For question involving exponential functions and questions on exponential growth or decay.
2
votes
2answers
87 views
How to solve $\sinh x = x$?
Does anyone have any thoughts on how to solve the following equation:
$$\exp(2x) - 2x\exp(x) - 1 = 0$$
If it helps, this equation is also equivalent to the following hyperbolic equation:
$$\sinh(x) ...
2
votes
4answers
80 views
Why the integral of $e^{-x}\;$ is $\;-e^{-x}$, and not $e^{-x}$?
I thought that the integral of $e^{x}$ is always $e^{x}$. Why does it change its sign to a negative when there is a negative exponent?
1
vote
3answers
75 views
Infinite Series
How can you show that
$$\left(1-\frac{2}{n^2}\right)^{n^2/2} \le \frac{1}{e}\:\: \qquad\forall n \ge 2$$
Any ideas? Infinite series have never really been my thing. Thanks
9
votes
2answers
319 views
Show that $ e^{A+B}=e^A e^B$
If $A$ and $B$ are $n\times n$ matrices such that $AB = BA$ (that is, $A$ and $B$ commute), show that
$$ e^{A+B}=e^A e^B$$
Note that $A$ and $B$ do NOT have to be diagonalizable.
0
votes
2answers
22 views
How do I simplify this expression
I have the expression $$e^{log(3)* log(7)\over log 2}$$
I know it can be simplified to $$3^{log(7)\over log(2)}$$ But I don't know how its done.
1
vote
2answers
50 views
How to show the double factorial isn't a polynomial
$(2n-1)!! = \dfrac{(2n)!}{2^{n} \times n!}$
I was wondering how you prove the double factorial is exponential.
I guess you have to prove that for all $m$ and $\alpha$ that there exists an $n$ such ...
2
votes
2answers
78 views
Why is $\frac{f'(x)}{f(x)}$ always a constant?
Today in class we learned that for exponential functions $f(x) = b^x$ and their derivatives $f'(x)$, the ratio is always constant for any $x$. For example for $f(x) = 2^x$ and its derivative $f'(x) = ...
0
votes
0answers
35 views
What is the fourier transform of this function?
With
$$ f(x) = \frac{1}{p} e^{-\pi x^2/p^2} $$
and $p>0$, I got an answer of $\displaystyle e^{-\pi p^2u^2}$.
I just wanted to make sure I got the right answer. If I didn't, I will work through ...
1
vote
1answer
25 views
Is it true that $\int t\frac{dF}{d \ln{t}} d \ln{t}=\int \frac{dF}{dt} dt$
It seems to be true that:
$$\int t\frac{dF}{d \ln{t}} d \ln{t}=\int \frac{dF}{dt} dt$$
For eg., this works with $\frac{dF}{dt}=\frac{1}{2} (\cos(\pi \ln{t})+1)$
But then there must be something ...
1
vote
2answers
81 views
Exponential equality
How do I can prove that
$$e\ :=\ \lim_{n\rightarrow\infty}\left(1 + \frac{1}{n}\right)^n\ =\ \lim_{n\rightarrow\infty}\sum_{j=1}^{n}\frac{1}{j!},$$
without use of derivatives?
23
votes
8answers
2k views
Do factorials really grow faster than exponential functions?
Having trouble understanding this. Is there anyway to prove it?
28
votes
3answers
827 views
If $e^A$ and $e^B$ commute, do $A$ and $B$ commute?
It is known that if two matrices $A,B \in M_n(\mathbb{C})$ commute, then $e^A$ and $e^B$ commute. Is the converse true?
If $e^A$ and $e^B$ commute, do $A$ and $B$ commute?
Edit: Addionally, what ...
3
votes
2answers
56 views
Inequality with a sum
I am reading Remarks on a Ramsey theory for trees: Janos Pach, Jozsef Solymosi, Gabor Tardos
http://arxiv.org/abs/1107.5301
I am stuck at inequality in proof of Lemma 6.
