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

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2answers
24 views

Find the inverse of the function

Find the inverse of the function $f(x) = -2 \cdot4^{2(x-3)} - 1$.
0
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1answer
49 views

Proof of simple interest formula

Can someone please prove to me that $I = PRT$, where $P$ is the principal, $R$ is the interest rate, and $T$ is the number of years/time. I have seen $I = P(1+TR) = P+PTR$ which does not equal $PRT$, ...
2
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0answers
41 views

Lie group - exponential Diffeomorphism

Let $G$ be a nilpotent, connected simply connected Lie group and $\mathfrak{g}$ its Lie algebra. It is known that the exponential map $\exp$ is a diffeomorphism. Now let $\mathfrak{g}_0$ be a Lie ...
2
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3answers
37 views

Programming Help - Solving for e(n)

I've been wrestling with this issue for a week and I just need some guidance on the math part of it. If I could just understand the math behind it I could piece together the functions to make it ...
1
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3answers
139 views

Prove that $f(x)=(e^x-1)(x^2+3x-2)+x$ has exactly one positive root, exactly one negative root and one root at $x=0$.

Prove that the function $f(x)=(e^x-1)(x^2+3x-2)+x$ has exactly one positive root, exactly one negative root and one root at $x=0$. My work so far: $f(0)=0$ Thus, $x=0$ is a root. For the ...
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0answers
22 views

What's the MLE of $\frac1\lambda$ for $f(x)=\frac1\lambda\exp − \frac x\lambda$?

What's the MLE of $\frac{1}{\lambda}$ for $f(x)=\frac{1}{\lambda}\exp \left(− \frac{x}{\lambda}\right)$? Is it equal to the mean of $x$ or the inverse of mean of $x$? Thanks,
-3
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1answer
31 views

Can I have an exponential function such that if x = infinity, y = 100?

I tried the most basic y = 100*constant^(1/x) assuming that 1/x = 0 when x is infinity, but it doesn't seem to work. This gives me a function that starts with a higher value of y and goes down till ...
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3answers
73 views

Prove that $e^x \gt 0$ for $x \in \mathbb{R}$ [duplicate]

This is a consequence of the exponential rule, but how do I actually prove it to be true?
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2answers
45 views

Expressing the sine function in terms of exponential

Prove $e^{iz} - e^{-iz} = \sin z$. I used $$\begin{align*} \sin z & = z - z^3/3! + z^5/5! - z^7/7! + \dots & (i) \\ e^{iz} & = 1 - z^2/ 2! - iz^3/3! + \dots & (ii) \\ e^{-iz} ...
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0answers
31 views

Continuous exponential functions

In my book, it makes it appear that any continuous exponential function, such as those regarding money, do not follow the traditional formula of $$\text{growth} = (1+\text{return})^x $$ Rather, it ...
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0answers
179 views

Laurent Series and Taylor Expansion of $ 1 / (e^z - 1) $

Could someone please assist me with the second part of the second paragraph, from "By expanding $f_1$..."? I am not convinced that my expansion for $f_1$ is right - I used the standard binomial, ...
1
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1answer
55 views

What's the MLE of lambda for $f(x)= \frac{1}{λ}\exp{\frac{−x}{λ}}$?

What's the MLE of $\lambda$ for $$f(x)= \frac{1}{λ}\exp\left({\frac{−x}{λ}}\right)$$ Values of x are 5,7,9,3,6,8 Is it just the mean of $x$? Thanks.
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0answers
40 views

What is the product of bessel functions of first and second kind when their arguments are same and tends to zero?

As we know, $\lim_{x \to 0} J_m(x)=0$ where $m\geq 1$ and $\lim_{x \to 0} Y_m(x)=\infty$ then what would be $\lim_{x \to 0}J_m(x)Y_m(x)$. Matlab shows the product is finite and $< 1$. What should I ...
1
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1answer
41 views

problem about population growth

At the beginning of the Gold Rush, the population of Coyote Gulch,Arizona was $365$.From then on ,the population would have grown by a factor of $e$ each year,except for the high rate of ...
0
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1answer
34 views

Find $t$ in $N = b \times g^t$.

The problem is the following: Find the value of $t$ in $N = b × g^t$. So for example "$512.000 = 2000 × 2^t$" I'm not really a mathematician so their may be a simple way or it could be hard.
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0answers
70 views

A question about exponential matrices

So here is my question, I would like to prove, If $R,S\in \mathcal M_{n\times n}(\mathbb R)$ are matrices such that, $$e^{t(R+S)}=e^{tR}e^{tS},\;\forall t\in\mathbb R$$ Then, $$RS=SR$$ And here is ...
1
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2answers
23 views

How do I find the inverse of this exponential function?

