For question involving exponential functions and questions on exponential growth or decay.
1
vote
0answers
23 views
How to solve polynomial-exponential equation
I'm trying to solve equations like the following one:
$$5 + 3x - 4x^3 = e^{x^2}$$
I've tried using the Lambert W function, but I didn't get any success. I must admit I'm relatively new to Lambert W ...
4
votes
0answers
29 views
Need help with $\int_0^2\frac{1}{2+\sqrt{3\,e^x+3\,e^{-x}-2}}dx$
Could you please help me to solve this integration problem?
$$\int_0^2\frac{1}{2+\sqrt{3\,e^x+3\,e^{-x}-2}}dx$$
Its approximate numeric value is $0.419197813818367...$, but I could not find an exact ...
25
votes
18answers
2k views
Simplest or nicest proof that $1+x \le e^x$
The elementary but very useful inequality that $1+x \le e^x$ for all real $x$ has a number of different proofs, some of which can be found online. But is there a particularly slick, intuitive or ...
0
votes
1answer
32 views
Integrate Exponential function with Integration Variable e (de)
I have the following problem: How do i integrate this: $ \int^1_0 e^{x^2+3 } de $ ?
Whats putting me off is the de, i don't know how to integrate over the e-function...
According to Wolfram Alpha, ...
0
votes
2answers
36 views
Taking derivatives of exponential function
Beware, this question might be silly and may contain mathematical fallacies.
$$
d/dt(e^{jwt}) = jwe^{jwt}
$$
$$
d/dt(e^{j \pi t}) = j \pi e^{j \pi t}
$$
$$
d/dt(e^{j 180 t}) = j 180 e^{j 180 t}
$$
...
3
votes
1answer
39 views
Easy proof that $\exp{Xt} = I \Rightarrow X = 0$
Let $X\in \mathbb{C}^{n\times n}$ and $I$ is identity matrix , than if:
$$
\forall t\in \mathbb{R}\quad e^{Xt} = I
$$
than
$$
X = 0.
$$
I'm looking for short and slick proof of this ...
3
votes
1answer
23 views
Working out an exponentially increasing value without a function
This may seem very basic, but I am stumped by this. I have been set this by my Math tutor and I have no idea what to do. I have checked everywhere, but I am pretty sure my tutor made these set of ...
0
votes
0answers
21 views
RSA finding message m
In regards to RSA,
If there exists the numbers...
$P=101$
Encryption exponent = 67
$36= m^e \pmod{P}$
Can you recover $m$ from just $P$ and $E$?
-1
votes
1answer
92 views
Relation between e and pi [on hold]
I found the following relation
$\pi=3+\frac{1}{5+\frac{1}{7+\frac{1}{9+\dotsb}}}$ known
and
$e=3-\frac{1}{5-\frac{1}{7-\frac{1}{9-\dotsb}}}$
Can we relate these directly?
0
votes
1answer
47 views
Relation of $e$ to other numbers…
I found the following result,
When i was working on my calculator
.
$$x^y < y^x \quad ,x < y \quad \text{ for } x,y<e$$
$$x^y > y^x \quad ,x < y \quad \text{ for } x,y>e$$
I can't ...
0
votes
1answer
16 views
Determining the parts of an exponential equation based on the graph.
I am given the following equation and asked to find $a$
$$s(t) = Ae^{-at}.$$
I know $A = 9$ and I know at $t = 1, s(1) = 3$ so I create the following equation
$$3 = 9e^{-a.1}$$
and reduce it down ...
1
vote
1answer
15 views
Simplification of an exponential equation.
I am currently in a class on sinusoids and exponential functions (that is confusing the hell out of me...).
My professor has the follow equation (I believe it is the general equaltion for ...
1
vote
2answers
54 views
How much caffeine is left in your system?
The half-life of caffeine in your system is 6 hours. If you drink 100mg of caffeine every day, what percentage is left in your system after a week?
If you drink 100mg caffeine every day for a month ...
0
votes
1answer
43 views
Exponential Equation $p \cdot a^x - q \cdot b^x \le c$
I have a hard exponential equation to solve:
Given $p,q,a,b,c$, I need to solve $x$ in the following equation:
$p \cdot a^x - q \cdot b^x \le c$.
Any suggestion is welcome. Thanks.
