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

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1answer
45 views

I dont understand how this Maclaurin series got manipulated into looking like this other Maclaurin series. Help Please

I have been reading a book on approximating $e$ and there is a couple lines that I am stuck on. Here they are: $x$ln$(1+\displaystyle\frac{1}{x}) = 1 - \displaystyle\frac{1}{2x} + ...
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1answer
24 views

Infinite summation of exponential $\sum_{n\in\mathbb{N}}e^{-n^k}$

For interger $k\geq 2$ is it possible to compute the sum and get an expression in terms of $k$? $\sum_{n\in\mathbb{N}}e^{-n^k}$
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1answer
23 views

Solution of $A = e^{\alpha t}\cos(\omega t + \phi)$

I would like to find the real roots of the function $$i(t) = \frac{\hat{V}}{R}\left(\frac{\omega^2}{(\alpha^2 + \omega^2)} \cos\left(\omega t + \tan^{-1}\left(\frac{\alpha}{\omega}\right)\right) + ...
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1answer
15 views

Exponential random variable with mean 1/gamma

If $X$ is an exponential random variable with mean $\frac{1}{γ}$, show that $\mathbb{E}[X^k]=\frac{k!}{γ^k},\,\, k=1,2,3,\cdots$ *Use the gamma density function $\mathbb{E}[X^k]=∫x^{k}γe^{-γx}dx$ ...
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1answer
34 views

Big-Oh of exponent of exponent

How does one whether an exponent of an exponent is the big-Oh of the other? For example, if I have $a^{b^n}$ and $b^{a^n}$, how would i determine and prove which is a big oh of another? I'm thinking ...
3
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1answer
93 views

When is there a vector $D$ with positive coordinates such that $e^{Ct}D$ has a negative coordinate?

Let $C$ be a $2 \times 2$ asymmetric matrix with real entries. Assume that $C$ has strictly negative, real eigenvalues. Fix $D\in\mathbb{R}^2$, where $D > 0$ (i.e., both coordinates are strictly ...
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1answer
23 views

Log arithmic Equation - Graph curved line

I'm recreating the graph picture below with equations. Using the online graphing tool "Desmos": These are all the equations I have done so far, with there restrictions top stop at specific points. ...
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1answer
27 views

Making a matrix invertible

Given $N$ distinct real numbers $x_1,\ldots, x_N$, how can I show that there exist real numbers $a_1,\ldots, a_N$ so that the following matrix is invertible? $$\begin{bmatrix} \exp(ia_1 x_1) & ...
2
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2answers
15 views

Roots of $i(t) = Ae^{\alpha t}cos(\omega t + \phi)$

I would like to find the roots of the function $i(t) = Ae^{\alpha t}\cos(\omega t + \phi)$ in the form $t = f(A, \alpha, \omega, \phi)$.
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1answer
261 views
0
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1answer
45 views

Graphing picture equations - Curve Lines

I'm basically trying to recreate the graph picture below. Using a online graphing tool "Desmos": I managed to create the equations for the straight lines and circles for the sunset picture. ...
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2answers
27 views

Exponential growth application

Corrosion is attacking the inside of a water tank. Today a 2cm x 2cm size patch is measured. We know the corrosion will grow at rate of doubling size every 5 days. What will its size be in sq/cms be ...
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4answers
92 views

Proof of inequality $\frac{2-a}{2+a}<e^{-a}$

How can I prove that $$\frac{2-a}{2+a}<e^{-a}$$ for all $a \geq 0$ ? For $a \geq 2$ it is clear, but how can it be shown for $0<a<2$ ?
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3answers
53 views

Solve $(e^{x+1} -2)(e^{2x} -4) = 0$ … but there is a problem!

I am a little bit confused. There is this problem: $$(e^{x+1} -2) (e^{2x} -4) = 0$$ I thought, i could just solve it like this $(a - b)(c - d) = 0 \therefore ac -ad -bc + bd = 0$ After few ...
2
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1answer
23 views

Evaluating $|a^b|$ when $a,b$ are complex

Here, $a^b=e^{b\log a}$ for some suitable (but fixed in advance) branch of the $\log$ function. What is the most general formula for $|a^b|$ when both $a$ and $b$ are complex, and what are the ...
1
vote
3answers
123 views

Definition of hyperbolic cosine and its relation with exponential function

Ordinary trigonometric functions are defined independently of exponential function, and then shown to be related to it by Euler's formula. Can one define hyperbolic cosine so that the formula ...
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2answers
31 views

Algebraic issues with the calculation of the second derivative of $(a+be^x)/(ae^x+b)$

