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

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1answer
36 views

Solving $Ae^x=Bx$ analytically, where $A$ and $B$ are constants?

This equation mixes both exponential terms and linear terms, something which I do not know how to deal with. Any pointers?
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0answers
33 views

Problem with commutator relations

part a) is fine. part b) is not. A commutator is defined as, for operators $A$ and $B$, $[A,B]=AB-BA$. [SOLVED]I get that $H(\lambda)=e^{-\lambda D}Ce^{\lambda D}$, $H'(\lambda)=-De^{-\lambda ...
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1answer
98 views

Double integrals of exponential functions

I need to find the double integral of $$e^{\frac{x}{y^2}}$$ bound by the $y\mbox{-axis}$, $x=y^2$, $y=1$, and $y=2$. The limits of integration were easy to find, but I am pretty confused about how to ...
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1answer
56 views

Exponentiation of imaginary operator

It is very easy to prove that if $D=\dfrac{d}{dx}$, then $(e^{nD}f)(x)=f(x+n)$ about $x=m$ in the real numbers. Proof: $$(e^{mD}f)=\sum^\infty_{n=0}\dfrac{D^nf}{n!}m^n\\ \implies ...
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4answers
70 views

What is the name of the answer to exponentiation?

What is the name of the answer to exponentiation? Adding two numbers produces a sum. Multiplying two numbers produces a product, but I cannot think of or find the name for the solution to ...
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1answer
20 views

Mean and STD of a sequence based on Exponential Random Variable

Say I have a sequence $S$ that is a exponentially random variable with mean $\mu$. Now say I create another other sequences from this: $T$ which is $T(n) = 2(S(n+1)-S(n))$. I know that theoretically, ...
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2answers
83 views

Integral of exponential using error function

I'm trying to solve some integrals below $$\int_{-\infty}^{\infty} {x^n e^\frac{-(x - \mu)^2}{\sigma^2}}dx$$ I am interested in the solutions where n = 0, 1, 2, 3, 4. I have learned that ...
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1answer
46 views

Exponential problems

A ship embarked on a long voyage. At the start of the voyage, there were 300 ants in the cargo hold of the ship. One week into the voyage, there were 600 ants. Suppose the population of ants is an ...
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1answer
82 views

Series proof for $e^x$.

Problem: Prove $$\sum_{n=0}^\infty \frac{1}{n!}x^n=e^x$$ I am a bit confused on how I should start this proof. Any pointers on how I should start would help.
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1answer
80 views

Gradient of matrix exponential function

Grateful if somebody could help me with the following. I am trying to find the gradient of the next expression: $$f(a_1, a_2, a_3, a_4)=\Vert R*y-x \Vert $$ where $y$ and $x$ are known 4x1 column ...
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1answer
68 views

Solving Exponential Function for termites vs spiders

The populations of termites and spiders in a certain house are growing exponentially. The house contains 120 termites the day you move in. After four days, the house contains 210 termites. Three days ...
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1answer
50 views

Rewriting in $y=A_0\cdot e^{at}$

How do you rewrite $y = −8(1.589)^{t − 3}$ in $y=A_0e^{at}$ form for appropriate constants $A_0$ and $a?$ For other problems I took the $\ln$ of the number inside the parenthesis. So for example I ...
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2answers
25 views

Exponential Growth Rates

So if you are given two different numbers to determine a growth rate, do you use to largest number compared to the value when x=0. For example the problem I am working on is: Your grandfather ...
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2answers
36 views

Logs of a complex number

Write a solution in Cartesian for of What should come next?
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1answer
15 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 ...
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1answer
66 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.
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8answers
272 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 ...
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1answer
31 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 ...
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4answers
175 views

How is the limit definition of e, actually equal to e? [duplicate]

$$\lim_{n\to\infty} (1+\dfrac{1}{n})^n = e.$$ To me this seems like $\dfrac{1}{n}$ goes to $0$, then you get $1^\infty$, which is equal to $1$. So why is it $e$?
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1answer
30 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 ...
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1answer
16 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} + ...
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1answer
24 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 ...
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2answers
68 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 ...
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1answer
26 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) = ...
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1answer
46 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 ...
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1answer
68 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 ...
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2answers
48 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
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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. ...
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1answer
42 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 ...
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0answers
20 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 ...
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1answer
32 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 ...
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3answers
62 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 ...
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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 ...
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1answer
59 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 ...
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2answers
34 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 = ...
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2answers
17 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 ...
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1answer
39 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 ...
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0answers
26 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 ...
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2answers
68 views

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

Is there any way of simplifying this integral? $$ f(x) = \int e^{2x^3}\,dx $$
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0answers
56 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 ...
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1answer
37 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?
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2answers
58 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) = ...
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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) = ...
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0answers
39 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. ...
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1answer
70 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 ...
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1answer
22 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 ...
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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 ...
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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 ...
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2answers
54 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 ...
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1answer
101 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 ...