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

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0
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2answers
36 views

Logs of a complex number

Write a solution in Cartesian for of What should come next?
0
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1answer
18 views

prove doubling time in an exponential function

I am currently working my way through Hughes-Hallet et al., Calculus- Single and Multivariable. I am having trouble with the following problem. Show algebraically that if P=P0a^t doubles between ...
-1
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1answer
67 views

how to solve equation $x^x=5$ [duplicate]

How can I calculate the equation $x^x=5$ Is it an exponential function? Thank you.
2
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8answers
289 views

How come $1^{\infty}$ = undefined, while $2^{\infty} = \infty$ and $0^{\infty} = 0$? [duplicate]

$1^\infty$ = undefined $2^\infty = \infty$ $0^\infty = 0$ Why is $1^\infty$ undefined? People were trying to explain to me that infinity isnt part of the Real numbers, yet, $2^\infty$ and ...
2
votes
1answer
33 views

Finding Complex Zeros

I have to find how many zeros $3e^z - z$ has in $abs(z) < 1$. I know a function has a zero of order m if $f(z) = (z-z_0)^mg(z)$, where $g(z)$ does not equal 0. I was thinking of maybe applying ...
0
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1answer
40 views

Rescaling, or finding logarithmic equivalent of exponential functions

I'm working on an algorithm that gets the weighted least squares of some data and outputs really big, or really small numbers. I'm talking about numbers like $3.55114473577E+256$. Now, part of the ...
0
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1answer
18 views

Basic Integration + Root + Exponential issue

I have the following question : $\int ( 27e^{9x} + e^{12x} )^{1/3} dx $ However when I solved it I simplified it first to: $\int ( 27e^{9x} + e^{12x} )^{1/3} dx = \int \sqrt[3]{27e^{9x} + ...
1
vote
1answer
25 views

%reduction in a decaying exponential function

I am working my way through a calculus book I purchased- Calculus- Single and Multivariable (3rd edition) by Hughes-Hallet et al. I am having issues with the following question "When the olympic ...
0
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2answers
72 views

Math notation that I am not familiar with

I am working on reverse engineering a game and have come across the following formula as a string in a config file: A*(B^xt)+C; xt=A2*x*(T>x)*(B2^x+C2) It ...
0
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1answer
28 views

Definition of Derivative And Exponential Functions

Given $f(x) = 5^{3x}$. Find $f'(x)$ using definition of a derivative. The definition of the derivative of $f(x)$ is $f'(x) = \lim_{h \to 0} \dfrac{f(x + h) - f(x)}{h}$ The derivative of $f(x) = ...
2
votes
1answer
65 views

Solving an equation with curly brackets and entering into excel

Hi I am sorry if this is a trivial question but I am trying to follow a book to create a mathematical model but I can not get the same result. The equation has curly brackets and I am not sure what ...
4
votes
1answer
73 views

Applications of the Exponential Integral?

this is my first time asking a question on here so please forgive me if I have made any formatting mistakes. I have the integral $f(x) = \int_0^\infty \frac{e^{-t}}{x + t} \; dt$ and I have shown the ...
0
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2answers
50 views

Is this exponential equation solvable? natural logarithms, exponential

$$\displaystyle{a=\frac{e^{-cos(\frac{b}{x})}-e^{-\frac{1}{x}}}{(1-e^{-\frac{1}{x}})}}$$ I'm trying to solve for $x$. $a$ and $b$ are constants. Any help is really appreciated. Thanks Ghassan
2
votes
2answers
68 views

How can I define $e^x$ as the value of infinite series?

I understand the definition of $e^x$ by limit. But I do not know how to come up with: $$e^x = \sum_{n=0}^{\infty}\frac{x^n}{n!}$$ without using Taylor series. more explicitly without using calculus. ...
1
vote
1answer
58 views

Biasing sigmoid curve

I wish to use the sigmoid function $1-{1\over1+e^{-x+c}}$ to obtain a value from 0 to 1 (to be used for a probability value), where $c$ is a constant. The higher this constant, the lower the ...
0
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0answers
23 views

Trying to reverse engineer a formula called “exponential_flat”

Here are the values I know: A = 200 B = 1.75 C = 0 A2 = 1.8 B2 = 0.93 C2 = -0.64 T = 14 It is possible that some of these values are not used. The other formula ...
0
votes
1answer
47 views

Simulation - Find the maximum of a function with exponential decay

I need to run a program to calculate the integral of the following function with exponential decay $$t(x) = \exp(-Lx)(a\sin(bx) + d\cos(ex))$$ and for a simulation purpose, I need to find maximum of ...
4
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3answers
88 views

Game With 21 Squares, How Many Possible Answers? Function Building

We played this game in our math class, okay, I'll explain how it's played. There are 21 squares in a straight line across, the first person shades in 2 adjacent squares. The next player shades in 2 ...
0
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1answer
23 views

Exponential Random Variables and Confidence

Assume that the amount of evidence against a defendant in a criminal trial is an exponential random variable X. If the defendant is innocent, then X has mean 1, and if the defendant is guilty, then X ...
0
votes
1answer
62 views

Finding time constants of a circuit?