I'm trying to work out the 2nd derivative of $\dfrac{a+be^x}{ae^x+b}$ I have $f''=\dfrac{(ae^x+b)^2(b^2-a^2)e^x-2ae^x(ae^x+b)(b^2-a^2)e^x}{(ae^x+b)^4}$ There are so many terms, and I'm seriously ...
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2answers
842 views

Integral of matrix exponential

Let us be given a square $n \times n$ matrix $A$. For a system \begin{align*} \dot{x}(t) = A x(t), \hspace{0.3 cm} x(0) = x_0 \end{align*} the solution is given by $x(t) = e^{At} x_0$. I am ...
7
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2answers
95 views

Why is this function a really good asymptotic for $\exp(x)\sqrt{x}$

$$f(x)=\sum_{n=0}^{\infty} a_n x^n\;\;\;\;\; a_n = \frac{1}{\Gamma(n+0.5)}$$ Why is this entire function a really good asymptotic for $\exp(x)\sqrt{x}$, where for large positive numbers, ...
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1answer
48 views

Using complex analysis to convert $b\cos \theta +a \sin \theta$ to a single trigonometric function

Using product $(a+bi)(\cos \theta+i \sin \theta) $ show that $$b\cos \theta +a \sin \theta=\sqrt{a^2 + b^2}\sin(\theta+\arctan(b/a))$$ and using this result show by induction that $$ ...
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0answers
15 views

MAP for exponential function (Maximum a posteriori)

I am trying to find the MAP for an exponential function of the form $p(y) = \theta.e^{{-\theta}y}$ Given that $\theta$ is constant, I want to estimate maximum $y$ = $p(y).p(X=x_i|y)$ for $i = 1..n$. ...
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2answers
103 views

Integration of $x^3 e^x$

I am beginner in calculus and I am struggling with this integral: $$\int x^{3}e^{x}\mathrm{d}x$$ If anyone could give me some hints, any help will be appreciated.
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0answers
29 views

Fourier transform of $e^{if(x)}$

I'm trying to find an explicit result for the following Fourier transform: $$\mathcal{F}\left[e^{if(x)}\right](k)=\int_{\mathbb{R}^n} e^{if(x)}e^{-ik\cdot x} dx$$ So far I could come up only with a ...
0
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1answer
25 views

Determining if a function decreases exponentially

Define a function: $f(x) = \sqrt{\frac{e^{-kx}}{1-e^{-kx}}}$ where $k > 0$. Does this function decrease exponentially? EDIT: Sorry, I meant to ask just if it decreases exponentially.
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1answer
3k views

Compound interest formula and continuously compounded interest formula derivation

My textbook gives the formula for compound interest as: $A\left( t\right) =P\left( 1+\dfrac {r}{n}\right) ^{nt}$ Where: P = The principal, r=the annual rate of interest, n= the frequency of ...
0
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1answer
95 views

Integration with exponential

$$\int y\,e^{x^2}\,dy$$ I begin with $$\int e^{x^2}y\, dy$$ let $u=e^x$, $du=e^x\, dx$ how do I continue?
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0answers
24 views

Find the partial derivative with respect to y of the function $f(x,y)=ye^{xy}$

My solution was $e^{xy} + xy e^{xy}$, but when I checked the solution manual it said the answer is $xy e^{xy} \log e + e^{xy}$. So I solved each function for $y$ by setting them each equal to $0$. ...
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3answers
214 views

Proving that a definition of e is unique

We can define $e$ as the number such that $\lim_{h \to 0} \frac{e^h-1}{h}=1$. However, of course we can only define $e$ this way if it is unique, i.e., there is no other value $c$ for which that is a ...
0
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1answer
20 views

How to find the inverse of a function involving e with a coefficient?

I was wondering how I would find the inverse of the following function, since the e has a co-efficient: $\frac{e^x}{1+2e^x}=y$ I got as far as $\ln y+\ln(2e^x) = \ln e^x$, which would be changed ...
1
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1answer
42 views

Expected values with exponentials

I've been stuck on this question for a while and it's annoying the hell out of me! I know it's a basic definition type of question, but I can't seem to understand it. Can any of you help? Question: ...
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3answers
64 views

$(x+y)^c\le x^c+y^c$ for $0<c\le1$ [duplicate]

The statement I'm trying to prove is: $(x+y)^c\le x^c+y^c$ whenever $0\le x,y$ and $0\le c\le1$. This comes up in the proof that $|x|_*^c$ is an absolute value whenever $0<c\le1$ and $|x|_*$ ...
2
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3answers
183 views

Simplifying expression and finding indefinite integral

(a) Simplify $$\Large \frac{e^{-4x} + 3e^{-2x}}{e^{-4x}-9} \quad.$$ (b) Hence find $$\Large \int\frac{e^{-4x} + 3e^{-2x}}{e^{-4x}-9} \mathrm{d}x$$ I tried to find a breakdown of the expression, but ...
2
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3answers
47 views