$x=-3(3^{-x})+9$ I know the steps up until a certain point. $x=-3(3^{-y})+9$ $x-9=-3(3^{-y})$ $\frac{(x-9)}{-3} = 3^y$ $ln (\frac{x-9}{-3}) = -y * ln 3$ Not sure what to do from here. I know I ...
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0answers
27 views

Generic Exponential curve base derivation

Alrighty so I am working on a computer program that forms ADSR envelopes including exponential curves for the attack, decay, and release segments. It uses the following equation for the exponential ...
-1
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2answers
32 views

Limit of n * ln(1+x/n)

How can you compute with the most primitive tools that: $$ \lim_{\stackrel{n \to \infty}{n > -x}}n \:\ln (1+\frac{x}{n})=x $$ Using l'hospital verifies this. However we hadn't proofed ...
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2answers
55 views

An exponential/polynomial inequality

Prove that there is at least $1$ real number $a>0$ with the property $$a^x\ge x^a $$ for any $x>0$.
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1answer
36 views

Prove this logarithm equation

I keep getting the wrong answer. Can someone please correct my working out a^x=b^(1-x) In(a)^x=In(b)^(1-x) xIn(a)=(1-x)In(b) xIn(a)=In(e)-xIn(b) xIn(a)+xIn(b)=In(e) x[In(a)+In(b)]=Ine ...
2
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2answers
42 views

why is $\lim_{t \to \infty}t^m e^{-\alpha t} = 0$ for every $m \in \mathbb{N}$ fixed and $\alpha \in \mathbb{C}$ with $Re(\alpha) \gt 0$?

I'm having trouble trying proving this fact: $\lim_{t \to \infty}t^m e^{-\alpha t} = 0$ for every $m \in \mathbb{N}$ fixed and $\alpha \in \mathbb{C}$ with $Re(\alpha) \gt 0$ I tried to use ...
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1answer
47 views

Inverse Function of Logarithm

The answer is A but I don't understand why! $ -2 \log_e (x^2) $ can be re-written as $ -4 \log_e(x) $ right? but why do these two graphs look different? the graph $-2 \log_e (x^2) $ is one to ...
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1answer
17 views

Maximum value of constant in logarithm problem

The first thing I did was: make: (x-1)^2 - k > 0 (x-1)^2 > k don't know what to do after this point... the maximum value of k is 9 i dont really understand what the maximum value of k is? ...
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0answers
33 views

Interchanging from exponential form to log form

Shouldn't the answer be x = loge(everything else in the bracket) why is the loge function divided by "k" ???
2
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5answers
413 views

Which is actually exponential?

I've heard the term "exponential" applied to two sorts of functions: $$n^x\text{, where $n$ is a constant (e.g., $2^x$)}$$ and $$x^2$$ Which is really exponential, and what do I call the other ...
0
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1answer
35 views

First order ODE with $f'(x) = 810(10)^x$

I'm trying to find an explicit form of the series $f(0) = 89.1,f(1) = 899.1,f(2) = 8999.1, \cdots$. My first though was to take the derivative and integrate it, which I've done before with a fair ...
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3answers
75 views

Calculate exponential decay

I know this has been asked a few times before, but I'm struggling to apply it to my scenario... I have some known values in a table: $$ \begin{array}{|c|c|} \hline \text{Degrees} & ...
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1answer
40 views

Length of a very basic exponential curve

I have the beginning points (0,1) and end points (180, 141.732) of a curve. The function I am currently using is f(x) = Ae^kx. However, when deriving the original function, I end up with 0 (from ...
2
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0answers
31 views

Showing the exponential and logarithmic functions are unique in satisfying their properties

The question asks to prove that there exists a unique function defined on $\Bbb R$ and satisfying the following conditions: 1) $f(1) = a$ $(a>0, a \neq 0)$ 2) $f(x_1) \cdot f(x_2) = f(x_1 + ...
2
votes
1answer
36 views

Evaluating a limit with two steps - Right/Legal?

$$\eqalign{ & \mathop {\lim }\limits_{n \to \infty } {\left( {{{4{n^2}} \over {(2n + 1)(2n - 1)}}} \right)^{1 - {n^2}}} = \mathop {\lim }\limits_{n \to \infty } {\left( {{1 \over {{{(2n + 1)(2n ...
0
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1answer
19 views