4
votes
1answer
56 views
Gamma Type Integral
I was hoping someone could help me with a question I came across recently: essentially it's a gamma type integral that your asked to evaluate/reduce:
...
0
votes
1answer
87 views
How to put exp function back in the integral?
Thanks to everybody in advance. I have this integral:
$$
e^{-at} \int_{0}^{t} e^{au}f(u) du
$$
where f(t) is something I can't integrate.
Is there a way to put the exponential function back in the ...
2
votes
1answer
108 views
Reflecting an exponential function over a y = 3 line.
How would you write the equation of $f(x) = 4^x$ that reflects over the line $y = 3$?
I've put in $f(x) = 3 + 4^{-x}$ which I thought was the right answer, but it isn't.
Thanks in advance!
0
votes
1answer
188 views
Equation containing modified bessel functions and exponential function
I'm trying to find a approximation solution for the following equation:
${e^{ - x}}\left[ {{I_o}\left( x \right) + {I_1}\left( x \right)} \right] = C$
where $I_0$ and $I_1$ is the modified Bessel ...
3
votes
0answers
42 views
How quickly can one compare exp(m/n) to a given rational?
For positive integers $\hspace{.06 in}m_{\hspace{.02 in}0}\hspace{.02 in},n_0\hspace{.02 in},m_1,n_1\:$, $\;$ how difficult is it to decide whether $$\exp\left(\hspace{-0.03 in}\frac{m_{\hspace{.02 ...
0
votes
1answer
149 views
matrix exponential is a rational or not?
I want to know whether following are true or false:
for any given natural number $n$, $T>0$ a rational, suppose that $Q_1, \cdots, Q_n$ are $m\times m$ matrices with rational entries, $t_1, ...
0
votes
0answers
24 views
splitting up summations of product
I want to check the validity of splitting up summations of products. I am using the DFT matrix and am trying to get a simplified expression of it . In essence, I am trying to prove the following lemma
...
2
votes
1answer
23 views
Exponential Function Shifts
I have some confusion about shifts concerning exponential functions. I can best describe my question with an example.
Take y = e^-(x-3). This graph has a reflection over the y-axis and is shifted ...
0
votes
0answers
14 views
Solution for a Mixture of Two Exponential Equations
I have a survival function that is a combination of two exponential functions. I would like to solve for the median survival time (ie when half the population is dead).
$$0.5 = \left( A(0) + A(0) ...
1
vote
2answers
32 views
Exponential Decay Help
Hi I don't mean to sound like a clueless student who should already know this subject but I am really stumped on the topic of exponential growth and decay. Namely this one question:
"When zombies ...
0
votes
0answers
16 views
Triple Exponential Smoothing Question
This website http://www.itl.nist.gov/div898/handbook/pmc/section4/pmc436.htm
seems to propose following:
L = season.length
mix = (x, y, a) -> x*(1-a) + y*a
...
1
vote
1answer
17 views
Exponential growth - apparently not getting something
"A certain bacteria population is known to quadruple every $90$ minutes. Suppose that there are initially $120$ bacteria.
What is the size of the population after $t$ hours?"
I've been using this ...
0
votes
1answer
20 views
Growth rate of $n^2$ vs $(\log_3(n))^3$
Which grows faster, $n^2$ or $(\log_3(n))^3$? How do I figure out which grows faster in general in these kinds of situations?
1
vote
1answer
41 views
Solution to Singular Integral
Does anybody know how to solve the following singular integral:
$$
\int_0^\pi \frac{\text{e}^{-i\cdot a\cdot \cos(x)}}{\cos(x)-b}\,\text{d}x
$$
with $x$, $a$ and $b$ being real-valued.
Do integrals ...
3
votes
2answers
124 views
The equation $x^y = y^x$.
The following equation has been of interest for a long time. I'm bringing this to your attention again:
$$x^y = y^x$$
How do you attack to find all real and imaginary solutions for this equation? or ...
1
vote
2answers
69 views
when does $f(a)^{f(b)}=f(a^b)$?
First $\text{f}\left( 1 \right)=1$ beacause
$\text{f}\left( a \right)^{\text{f}\left( 1 \right)}=\text{f}\left( a \cdot 1 \right)$, and
$\log_{\text{f}\left( a \right)} \text{f}\left( a ...
