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

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2
votes
2answers
116 views

Integral of $\sqrt{ {\rm ln}^2 4 \cdot 4^{2 x} + 1}$

I'm currently taking calculus, and have hit a problem that is causing me confusion. I have the answer to the problem, I just have no idea how to arrive at that answer. The problem is as follows: $$\...
5
votes
3answers
73 views

$f(x)$ is an analytic function in $\mathbb{R}$ such that $f(-x)f(x)=1$. What else can we find out about $f(x)$?

Well, I know that there are some easy things we can say immediately: $f(0)= \pm 1$, follows immediately $f(x)=\pm 1$ is the obvious solution, so let's look for other solutions. Moreover, let's ...
1
vote
5answers
140 views

Integrating $\int xe^{-x} dx$ without parts

Can $\int xe^{-x} dx$ ever be solved by integration by substitution without using parts. Or does, as I suspect, substitution fail to yield a solution in this case. Seems that we can't get a ...
0
votes
2answers
62 views

Problem with the definition of $e$?

I have an issue understanding one of the definitions of $e$ that I found in a textbook I am using. They defined e as the limit of $(1+x)^{1/x}$ as $x\to 0$. But as $x$ approaches $0$ it can come in ...
1
vote
4answers
80 views

How would one solve the following equation?

This equation is giving me a hard time. $$e^x(x^2+2x+1)=2$$ Can you show me how to solve this problem algebraically or exactly? I managed to solve it using my calculator with one of its graph ...
0
votes
1answer
41 views

How can I solve this equation: $xe^{ax}=b$? [duplicate]

How can I solve this type of equation? $xe^{ax}=b$?
10
votes
1answer
142 views

Nontrivial integral representations for $e$

There are a lot of integral representations for $\pi$ as well as infinite series, limits, etc. For other transcendental constants as well (like $\gamma$ or $\zeta(3)$). However, for every definite ...
0
votes
2answers
50 views

Solving an equation with trigonometric and exponential functions

While tutoring about integration and derivatives of functions, we needed to determine what the biggest distance between two functions, one trigonometric $f(x)$, the second exponential $h(x)$ for value ...
2
votes
2answers
65 views

Has $e^x = ax^2$ a general solution for all $x$?

I was fiddling around with some math and stumbled upon $\exp(x) = a x^2$, finding myself unable to find a solution. Does it even have a general solution $a$ for all $x$? Some googling brought me to ...
1
vote
2answers
48 views

About Euler's formula $e^{ix}=\cos x+i\sin x$ [closed]

I think I probably miss something. Can you tell me what it is? In my assumption, that any given 'x' value, $$e^{ix}=\cos x+i\sin x$$ But, why don't I get the same value in the equation when I ...
0
votes
1answer
20 views

What formula represents how to compute a growing stream to reach a fixed total?

Given a growing sequence of values, where each next element in the sequence is the prior value times a constant growth factor, what formula allows the computation of the starting value in order to ...
2
votes
2answers
68 views

Limits problem without L'Hopital

I am prompted to solve the following limit $$\lim_{x\mapsto 0} (\cos(x)^\frac{1}{x^2})$$ I try to approach this problem by doing $$\lim_{x\mapsto 0} (-1+(1+\cos(x))^\frac{\cos(x)}{\cos(x)x^2})$$ ...
3
votes
4answers
101 views

Determine drug concentration over time, given its halflife and dosage

I want to calculate which of two doses is going to have the most active ingredient over the total time of an experiment. So as an example let's say I have a drug which has a halflife of 5 hours, and ...
0
votes
1answer
65 views

exponential function with polynomial exponent

hi guys can anyone help me, I am currently working with some integrals and i am tangled with $$\int_0^1 x^2(1-x)e^{-ax^2+b(1-x)^2}dx$$ and $$\int_0^\infty x^{-1}e^{-ax^2-bx^{-2}}dx$$ I had tried ...
8
votes
4answers
313 views

What is process/function to cancel base (in value with exponential)?

I have this equation (identity?): $$3^{3x}=3^{2y+1}$$ I understand that this simplifies to: $$3x=2y+1$$ So, the base (of the exponentials) cancel out. I want to know, what the process/function of ...
4
votes
1answer
67 views

Are the functions $e^{z^n}$ all algebraically independent?