$n\geq 8$, $k=2\lfloor ...
1
vote
1answer
85 views
Sum the series : $\sum_{n\geq 1} \frac{C(n,0)+C(n,1)+C(n,2)+\cdots+C(n,n)}{P(n,n)}$
Sum the series $$\sum_{n\geq 1} \frac{C(n,0)+C(n,1)+C(n,2)+\cdots+C(n,n)}{P(n,n)}$$
The answer is given as $e^2-1$.
For getting that answer, $C(n,0)+C(n,1)+C(n,2)+\cdots+C(n,n)$ should be equal ...
0
votes
0answers
55 views
Very quick question related to logarithmic spiral bug/mice/dog problem.
http://activityworkshop.net/puzzlesgames/fourdogs/solution.html
Hey guys, I'm sure a lot of you have seen this problem before. I've been working on understanding the solution given here for a while ...
1
vote
1answer
56 views
Non-trivial solution for $2*a^k = b^k + c^k$
I have a data set where I want its median to be the arithmetic average of maximum and minimum by multiplying every value with a factor $k$ and then applying the exponential function. This leads to the ...
0
votes
1answer
15 views
I want to estimate the incremental increase in turnover for an extra product option
For example:
A shop sells sandwiches. They find that if they sell $3$ types of sandwiches they sell more than they would if they only sold $2$ types of sandwiches. However the increase in turnover ...
1
vote
2answers
72 views
Is it possible to have a positive exponential function that starts below zero?
I'm working on a project for my math class. We need to make an image on our calculators (Texas Instruments) using the DrawF function (which graphs functions as ...
0
votes
1answer
27 views
Continuous exponential growth and misleading rate terminology
I'm learning about continuous growth and looking at examples of Continuously Compounded Interest in finance and Uninhibited Growth in biology. While I've gotten a handle on the math, I'm finding some ...
5
votes
2answers
96 views
Help finding inverse of $f(x)=\frac{e^x-e^{-x}}{2}$
I'm trying to find the inverse of $f(x)=\frac{e^x-e^{-x}}{2}$. My textbook says $f^{-1}(x)=\ln(x+\sqrt{x^2+1})$, but I haven't been able to get that answer. Switching $x$ and $y$, I tried solving for ...
2
votes
2answers
106 views
Exponential function formula proof
How does one arrive at $e^4$ from
$$\sum_{x=0}^{\infty}\frac{ 4^x}{x!}$$
3
votes
1answer
101 views
$\iiint e^{-x^2-2y^2-3z^2}dV$
I was given this question in class but I don't understand how to do it.
Evaluate the triple integral in $\mathbb{R}^3$ of $\iiint e^{-x^2-2y^2-3z^2}dV$.
The hint was to use the idea that $\int ...
1
vote
2answers
27 views
Why use exponential equation in a (simple) rate problem?
I have a problem:
In the beginning there were 4. When time equaled 5, there were 20. How many would there be when time equaled 40?
To begin with, I really don't like this problem for its lack of ...
8
votes
4answers
166 views
prove that $\frac{1-e^{-x^2}}{x}\le 2\sqrt{2} , \ x>0$,
Can you show very easy methods? I hope I'll see many methods. Thank you everyone.
Prove that:
$$\frac{1-e^{-x^2}}{x}\le 2\sqrt{2} \ \ \ \qquad \forall x>0.$$
1
vote
1answer
33 views
Stuck on solving for x in exponential to find variance
The problem seems simple:
Let X be an exponential random variable such that $P(X \le 2) = 2P(X > 4)$.
Find the variance of X.
Easy, right?
$
P(x \le 2) = 1 - e^{-2\lambda} $
and
$ P(x > 4) = ...
