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

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1answer
28 views

Following flash, a camera's battery begins to recharge the flash’s capacitor, which stores electric charge given by $Q(t) = Q_0(1 − e^{−t/a})$

(The maximum charge capacity is $Q_0$ and $t$ is measured in seconds). (a) Find the inverse of this function and explain its meaning. (b) How long does it take to recharge the capacitor to 90% of ...
0
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2answers
29 views

How to write $a^{ix}$ in terms of $\sin(x)$ and $\cos(x)$?

We know that $e^{ix} = \cos(x) + i\sin(x)$ and the plot of $2^{ix}$ seems to have sinusoidal behavior. http://goo.gl/Xfg2wp Can we claim that we can write $a^{ix}$ in terms of $\sin(x)$ and ...
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0answers
17 views

Changing a negative definite matrix to a positive definite matrix

Consider a negative definite matrix $X$,then $(I-e^X)$ is a positive definite matrix. What condition should matrix $X$ satisfy?
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1answer
49 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
27 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
16 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
35 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 ...
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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) & ...
0
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1answer
24 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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2answers
16 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
24 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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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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1answer
46 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. ...
-2
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4answers
93 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
54 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 ...
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2answers
32 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 ...
7
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2answers
97 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, ...
0
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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 $i = 1..n$. ...
3
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1answer
101 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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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 ...
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2answers
108 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.
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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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 $0$. ...
8
votes
3answers
215 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 ...
0
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1answer
96 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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3answers
65 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 $|x|_*$ ...
1
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1answer
43 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: ...
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 ...
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3answers
24 views

Solve for two variables, two equations with exponents [closed]

Solve for both k and x, where $5=k(300)^x$ and $80=k(600)^x$
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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 ...
5
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1answer
179 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 ...
0
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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} = ...
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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
56 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 ...
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.
2
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1answer
52 views

Why does a heating model work?

I am referring to: $T=T_0 e^{kt}$ where T=temperature,t=time and k=constant. It seems to work, I as just curios to why it works?
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2answers
120 views

Is there any proof for this formula $\lim_{n \to ∞} \prod_{k=1}^n \left (1+\dfrac {kx}{n^2} \right) =e^{x⁄2}$

Some times ago, In a mathematical problem book I sow that this formula. I don't no whether it is true or not. But now I'm try to prove it. I have no idea how to begin it. Any hint or reference would ...
2
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3answers
73 views

If $\ln x$ is defined via an integral and $e$ defined from $\ln x$, how would you prove that $\ln x$ is the inverse of $e^x$?

This is a somewhat technically specific question about the relationship between $\ln x$ and $e^x$ given one possible definition of $\ln x$. Suppose that you define $\ln x$ as $$\ln x = ...
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4answers
45 views

Proving the Derivative of $f'(x) = b^x$

Given $f(x) = b^x = e^{x\ln b}$ for $b > 0$, can someone show me how $f'(x) = \ln b e^{x\ln b}$ ?
1
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1answer
56 views

Constrained Newton-Raphson method

Peace be upon you, I want to solve a system of two equations in which the existence of $ln\left(\frac{\alpha}{\alpha+\beta}\right)$ function makes some limitations in iterations of the Newton-Raphson ...
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0answers
22 views

Overflow and underflow of a probability value

I am evaluating the probability that the minimum of a process is a above a a barrier $\log(H)$. The probability is given by $$P_i=1-\exp\left(-2\frac{(\log(H)-x)(\log(H)-x_b)}{\tau\sigma^2}\right).$$ ...
4
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0answers
55 views

Exponential of a polynomial of the differential operator

Given that $$\exp(aD)f(x)=f(x+a)$$ where $\exp(D)$ is the exponential of the differential operator $D$, is there a similar closed-form, general expression for $\exp(g(D))f(x)$, where $g(D)$ is a ...