

Given a tree that has three nodes each level I want to find the formula that predicts the number of all nodes with a given tree height. I fitted the data into Numbers with an exponential function ...
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1answer
31 views

Exponential equation and point on curve

I have two points: $$A(X_0,Y_0) $$ $$ B(X_1,Y_1)$$ And I need to find the function that creates an exponential growth between the two point. The fuction for exponential growth is $y = ab^x$ and ...
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2answers
94 views

Evaluating $\int _{-1}^{e} \frac{1}{x}dx$

Here very easily by the Fundamental Theorem of Calculus $$\int _{-1}^{e} \frac{1}{x}dx=\ln(e)-\ln(-1)$$ From Euler's identity $e^{i \pi}$=-1 we can easily deduce that $\ln(-1)=i \pi$. Thus the ...
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2answers
130 views

How do I solve this exponential function? $2^{-100x} = (0.5)^{x-4}$

How do I solve for $x$? $2^{-100x} = (0.5)^{x-4}$
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2answers
55 views

How do you take this limit algebraically (Not using the graphing calc)

$$\lim_{x\to0}{\frac{e^x-1}{x}}$$ I determined the limit by graphing this and seeing that the graph approaches 1 as x approaches 0. But, is there a way to algebraically determine this limit?
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2answers
82 views

Show that $\lim_{\delta \to 0}(1-\lambda \delta)^{1/\delta} = e^{-\lambda}$

My professor said that $$\lim_{\delta \to 0}(1-\lambda \delta)^{t/\delta}=e^{-\lambda t}$$ can be shown with L'Hospital's rule. I don't know what he meant. What is the best way to show this (or, ...
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0answers
27 views

Explanations of the Euler's continued fractions to compute exponential

After looking for explanations of the Euler's continued fractions to compute exponential on internet and after reading Euler's explanations about, I still don't understand how Euler find this ...
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3answers
60 views

Find the roots of 2 equations

Show that the equation $e^{-x} = x^2$ has a root between $x=0.70$ and $x=0.71$. I think you have to use natural logs to get rid of the $e$ however after that, i'm not sure how to solve for $x$
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1answer
49 views

Finding Taylor approximation for $x^4e^{-x^3}$

I'm trying to find Taylor approximation for the function: $$x^4e^{-x^3}$$ I started taking the first, second, third, etc. derivatives but the expression for it seems to explode with terms. I was just ...
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2answers
91 views

Accurate computation of $\exp(a x^2) Q(x)$ for big values of $x$?

I was wondering how one can accurately compute the value of $\exp(a x^2) Q(b x)$ for large values of $$x \left(Q(x) \triangleq \frac{1}{\sqrt{2\pi}}\int_x^{\infty} e^{-\frac{u^2}{2}} du \right)?.$$ ...
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1answer
588 views

where do exponential and logarithmic functions intersect?

If $0<a<1$, then the graphs of $y=a^x$ and $y=\log_a(x)$ intersect at some point $(t(a),t(a))$. Does this function $t(a)$ have any nice expression? How much do we know about this function, ...
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1answer
85 views

Integral of a function and a derivative with respect to the same variable

So, I have the following solution to a differential equation: $$ y(t) = \frac{k\int \! e^{-pt} \frac{\mathrm{d}x(t)}{\mathrm{d}t} \mathrm{d}t}{e^{-pt}} $$ But I am not sure if I can cancel out the $ ...
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2answers
184 views

Conjectural closed form for $\int_0^\infty\sqrt[3]z\ \operatorname{Ei}^2(-z)\,dz$

While trying to answer the question "A closed form for $\displaystyle\int_0^1\frac{\ln(-\ln x)\ \operatorname{li}^2x}{x}dx$", I came up with a conjecture: $$\int_0^\infty\sqrt[3]z\ ...
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1answer
159 views

Bound on the norm of a matrix exponential in Jordan Form

I'm looking to prove the following lemma: Let $A$ be a matrix in $\mathbb{R}^{n\times n}$. Then for any $\lambda^* > \max_{\lambda} \; \mathrm{Re} \; (\lambda)$ such that $ \lambda \in\sigma (A)$, ...
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0answers
52 views

How general is the convergence of the exponential function's power series?

Let $\mathbf{V}$ be a Fréchet space whose underlying set is $V$. Let $\;\; \beta \: : \: V\times V \: \to \: V \;\;$ be a continuous bilinear map that has an identity element and is ...
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1answer
27 views

Exponential growth/reduction of Laser Intensity question?

Hi i have a pretty simple question but I am not quite sure on how to solve/approach it. THe question: "The Intensity of a laserbeam declines with the penetrationdepth into matter exponentially. At ...
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1answer
26 views

How to understand the solution to an exponential variable equation?

$5(2^{n−1} + 5 ·3^{n−1}) − 6(2^{n−2} + 5 · 3^{n−2}) = 2^{n−2}[10 − 6] + 3^{n−2}[75 − 30] = 2^{n−2} · 4 + 3^{n−2} · 9 · 5 = 2^n + 3^n · 5 $ There are enormous leaps in my understand between each ...
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1answer
40 views

Calculate the Burning Time for a Lamp

If you have a lamp with burning time 4000 hours. If the time goes forward until the lamp will be destroyed the exponential distribution is 3675 hours, what is the probability of a lamp to be working ...
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0answers
520 views

How to solve polynomial-exponential equation

I'm trying to solve equations like the following one: $$5 + 3x - 4x^3 = e^{x^2}$$ I've tried using the Lambert W function, but I didn't get any success. I must admit I'm relatively new to Lambert W ...