3
votes
3answers
89 views
Proof of the Series Representation of e…stuck
Bottom Line: Prove that $e = 1+1+\frac{1}{2!}+\cdots$
Define $e = \lim\limits_{n \rightarrow \infty} \left(1+\frac{1}{n}\right)^n$
I would like to do it by expanding $\left(1+\frac{1}{n}\right)^n$ ...
0
votes
0answers
28 views
Normalizing sum of radial basis functions
I am trying to simulate decision making using a sum of 3-dimensional radial basis functions.
I have $n\in [1,\infty]$ radial basis functions like
$f_i(x,y,z) = A_i\ ...
2
votes
2answers
138 views
How to calculate $e^{-0.4}, ~e^{-1.67}, ~e^{-5}, ~e^{0.6},~e^{1.23}, ~\text{and} ~e^{7}$ by hand
How can we calculate $e^{-0.4}, ~e^{-1.67}, ~e^{-5}, ~e^{0.6},~e^{1.23}, ~\text{and} ~e^{7}$ by hand (without using a calculator)?
0
votes
0answers
16 views
Formula for price decrease of a service over the time it takes to complete
A translation job done in $time=1$, costs $55$ per $1800$ characters.
The same translation job done $time=10$ costs $35$ per $1800$ characters.
What is the formula to show this whereby variables ...
1
vote
1answer
20 views
Creating a function $f(x)$ such that $f(0) = 10$ and the instantaneous percentage growth rate of $f(x)$ is a steady $25\%$?
The function I came up with is $10e^{0.25x}$. I took the initial condition of $f(0) = 10$ and placed it into an exponential function:
$$ae^{0.25x} = 10$$
When $x = 0$, $a = 10$ as well.
...
3
votes
2answers
42 views
Complex representation of sinusoids question
A sum of sinusoids defined as:
$$\tag1f(t) = \sum_{n=1}^{N}A\sin(2\pi tn) + B\cos(2\pi tn)$$ is said to be represented as:
$$\tag2f(t) = \sum_{n=-N}^{N}C\cdot e^{i2\pi tn}$$
which is derived from ...
0
votes
1answer
73 views
upper bound for $e^{ax^2}$
I want to find a upper bound for $$e^{ax^2}\leqslant \: ?$$
"a" is a constant and $a\geqslant 0$ .
x is a variable.
I prefer to have a polynomial function or power function (like $ x^{k}$)
is there ...
5
votes
0answers
104 views
integral of the product of a trigonometric and an exponential function
Since tan has an odd power I would normally aim to sub $u=\sec(x)$, but I cant get rid of the $2^x$.
$$\int 2^x \tan^9(x^2)\sec(x^2)dx$$
I also tried integrating by parts but it got more complicated. ...
3
votes
2answers
186 views
Diagonalise a matrix and show the formula
I have diagonlised P to get
$$P=\left(\begin{matrix}
-1 &0 &0\\
0 &0 &0\\
0 &0 &1
\end{matrix}\right)$$
however am unsure on how to proceed, would appreciate any help!
By ...
9
votes
2answers
165 views
Is this a new expression for $e$?
Let $f(x) = x^x$.
Then, let us define a function $p(x)$ such that:
$$p(x) = \frac {f(x+1)}{f(x)} - \frac {f(x)}{f(x-1)}$$
As the value of $x$ increases, $p(x)$ approaches $e$, or (I think),
...
9
votes
3answers
177 views
Prove that $(e+x)^{e-x}>(e-x)^{e+x}$
I get stuck with proving that $$(e+x)^{e-x}>(e-x)^{e+x}$$ for $x \in (0, e)$. All I know, is that it is doable with Jensen inequality, and I started with defining $$f(x)=(e+x)^{e-x}$$ and further ...
2
votes
4answers
140 views
How to use the power series of $e^x$ to find the derivative of $e^x$?
I am currently reading Roger Penrose's The Road to Reality and in the book, the author poses various problems with three different levels of difficultly easy, hard and really hard, according to the ...