I'm currently writing about Riemann surfaces and the algebraic dependence of any two meromorphic functions on a compact surface. I'm trying to think of an example of how this result fails for a ...
0
votes
0answers
40 views

show by calculation that the derivative of the fermi function (logistic function) can be expressed by the function itself

I'm taking a course on Neural Networks. one of the questions on our exam will (likely) be: ...
1
vote
0answers
29 views

Solve for deceleration in exponential decay equation

$$y = y_0 + v_0\cdot d\frac{1 - d^{t}}{1 - d}$$ $y$ = final position $y0$ = initial position $v0$ = initial velocity $d$ = deceleration $t$ = elapsed time How do I solve for $d$? Specifically, ...
1
vote
5answers
71 views

Solving $3^x \cdot 4^{2x+1}=6^{x+2}$

Find the exact value of $x$ for the equation $(3^x)(4^{2x+1})=6^{x+2}$ Give your answer in the form $\frac{\ln a}{\ln b}$ where a and b are integers. I have tried using a substitution method, i.e. ...
-6
votes
3answers
60 views

Is this property true? $\exp(x)=-\exp(-x)$ [closed]

I'd like to know if the following equality is true : $$\exp(x)=-\exp(-x)$$ Thank you.
0
votes
1answer
20 views

Solving an integral expression related to the circle method and complex exponential function

While reading a paper on the circle method, I came accross the following exercise: Let $e(x) = e^{2 \pi i x}$, prove that for $n,m \in \mathbb {Z}$, $\int_0^1 e(nx)e(-mx)=\delta _{mn}$ (Kronecker ...
6
votes
5answers
275 views

Arithmetic growth versus exponential decay

I have a kilogram of an element that has a long half-life - say, 1 year - and I put it in a container. Now every day after that I add another kilogram of the element to the container. Does the ...
2
votes
2answers
46 views

Intersection of a power function with a line: how to compute?

How to compute $x$ from $$q x^p = 1 - x$$ where $x$ and $q$ are positive, while $p$ is a real number? When $p > 0$: it's two monotonic functions, one increasing and one decreasing, and having ...
1
vote
0answers
46 views

Taylor of $\ln(f(exp(x))))$?

Let $ f(x) = \sum a_n x^n$ Such that The $a_n$ are real and $f(a),f ' (a) , f " (a) > 0 $ for any real $a > 0$. Let $ \ln(f(exp(x))) = \sum b_n x^n $. Let $c_n = a_n - b_n$. For a given $f$ ...
3
votes
1answer
77 views

number of permutations of [n] for which all cycles have even length

I'm looking to find the number of permutations of [n] for which all cycles have even length, call that number $f_n$. I've seen here: Number of permutations of a specific cycle decomposition that the ...
1
vote
0answers
39 views

exponential generating function for the number of ways to arrange marbles in a line

Say we have red, green, and blue marbles that we are arranging in a line of length n. We need to use an even number of blue marbles, at least two red marbles, and at most two green marbles. I am ...
1
vote
2answers
50 views

On Proving that $e^x$ is continuous at $0$ utilizing a limit result.

I was assigned the task to prove that $e^x$ is continuous in $x=0$ utilizing the fact that $$\lim_{x \rightarrow 0} \frac{e^x - 1} {x} = 1 $$ I think I am supposed to show that from the fact that for ...
10
votes
4answers
1k views

Showing that the exponential expression $e^x (x-1) + 1$ is positive

I'm looking at $$ f(x) = e^x (x-1) + 1$$ I'm having the feeling (based on the application where I am using it), that $f(x)$ should be strictly positive for $x > 0$. Indeed, Wolfram Alpha plots ...
0
votes
4answers
63 views

How to fit data to an asymptotic exponential?

I have 3 points that I must adjust to the following formula: $$ C = a' \cdot (1-e^{\alpha \cdot t})$$ The magnitudes I know are $C$ and $t$, and I have to obtain $a'$ and $\alpha$. I know that ...
1
vote
1answer
65 views

Solving $\operatorname{cis} x \operatorname{cis} 2x \operatorname{cis} 3x \dots \operatorname{cis} nx=1$

Given the equation: $$\operatorname{cis} x \operatorname{cis} 2x \operatorname{cis} 3x \dots \operatorname{cis} nx=1$$ How can I solve it? I know that $\operatorname{cis} x=\cos x+i\sin x$, but I ...
0
votes
0answers
20 views

Solving a complex exponential equality for all solutions

Since I learned how to take complex exponents and write the solution in $a+bi$ form, I've wanted to solve the following problem: $$x^n=y^m$$ Where we have $x,n,y,m\in\mathbb{C}$. Rewriting, this ...
0
votes
3answers
71 views

Does the logistic function have a relation with arctan(x)

The logistic function is: $$f(x)=\frac{L}{1+e^{-k(x-x_0)}}+B$$ It's plot looks similar to the plot of $arctan(x)$. Therefore I was wondering whether there is a relationship between these two ...
4
votes
2answers
131 views

Show that $f'(0)= \lim_{\Delta x \to 0}\frac{f(\Delta x)-1}{\Delta x} = 1$

This question is related to another question I asked here. Specifically, using the definition of $e$ I gave in that question: There exists a unique complex function $f$ such that $f(z)$ ...
0
votes
3answers
115 views

For a fixed complex number $z$, if $z_{n}=\left( 1+\frac{z}{n}\right)^{n}$. Find $\lim_{n \to \infty}|z_{n}|$

This is a step on the way to proving $\lim_{n \to \infty}\left(1 + \frac{z}{n}\right)^{n} = e^{z}$. Please do not mark this question as a duplicate. I am not asking the same thing other people are ...
1
vote
1answer
91 views