So this is a homework question and I am having trouble figuring out what they are asking. 'The potential difference (voltage) across the capacitor at time t > 0 is given by $V_C(t) = q(t)/C$. The ...
1
vote
2answers
36 views

Integrating two exponentials produces a cosine integral? Can somebody explain?

I discovered the following conversation that I do not understand. It reads: $$\int_{-U_1}^0 {(\frac {u_1} {U_1}+1)e^{-j\omega_1u_1}}~du_1+\int_0^{U_1} {(-\frac {u_1} {U_1}+1)e^{-j\omega_1u_1}}~du_1 = ...
2
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2answers
25 views

Exponential growth precalc population

The population of City A increases by 8% every 10 years. The population of City B triples every 120 years. The two cities had equal populations of 10,000 residents each in the year 2000. In what year ...
1
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1answer
46 views

What is the integral containing decaying exponential function?

I am trying to figure out properties of the following integral: $$p(t)=\int_{0}^{t} e^{\alpha(t-t')} f(t')dt', \hspace{1 cm} t>t'$$ I would google and read more info about this integral but I do ...
2
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0answers
27 views

Yacov Perelman Nepero game

This is my first question, so sorry if I'll make any mistake in using the site formatting. I found this game on a book by Yacov Perelman and I thought it could be nice to introduce Nepero number to ...
-1
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2answers
68 views

Integration of $g(x) = e^{f(x)}$ [closed]

Is there any way of simplifying this integral? $$ f(x) = \int e^{2x^3}\,dx $$
1
vote
0answers
83 views

How to compute time ordered Exponential?

So say you have a matrix dependent on a variable t: $$A(t)$$ how do you compute $$e^{A(t)}$$ It seems Sylvester's formula, my standard method of computing matrix exponentials can't be applied ...
1
vote
1answer
38 views

How to evaluate the integral $\int_0^{\ln3} e^{x-e^x}\,\mathrm dx$?

How to evaluate the following definite integral? $$\int_0^{\ln3} e^{x-e^x}\,\mathrm dx.$$ Should I use some sort of U Substitution?
2
votes
2answers
62 views

Is this a valid proof for eulers formula?

I am wondering whether this proof is a valid proof of Eulers formula: $e^{ix}=i\sin(x)+\cos(x)$ $$\frac{d}{dx}e^{ix} = i(e^{ix})$$ $$\frac{d}{dx}(i\sin(x)+\cos(x)) = i\cos(x)-\sin(x) = ...
0
votes
1answer
30 views

Apply the Fourier Transform to $A\cdot e^{-a|k - k_0|}$

I have the following problem: The task is to show that $$f^*(k) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^\infty f(k) e^{ik(x-vt)} dk$$ with $f(k) = A\cdot e^{-a|k - k_0|}$ equals $$f^*(k) = ...
0
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0answers
41 views

Removing backups in an exponential fashion

Background: I want to create a backup system that utilizes the full space of a hard-disk. Given that all backups are approximately equal in size this means that I can save a fixed amount of backups. ...
0
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1answer
88 views

Rate of exponential decay

Good day all I have this curve (it is a solution of a partial differential equation that am working on) and I want to calculate numerically the rate of exponential decay but I don't know how to go ...
0
votes
1answer
29 views

Not understanding one step in derivation of Dirichlet kernel

I was reading some notes on the Dirichlet Kernel and they have a proof of how it reduces to $\sin(2\pi(N+ 1/2)t)/\sin(\pi t)$. I could follow the steps except for one early step which is the ...
1
vote
1answer
51 views

Confusion about an algorithm making a choice between two options, with probabilities.

I am totally puzzled at grasping the meaning of "we move to B with probability P1 OR we move to C with probability P2" in the following scenario. A,B,C are points in a 64-dimensional space. Reading ...
1
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1answer
26 views

Prove $e^c>c^e$ if $c>0$ and $e \neq c$ using graph.

I am on this question where it tells me to show $e^c>c^e$ if $c>0$ and $e \neq c$ using the graph of $\dfrac{(log(x))}{x}$. Now it is obvious that the graph reaches a maximum at $x=e$ but how ...
1
vote
2answers
55 views

Are these derivatives correct??

