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

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2
votes
2answers
69 views

Does e = limit as x tends to negative infinity hold true?

Does $$e=\displaystyle\lim_{x \to -\infty}\left(1+\frac{1}{x}\right)^x\qquad\quad?$$
2
votes
2answers
67 views

$(1-x)^y ≈ e^{-xy}$

Here is an approximation I often see in biology articles but don't really understand: $$(1-x)^y ≈ e^{-xy}$$ I think this $e^{-xy}$ closely approximates $(1-x)^y$ whenever $x$ is small. Can you help ...
4
votes
1answer
150 views

Multiple integral over a disc

I would need some help on this integration problem: $$I=\int_0^{2\pi}\int_0^{R}\int_0^{2\pi}\int_0^{R}\exp(-a\ r_{12}) \ r_1 \ r_2 ...
14
votes
3answers
2k views

Is this question too easy or am I getting it wrong?

In my homework, I am asked to find the limit $$\lim\limits_{x\to0}{\frac{x}{e^x}}$$ But obviously, you could just substitute $x = 0$: $$\lim\limits_{x\to0}{\frac{x}{e^x}} = ...
0
votes
2answers
33 views

Definition of e, how to relate that to other interest rates

I understand that one way of understanding the meaning of the number $e$ is to form a compound interest formula, $A = \left(1+\frac{1}{n}\right)^{nx}$ and then let $n\rightarrow \infty$ for which this ...
0
votes
2answers
57 views

Solve an Exponential Equation

We have: $$16^x = 12^x + 9^x$$ Just by visual inspection one can say that the answer lies somewhere between $1$ and $2$. I gave the starting point of the iteration as $2$ and plugged the function ...
1
vote
1answer
30 views

A problem with progressive percentage incrementation

I have absolutely no clue if any of the terms I used in the title actually exist or make sense. I'm usually good at math (relatively) but I am facing this problem today that I just cannot solve. John ...
2
votes
2answers
43 views

How to use exponential function?

I know how to use exponential function when required in computer calculator but how does it work? I am still studying and our textbooks are not so detailed which gives us the idea how it works. I am ...
0
votes
2answers
35 views

A High School Exponential Decay Question

I've been teaching a student, when we encountered a question that I just couldn't work out. It goes like this: A city has a population of 10 million, decreasing by 1% every year. In a hundred ...
0
votes
3answers
46 views

Exponential problem

$\$10,000$ increases every day by $1\%$. How long until it doubles? I tried doing it by multiplying by $1\%$ for each day and got $10$ days but I don't think that is right. I know there must be an ...
0
votes
1answer
25 views

Exponencial function where I give $x$ to $x$ and it'll return me an exponential function between $0$ and $1$.

Sorry my enlgish isn't very good. I'm looking for a function that if, for example, I want $x=$ from 300 to 24 and it'll give me y between $0$ and $1$ exponentially.
6
votes
3answers
450 views

Evaluate a limit (probably involving L'Hôpital rule)

Evaluate the limit: $$\mathop {\lim }\limits_{x \to \infty } x\left( {{{\left( {1 + {1 \over x}} \right)}^x} - e} \right)$$ My attempts didn't yield a result. I'd be glad for a guidance. Thanks!
0
votes
1answer
66 views

What is the outdoor temperature? Working included.

Is my working correct in regards to this question? I'm quite stuck on it and I'm not too sure if I am in the right direction. Any advice is appreciated. Thank you. Question: A thermometer that has ...
5
votes
1answer
61 views

Solutions to $x+e^x=k$

So I am trying to solve $x+e^x=k$ and here is what I have done: $$x+e^x=k$$ $$e^{x+e^x}=e^k$$ $$e^xe^{e^x}=e^k$$ Now, if we use the lambert W function which has the identity such that if $y=xe^x$ ...
2
votes
3answers
52 views

Determining $\lim_{n\to\infty}\left(n^{\tfrac{1}{n}}-1\right)^n$ with only elementary math