Proof of existence and uniqueness of the exponential function using ODEs

In our lecture notes for our complex analysis class, we were given the following theorem: Theorem: There exists a unique complex function $f$ such that $f(z)$ is a single valued function $...
0
votes
1answer
31 views

calculating logarithmic equation

I have an equation , which is like this $64n\log n < 8n^2$ . (the base of logarithm is 2) I know how to solve the logarithmic equations . I am a programmer , so I wrote a simple program and ...
0
votes
1answer
40 views

Waiting Times for Next Person in Line

I'm working on a problem and came up with two different solutions! I'm not sure which is correct. Problem: Two clerks with service time $exp(1)$ are helping 2 customers while a third waits. What is ...
1
vote
1answer
18 views

Mutidimensional integration with cosine rule in exponent

I have been attempting to find a way to simplify the following multi-dimensional integration form: $\int\limits_{0}^{\infty}\int\limits_{0}^{\infty}\int\limits_{0}^{\pi} e^{\left(-a-b+\sqrt{a^2+b^...
37
votes
1answer
691 views

Mirror algorithm for computing $\pi$ and $e$ - does it hint on some connection between them?

Benoit Cloitre offered two 'mirror sequences', which allow to compute $\pi$ and $e$ in similar ways: $$u_{n+2}=u_{n+1}+\frac{u_n}{n}$$ $$v_{n+2}=\frac{v_{n+1}}{n}+v_{n}$$ $$u_1=v_1=0$$ $$u_2=v_2=...
0
votes
4answers
148 views

Are there real solutions to $x^y = y^x = 3$ where $y \neq x$?

I need to solve the following equation for (x,y) $$x^y = y^x = 3$$ Everytime I run a numerical method for this problem, I get $$ (x,y) = (1.82546...,1.82546..) $$ I expect there to be a solution ...
1
vote
1answer
21 views

Exponential equation with same, unknown bases

I have the following equation: $(x-3)^{(x^2-x)} = (x-3)^2$ The book says solutions are: $x_1 = -1, x_2 = 2, x_3 = 3, x_4 = 4$ I was only able to get -1, 2, 4 by doing this: $(x-3)^{(x^2-x)} = (x-3)^...
2
votes
1answer
66 views

How to manipulate the knee of the curve of an exponential function?

I want to be able to manipulate the point of inflection of an exponential curve equation:$$a\exp\{xb\}.$$ could somebody tell me which parameter I may introduce in such a formula in order to make the ...
1
vote
5answers
64 views

Proof problem: show that $n^a < a^n$ for all sufficiently large n

I would like to show that $n^a < a^n$ for all sufficiently large $n$, where $a$ is a finite constant. This is clearly true by intuition/graphing, but I am looking for a rigorous proof. Can ...
0
votes
1answer
28 views

How to interpret b in $y=x^{e^{bz}}$ in nonlinear regression?

What is the correct way to interpret b in this nonlinear equation $y=x^{e^{bz}}$? I've estimated the model and b seems to be the percent change in y with a unit change in z, but I am unsure how to ...
0
votes
1answer
16 views

Find the value of $k$ in the exponent

I am trying to calculate the $k$ value in this equation: $\dfrac1{n^c} \le \left(1 - \dfrac2{n(n-1)} \right)^k$ by using the logarithm, I am getting for $k$: $\log_{1- 2/n(n-1)} n^{-c} \le k$ is ...
5
votes
1answer
68 views

Show that $ \exp \left(SL(2,R)\right)$ is the set of all matrices with positive trace $\geq -2$

Using the fact that every matrix in $SL(2,\mathbb{R})$ is conjugate in $SL(2,\mathbb{R})$ to one of the following matrices: $$ \left(\begin{array}{rr} a & 0\\ 0 & \frac{1}{a} \end{array}\...
1
vote
4answers
52 views

Finding the limit of the sequence $x(n) = (1+2/n)^n$ [duplicate]

What we are allowed to use - 1) The fact that limit of $(1+1/n)^n$ exists and assumed to be some real number $e$ 2) Subsequencial properties of limits of sequences 3) Basic properties of limit In the ...
1
vote
2answers
39 views

Solve equation with exponentials

I'm trying to obtain $x$ in the following equation: $$ 1= ae^{bx}+ce^{dx} $$ with a,b,c,d known. What I did was to take the ln of all the equation, so I have: $$ \log\frac{1/ac}{b+d} = x $$ ...
1
vote
1answer
41 views

Gradually and eventually slow and exponential [closed]

I am not a math expert. Thats exactly why I am here. Please help. I need an equation for the below use case. Consider a lottery. Winner always gets 1$ (constant). No more. As the number of users-...
8
votes
3answers
122 views

General formula for the higher order derivatives of composition with exponential function

Suppose I have a function $x:\mathbb{R} \to \mathbb{R}$ and consider: $$g(t) = e^{x(t)}$$ When I start differentiating with respect to $t$ I obtain: \begin{align} g'&=e^xx'\\ g''&=e^x((x')^2+x'...