0
votes
2answers
34 views
Solving exponential (decay) for x
Well seems like I have a mathematical breakdown at the moment.. But I'm wondering, how CAN you actually solve a function of the form
$y = y_0 + Ae^{Bx} + Ce^{Dx}$
(Where B & D are negative, non ...
3
votes
0answers
68 views
Prove (*) by induction on k.
Challenge: For linear systems with constant coefficients, in some sense we "never need more" than the so-called exponential polynomials, meaning expressions in the form
$$\sum_{i=1}^m ...
1
vote
1answer
73 views
Rate of growth of exponential functions
I have difficulties about proving the following:
Prove that exponential functions $a^n$ have different orders of growth for different
values of base $a>0$.
It looks obvious that when $a=3$ it ...
1
vote
1answer
32 views
Calculate how long it take to reach a goal
Given a growth rate for a period and a goal. How can I calculate how many periods that it will take to reach that goal.
Example:
An investment currently valued at $400 grows at 30% per week.
...
3
votes
2answers
79 views
Computing an integral with exponent and sqrt
Can someone help me to evaluate the integral
$$
\int_{b}^\infty \sqrt{x} e^{-ax}dx.
$$
I guessed that it has no elementary anti-derivative, and indeed substituting $x=t^2$ and then applying ...
1
vote
1answer
78 views
Need help understanding $a^x$ defined as a limit
I'm doing a big fat calculus review, going through Paul Garrett's Calculus Refresher. So far it's very clear and concise, but I just got stuck at one point. He lays out a bit of review on exponents ...
2
votes
1answer
29 views
Determining constant values from 3 equations
I have the following three equations:
$$\begin{align*}
k_1 + k_3 &= 0\\
k_1e^{k_2(0.1)} + k_3 &= 1\\
k_1e^{k_2(1)} + k_3 &= 100
\end{align*}$$
How do you go about solving for values ...
1
vote
2answers
78 views
Why Am I getting wrong answer ( differentiating exponential function)
I have $f(x) = \displaystyle\frac{2e^{2x}}{x^2}$, and I 'm looking for minimums/maximums of the function. Now for some reason, I differentiate and only find $x=1$, while in the solution page it says ...
0
votes
3answers
148 views
What is the name of the function $f(x)=\frac{1}{x}$?
I'm facing this function:
$$f(x)=\frac{1}{x}$$
What I know is that the above equation is one of the simplest forms of "rational functions", where the numerator is $1$ and the denominator is $x$.
Is ...
3
votes
4answers
152 views
Compound interest derivation of $e$
I'm reviewing stats and probability, including Poisson processes, and I came across:
$$e=\displaystyle \lim_{n\rightarrow \infty} \left(1+\frac{1}{n}\right)^n$$
I'd like to understand this more fully, ...
5
votes
3answers
79 views
Stuck on this integral involving exp and the floor function
Here is the integral
$$\int_0^\infty \lfloor x \rfloor e^{-x}dx$$
Here is what I have so far:
$$I = \sum_{n=0}^\infty \int_n^{n+1} n e^{-x}dx$$
$$ = \sum_{n=0}^\infty -ne^{-n-1} + ne^{-n}$$
$$ = ...
3
votes
4answers
92 views
$X$ is $\text{Exp}(\lambda)$, $Y$ is $\text{Uniform}(0,X)$. How can I find $\Bbb E[Y]$ and $\text{Var}(Y)$?
$X$ is $\text{Exp}(\lambda)$, $Y$ is $\text{Uniform}(0,X)$. How can I find $\Bbb E[Y]$ and $\text{Var}(Y)$?
I did tried to plug it like double integral of $\Bbb E[Y]$ from 0 to X which $f(t)$ is ...
1
vote
2answers
93 views
Entire function which equals exponential on real axis
I need to find all entire functions $f$ such that $f(x) = e^x$ on $\mathbb{R}$.
At first it seems that, since the function $f$ will be real analytic on $\mathbb{R}$ and will have a power series ...