Take take the function defined as $$f(x) = \left\{ \begin{array}{ll} exp(\dfrac{-1}{x^{2}}) & \mbox{if } x \neq 0 \\ 0 & \mbox{if } x = 0 \end{array} \right. $$ Now I am asked to check ...
1
vote
1answer
111 views

2.71828. And then another 1828.

This may qualify as the silliest math.SE question ever, but am I really the first person ever to worry about this? The decimal expansion of $e$ has a 2. And then a 7. And then a 1828. And...well, then ...
0
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2answers
62 views

No. of real solutions of the equation $2 \cos (\frac{x^2 + x}{6}) = 2^x + 2^{-x} $

How many real solutions are there of the equation $2 \cos (\frac{x^2 + x}{6}) = 2^x + 2^{-x} $? Please illustrate.
3
votes
1answer
71 views

On the equation $\exp(a x+b)=\ln(x)$

I am confronted with: $$\exp(a x+b)=\ln(x)$$ for $a,b$ reals and $a<0$, $b>0$. I need the (unique) solution for $x$. My first target is (if it exists) an analytic solution in terms of ...
1
vote
3answers
64 views

Can you help me to get the integration of $\frac{1}{x} \exp(-x)$?

I need the solution of integral like $$\int^\infty_a \frac{1}{x} e^{-x} \,dx.$$ Thank you
0
votes
1answer
45 views

No. of real solutions of the equation $\big(\!\frac{9}{10}\!\big)^x = - 3 + x - x^2$

How many real solutions are there of the equation $\left(\dfrac{9}{10}\right)^x=-3+x-x^2$ ? Please illustrate.
0
votes
0answers
17 views

Scaling model output to be between 0 and 1

I have fitted Cox model and the output is generated as: $e^{\beta x}$, where $\beta$ is the coefficient. Now, I would like to have the model output ranging between $0$ and $1$. I'm currently using ...
2
votes
0answers
21 views

Show that $x' = Ax$ is an attractor if end only if there is a quadratic form $q$ positive definite such that $Dq(x) . Ax < 0$ for all $x \neq 0$

Show that $x' = Ax$ is an attractor if end only if there is a quadratic form $q$ positive definite such that $$Dq(x) . Ax < 0$$ for all $x \neq 0$ Definition: a linear system $x' = Ax$ called ...
4
votes
2answers
96 views

Show that $e^{t(A+B)} = e^{tA}e^{tB}$ for all $t \in \mathbb{R}$ if, and only if $AB = BA$.

Let A,B real or complex matrixes. Show that $e^{t(A+B)} = e^{tA}e^{tB}$ for all $t \in \mathbb{R}$ if, and only if $AB = BA$. I demonstrated the reciprocal: $\Leftarrow )$ The two equations are ...
7
votes
2answers
93 views

Is $\int_x^{\infty}e^{-\frac{t^2}{2}} < \frac{1}{x}e^{-\frac{x^2}{2}}$?

While solving a problem in real analysis, I got stuck. I need to prove $$\int_x^{\infty}e^{-\frac{t^2}{2}}dt < \frac{1}{x}e^{-\frac{x^2}{2}} $$ Clearly I have to use some kind of inequality, but ...
8
votes
7answers
289 views

How to prove continuity of $e^x$.

I simply want a proof that $e^x$ is continuous. I have never really been able to find something satisfying these points: $e$ is defined to be the limit $\lim_{n\to\infty}\left(1+{1\over ...
4
votes
4answers
103 views

How to calculate $\exp\left(t\begin{bmatrix}0 & z\\z^* & 0\end{bmatrix}\right)$?

or, in a more general case: $e^{\begin{bmatrix}0 & v\\w & 0\end{bmatrix}}$, where: $v, w \in \mathbb{C}$
0
votes
1answer
19 views

finding limits as function approaches zero

Set $n=1+\epsilon$ and let $\epsilon$ tend to zero. $$ \begin{align} c_1 &= \frac1{2\pi} \left[\frac{e^{i\pi (1+\epsilon)}-e^{-i\pi(1+\epsilon)}}{(1+\epsilon)^2-1}\right]\\ &= ...
1
vote
1answer
44 views

How do the steps of this definite integral work?

Sorry if this is a really basic question but I can't seem to get my head around the steps involved in this integration at all. My equation to be integrated is as follows: ${ds \over s}=\mu dt$ ...
1
vote
2answers
150 views

Fourier Transform the following exponential and cosine function: $f(x) = e^{-a^{2}x^{2}}cos(bx)$

I have a previous exam here for my course (Provided by the professor) that requires me to do a Fourier Transform of the following equation. Here is the function: $f(x) = e^{-a^{2}x^{2}}cos{(bx)}$ ...
2
votes
0answers
27 views

Area of intersection of polynomial and exponential functions

I was inspired to explore this by a recent post on the math subreddit, which to my knowledge went nowhere. Consider the families of functions $x^y$ and $y^x$. Given some $y \in \Bbb R$, the roots ...