Roots of Unity - $x^3 = -i$

I need to find the roots of unity for the following equation: $$x^3 + i = 0$$ Thus, $x^3 = -i$. I know that $-i = \exp[i(\frac{3\pi}{2} + 2n \pi)]$ however I do not know how to get all roots. ...
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1answer
17 views

Order of growth in uniform distribution

Consider an i.i.d. sample $\{X_1, \ldots , X_n\}$ from the uniform distribution on $[ 0,\theta]$ and the estimator $$M_n = \max\{X_1,X_2,\ldots,X_n\} $$ What does the above statement mean? I ...
0
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2answers
304 views

Find Log equation from data points

I have the following data points, (left hand column goes from 0-127, right hand column goes from 30-22000 hz. Is there any calculator I can use to find a "log" function of this data, so that it comes ...
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3answers
24 views
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0answers
13 views

Exponential and Logarithmic Differentiation.

Q. If $xe^{xy}=y+sin^2x$, then find $\frac{dy}{dx}$ at x=0. If we differentiate the function directly as follows: $e^{xy}+xe^{xy}\left[y+x\frac{dy}{dx}\right]=\frac{dy}{dx}+sin\left(2x\right)$ At ...
2
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1answer
313 views

Exponential Function Shifts

I have some confusion about shifts concerning exponential functions. I can best describe my question with an example. Take y = e^-(x-3). This graph has a reflection over the y-axis and is shifted ...
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1answer
178 views

Finding every triplet $(n,a,b)$ such that $n!=2^a-2^b$

Question : Let $n,a,b$ be positive integers. Are there infinitely many triplets $(n,a,b)$ which satisfy the following equality?$$n!=2^a-2^b$$ If Yes, then how can we prove that? If No, then how ...
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3answers
32 views

Show an exponential function has a valid density.

Given: Let $X$ be exponential with parameter $\lambda$, that is $$ f_X(x) = \begin{cases} \lambda e^{-\lambda x} & \text{if }x> 0, \\ 0 &\text{for }x\leq 0. \end{cases} $$ where ...
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1answer
32 views

Why does this proof make sense and theorems required

I saw the proofs on the derivative of $\frac{d e^x}{dx}=e^x$ from here and the one that was intriguing was this : $$e:=\lim_{n\to\infty}\left(1+\frac{x}{n}\right)^n \implies \frac{d(e^x)}{dx} = ...
3
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1answer
55 views

Details from a Proof that a Tournament has Property $S_k$

(Edit: While the context is not central to my question, I decided to include it anyway to make the question a little more searchable.) Some technical details are omitted from a theorem in Alon and ...
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0answers
29 views

How does this exponential equation add up?

Here is the equation. How do we add up and get such a value? I can convert to a non-rad value, but I can't understand how 0.327 or -1.18 is gotten
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1answer
94 views

Generalisation of Lambert W function?

I want to solve an equation of the form: $\exp(C / x) - 1 = D / (x + a)$ This seems to be almost in a form where I can express solutions in terms of the Lambert W function but I can't seem to figure ...
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1answer
104 views

Lambert W function?

I need to solve the next equation: $$ax^{bx+c}=d$$ where a, b, c and d are positive real values. Do I need to use Lambert W function, or there is some other method? Thanks!
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2answers
361 views

Problem Solving ( Sequences and series)

After injection of a dose $D$ of insulin, the concentration of insulin in a patient's system decays exponentially and so it can be written as $D\exp^{-at}$ where $t$ represents time in hours and $a$ ...
0
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1answer
33 views

the absolute value of $\frac{1}{e^{i\omega t}-1}$

I am told to get the absolute value of $$\frac{1}{e^{i\omega t}-1}$$ I sense that there's something ridiculously simple about this, but I tried working from the fact that if I square it, the absolute ...
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2answers
40 views

How to solve exponential inequality with $x$

I need to solve the following inequality. $$\ln(x) - x > 0.$$ I oddly remember that it can only be done by using the graph... Is it true? I have the same problem with $$e^x(x-1)>-2.$$ ...
2
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1answer
78 views

Summation of exponential series [duplicate]

Evaluate the limit: $$ \lim_{n \to \infty}e^{-n}\sum_{k = 0}^n \frac{n^k}{k!} $$ It is not as easy as it seems and the answer is definitely not 1. Please help in solving it.
8
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1answer
194 views

Closed form of $\int_0^2\frac{1}{2+\sqrt{3\,e^x+3\,e^{-x}-2}}dx$

Could you please help me to solve this integration problem? $$\int_0^2\frac{1}{2+\sqrt{3\,e^x+3\,e^{-x}-2}}dx$$ Its approximate numeric value is $0.419197813818367...$, but I could not find an exact ...