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

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2answers
34 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
107 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
49 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 $$ ...
0
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0answers
22 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
votes
1answer
106 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 ...
0
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2answers
113 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
26 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.
0
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0answers
27 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
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3answers
224 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
votes
1answer
23 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
votes
1answer
103 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?
1
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3answers
67 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
vote
2answers
50 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
votes
3answers
188 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
votes
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. ...
1
vote
1answer
20 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
25 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
15 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 ...
6
votes
1answer
193 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 ...
1
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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
45 views

Proving that the exponential function is its own derivative, using the limit definition of $e$

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^x:=\lim_{n\to\infty}\left(1+\frac{x}{n}\right)^n \implies \frac{d(e^x)}{dx} = ...
0
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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
3
votes
1answer
62 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
votes
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 ...
0
votes
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
84 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
votes
1answer
54 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?
1
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2answers
128 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
votes
3answers
76 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 = ...
1
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4answers
50 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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2answers
93 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 ...
0
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0answers
25 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
58 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 ...
1
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2answers
21 views

Special vs. General Case in Basic Algebraic Notation

From Rogawski, Jon (2011-04-01). Single Variable Calculus (Page 343). W. H. Freeman. : To compute the derivative of a function of the form eg(x), write eg(x) as a composite eg(x) = f(g(x) ), where ...
0
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1answer
88 views

Find exponential decay equation for tiger population model

I've forgotten how to do it first it starts.. In 1900, there were 100,000 wild tigers worldwide; in 2010 the number was 3200. (a) Assuming that the tiger population has decreased exponentially, find ...
2
votes
2answers
105 views

Solving integral that contain exponential function and lower incomplete gamma function

I have the following integral; $$y=\int_0^\infty\frac{e^{-xf}}{m+x}\gamma(a,hx)~dx$$ where $f,m,h\in\mathbb{R}^+$ , $a\in\mathbb{N}$ , $\gamma\left(a,h x\right)$ is the lower incomplete gamma ...
5
votes
1answer
87 views

If $e^{i\theta}=e^{i\varphi}$, then $\theta-\varphi=2k\pi$

This is pretty easy I think but I am having a tough time trying to prove this in a satisfying way to me. I am trying to show that $$e^{i\theta}=e^{i\varphi} \Rightarrow \theta-\varphi=2k\pi,\, \text{ ...
2
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4answers
186 views

The definition of e by limits of $(1+1/n)^n$ through series expansion

I think the problem I have is due to not being knowledgeable about limits. If I use binomial expansion to expand $(1+1/n)^n$ to $1 + \frac{n!}{(n-k)!k!}*(1/n)^k + ...$, I can imagine replacing $n$ ...
3
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0answers
46 views

Why is the base of an exponential function limited to the set of real numbers greater than zero?

From Rogawski, Jon (2011-04-01). Single Variable Calculus (Page 339). W. H. Freeman. An exponential function is a function of the form $f (x) = b^x$, where $b > 0$ and $b \neq 1$. Why is $b$ ...
0
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2answers
58 views

What is the method to correctly isolate $y$ as the dependent variable for $x = e^y$?

In this youtube video about 5:00 minutes in, the instructor makes the point that you can simply exchange the $x$ and $y$ values of the exponential form $x = e^y$ of the equation $y = ln x$ to make $y$ ...
0
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0answers
103 views

Complex exponential integral: Mathematica and MATLAB give unexpected results

I currently compare analytical vs. numerical evaluation of the complex exponential integral and find mismatches: The imaginary part differs by $\pm \pi$ and the real part has a large error when ...
0
votes
2answers
60 views

Matrix Exponent - equivalent of a rotation matrix

Every Rotation Matrixcan be represented as a power of e with exponent a skew symmetric matrix. In particular, if we have a rotation matrix ${R}\in\mathbb R^{3 \times 3,}$ then there will be a skew ...
2
votes
2answers
64 views

Evaluating exponential integral

I am struggling for some time to solve the following integral: $$ \int_{-n}^{N-n} \left( \frac{e^{-j\pi(\alpha-1)\tau}}{\tau} - \frac{e^{-j\pi(\alpha+1)\tau}}{\tau} \right) d\tau $$ $N$ is a ...
5
votes
3answers
120 views

Is $x^x$ a polynomial, an exponential or both?

If $c$ is a constant, and $x$ is a variable, we'd say that $f(x) = x^c$ is a polynomial function of order $c$. Conversely, the function $f(x) = c^x$ would be called an exponential function. Is there ...
0
votes
2answers
38 views

Solving equations having both log and exponential forms

How can one Solve equations having both log and exponential forms: For eg... $e^x$ $=$ $\log_{0.001}(x)$ gives $x=0.000993$ (according to wolfram-alpha ...
1
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0answers
93 views

Why is sequence $(1+\frac{1}{n})^{n+1}$ descending? [duplicate]

I was studying the proof of $e$ number when I noticed something: Why is the sequence $(1+\frac{1}{n})^{n+1}$ descending? It starts ascending with grater n but in one moment it starts descending? Why ...
0
votes
3answers
29 views

How do I rewrite a logarithm in exponential form, so as to plot it? $f(x) = 2\log x$

How do I write $f(x)=2\log x$ in exponential form? Is $2(10)^y=x$ correct?
-1
votes
1answer
32 views

Exponential function, domain of definition

I have the function $\displaystyle f(x,y)=x^2e^{-x^2-y^2}$ with the domain of definition = $\{(x,y) \mid x^2+y^2=2\}$ The task is to decide $f$'s maximum and minimum value and the range. How do I get ...
0
votes
1answer
26 views

Integrals with an imaginary linear term in the argument of the exponent

in this entry on Wikipedia stays $$ ...