I am trying to find this limit: $$\lim_{n\to\infty}\left(n^{\tfrac{1}{n}}-1\right)^n,$$ I tried using exponential function, but I see no way at the moment. I am not allowed to use any kind of ...
1
vote
1answer
58 views

Is my working correct? Exponenial decay

Is my working correct? If not, please let me know where I have gone wrong. Thank you for taking the time to check! Question: A thermometer that has been stored indoors where the temperature is 22 ...
0
votes
1answer
21 views

An efficient technique to test if an exponential of logs gives an integer

Is there an efficient way of testing if the resulting value of an exponential gives an integer without actually expanding the equation. For example: $ {\log(12) - \log(4)}=1.09861\ldots $ and is a ...
1
vote
1answer
41 views

Rate of convergence of an exponential function

If I have a function $$f = \exp(\sqrt{n} \cdot \frac{\sqrt{\log{n}}}{\sqrt{n}-\sqrt{\log n}}),$$ I can notice, that $$\lim_{n \to \infty} f = \infty,$$ but also I can notice that it goes very slowly ...
0
votes
1answer
31 views

Sketching the graph of $y =\ln(4-x)$

$y = \ln(4 - x) $ This graph has two operations applied to the $\ln x$ graph - a reflection and a translation. If you reflect the graph in the $y$-axis first, and then shift the graph 4 units to ...
0
votes
1answer
31 views

Why is $\left(1-\frac{1}{k}\right)^t < e^{-t/k}$?

I came across this statement, but can't see why it holds: $\left(1-\frac{1}{k}\right)^t < e^{-t/k}$ I'm sure it's something simple, but I don't have a great deal of mathematical experience. I ...
2
votes
1answer
66 views

The only differentiable function $f \colon \mathbb R \to \mathbb R$ such that $f^\prime(x)=f(x)$ is $f(x)=ce^{x}$

Prove that the only differentiable function $f \colon \mathbb R \to \mathbb R$ such that $f^\prime(x)=f(x) \mspace{1ex} \forall x\in \mathbb R$ is $f(x)=ce^{x}, \forall x\in \mathbb R$, and for some ...
2
votes
2answers
41 views

What's the density of $Z=\max(X,Y)-\min(X,Y)$ with $X,Y$ exponentials of parameter $\lambda$?

Let be $X,Y$ two independent exponential random variables with parameter $\lambda$. What is the pdf of $Z=\max(X,Y)-\min(X,Y)$? Thanks for your help.
4
votes
2answers
65 views

Operators $A$ such that $e^A$ is norm preserving

Let $X$ be a Banach space. $A$ a bounded operator. We can define the exponential of $A$ by $$e^{A}=\sum_{n=0}^{+\infty}\frac{A^n}{n!},$$ which is also a bounded operator. Is there any sufficient ...
3
votes
1answer
113 views

Is the function $f(x)=1^x=1$ considered an exponential function?

I am confused about the following: The exponential function (by definition) is a function of the form $f(x)=a^x$ where $a>0$. However, when $a=1$, we get the constant function $f(x)=1^x=1$. Is the ...
2
votes
4answers
59 views

Can somebody explain to me why these terms are equal?

I just read a proof on ProofWiki that proves Euler's formula, but I can't seem to understand what is done in this following step: ...
0
votes
1answer
74 views

What is the outdoor temperature? Help please!

Does anyone know how I would go about answering this question? Any feedback is appreciated! A thermometer that has been stored indoors where the temperature is 22 degrees Celsius, is taken outdoors. ...
0
votes
1answer
35 views

trouble with infinite values from exp() and log()

I'm writing a function for Gaussian mixture models with spherical covariance structures--ie $\Sigma_k = \sigma_k^2 I$. This particular function is similar to the ...
4
votes
1answer
139 views

What $\alpha$ such that if $xy=\alpha$, then $e^{-x}+e^{-y}\geq 2e^{-\sqrt \alpha} $?

For every $ x,y \gt 0$, if $ xy=\alpha$, then we have $$e^{-x}+e^{-y}\geq 2e^{-\sqrt \alpha} $$ What are the possible values of $\alpha$? $2 < e^{1/(n+1)} + e^{-1/n}$ led to this problem. ...
0
votes
2answers
38 views

How to prove the following? $\frac{d}{dx}a^x=(\ln a)a^x$

How to prove that the following holds? $$\frac{d}{dx}a^x=(\ln a)a^x.$$ Just a hint will do it.
3
votes
2answers
96 views

Show that $f(x)=e^x$

In this case $f(x)=1+x+x^2/2!+x^3/3!+x^4/4! + ... = \sum_{n=0}^\infty \frac{x^n}{n!}$. I understand it conceptually in terms of the Taylor series, but I have no idea how to prove it rigorously.
26
votes
3answers
854 views

Limit with a big exponentiation tower

Find the value of the following limit: $$\huge\lim_{x\to\infty}e^{e^{e^{\biggl(x\,+\,e^{-\left(a+x+e^{\Large x}+e^{\Large e^x}\right)}\biggr)}}}-e^{e^{e^{x}}}$$ (original image) I don't ...
3
votes
2answers
73 views

Non-integral power of a singular matrix

I know, that if $A$ is nonsingular matrix, so $\det{A} \ne 0$, then $A^p=\exp\left(p\ln A\right)$ is true for any real exponent, but what about if $A$ is singular? Then $A$ has a zero eigenvalue, so ...
0
votes
2answers
45 views

Compound interest coumpounded n time per year formula. $A=P\left(1+\frac{r}{n}\right)^{nt}$ intuition behind it.

I know that the compound interest formula for the interest compounded annually is given by $$A=P(1+r)^t$$ I know the intuition behind it. But why the compound interest formula for the interest ...
0
votes
1answer
29 views

Exponential percentage decrease based on time

I have a bar that shows the time left for a task to finish and I want it to decrease faster as it gets closer to the end time. Example: Let's assume that the total time required for Task A to ...
1
vote
2answers
37 views

How do I transform an r.v. using the floor function? (exponential distribution)

Just had a bash at this question for my Intro to Maths Stats module...I got to the end with a probability density function rather than a probability mass function, namely $f_Y(y) = \lambda a ...
1
vote
1answer
16 views

Mean and STD of a max/min of an exponentially distrubuted iid random variable

Let $S_1, S_2, S_3, ...$ be a sequence of independent, identically distributed (iid) random veriables, each exponentially distributed with a mean of $\mu_S$ (hence $\sigma_S = \mu_S$). Let $M_n = ...
3
votes
1answer
49 views

Definite integration of a exponential function mixed with rational functions

Suppose $a>0$ , I am interested in a solution of the following definite integral: $$\int_{1}^{\infty}\frac{\exp({-az})}{z \sqrt{z^2-1}}{\,dz}$$ Thank you.
3
votes
2answers
67 views

Calculation of integral $\int\exp \left(-\alpha \sin^2 \left(\frac{x}{2} \right) \right) dx$

Given $\alpha$ is a constant. How to calculate the following integral? \begin{equation} \int \exp \bigg(-\alpha \sin^2 \bigg(\frac{x}{2} \bigg) \bigg) dx \end{equation} Thanks for your answer.
1
vote
3answers
95 views

Prove by induction that (5^(n))-1 is divisible by 4 for all natural numbers n.

Prove by induction that $5^n-1$ is divisible by $4$ for all natural numbers $n$. I got $P(k+1)=5^{k+1}-1$ but I don't where to go now.
1
vote
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?
0
votes
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 ...
1
vote
1answer
87 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 ...
2
votes
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 ...
0
votes
4answers
66 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 ...
1
vote
1answer
19 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, ...
2
votes
2answers
81 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 ...
1
vote
1answer
43 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 ...
0
votes
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.
2
votes
1answer
78 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 ...
1
vote
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 ...