Determining the gradient of a decaying function

I have data that can be fitted using an exponentially decaying function: $y = e^{-t/\tau}$ and I want to determine the value of $\tau$. I see that if I make the t-axis logarithmic I get a straight ...
1
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2answers
332 views

Problem Solving ( Sequences and series)

After injection of a dose $D$ of insulin, the concentration of insulin in a patient's system decays exponentially and so it can be written as $D\exp^{-at}$ where $t$ represents time in hours and $a$ ...
0
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0answers
67 views

Expected waiting time of the process in the queue given that the process is served

Consider the following situation: There is one server with exponential service time with parameter $\lambda$. One process is waiting in the queue. The waiting time is exponential with parameter ...
0
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1answer
24 views

solving exponential functions

$(27^{x - 1})(3^x) = 9^{2x-3}$. I apologize if you do not understand the equation. I was unsure on how exactly to represent it correctly. I have gotten to the step in the equation where it is ...
1
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1answer
38 views

derivative of a definite integral with base e

$$\frac{d}{dx} \int_3^{x^2} e^{t^3} dt$$ I can sorta figure out how to solve problems like this, if it was an indefinite integral...
1
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1answer
26 views

Sketching Logs with Quadratic Terms

$\log(x^2+1) = y$ asymptote at $x^2+1 > 0$ and so there is no asymptote $x$ and $y$ intercept at $(0,0)$ How do you know that the function goes both directions, and has a dip in the middle? ...
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1answer
40 views

Can the variables of $y = A + B \mathrm{e}^{C t}$ be solved analytically given 3 sets of points?

Given the non-linear equation $y = A + B \mathrm{e}^{C t}$ and 3 sets of points: ($y_1$, $t_1$), ($y_2$, $t_2$), ($y_3$, $t_3$), can the variables $A$, $B$, and $C$ be calculated analytically? ...
0
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3answers
100 views

Continuity proof for exponential

Show that $f(x) = e^x$ is continuous using the epsilon-delta definition. I can't seem to write down anything meaningful...
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1answer
36 views

Find the inverse function about a exponential related function

Here is the function:$$y = 4x + {x^m},where{\text{ 0 < m}} \leqslant {\text{1;}}$$ Approximately results is acceptable.
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4answers
350 views

Absolute value of complex exponential

Can something explain to me why the absolute value of a complex exponential is 1? (Or at least that's what my textbook is telling me.) For example: $$|e^{-2i}|=1, i=\sqrt {-1}$$
8
votes
3answers
122 views

Showing $n!<e(\frac{n}{2})^n$

I'd like to prove that $n!<e(\frac{n}{2})^n$. What I have so far: $\sqrt[n]{n!} = \sqrt[n]{1\cdot 2 \cdot \ldots \cdot n} \leq \frac{1+\ldots +n}{n}=\frac{(n+1)n}{2n}=\frac{(n+1)}{2}$. Thus ...
1
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1answer
60 views

Product of exponential distributions

Suppose $X_1$ is $\mathrm{Exp}(\lambda_1)$ and $X_2$ is $\mathrm{Exp}(\lambda_2)$. $X_1$ and $X_2$ are independent. Let $Y = \min (X_1, X_2)$ and $Z = \max (X_1, X_2)$ and $W = ZY$ . Compute the ...
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2answers
44 views

Solving an equation with both linear and exponential terms

Can I find an algebraic solution for the equation below? Thank you. $$ x+e^{x}(x+a)=b $$
2
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1answer
26 views

n events of one process occuring before m events of another process

Assume that you have two independent Poisson processes, N1( t ) with rate λ1 and N2( t ) with rate λ2 . What is the probability that n events occur in the first process before m events occur in the ...
1
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1answer
33 views

Problems with another characterization of exponential functions

As in other two of my questions, which are already answered by myself, I am treating exponential function again. Now, from the perspective of continuity only. These means, I can not use any single ...
7
votes
5answers
140 views

If $\frac{d}{dx}e{^x} = e{^x}$, then why does $\frac{d}{dx}e^{-14}$ = 0?

If $\frac{d}{dx}e{^x} = e{^x}$, then why does $\frac{d}{dx}e^{-14}$ = 0? Why doesn't $\frac{d}{dx}e^{-14}$ = $e^{-14}$? I don't understand.
4
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3answers
270 views

Find a closed form from the given power series

I have the power series $\sum_{n=0}^{\infty} {z^{2n}\over{n!}}$, how do I find the closed form for this power series. I am aware that $e^z=\sum_{n=0}^{\infty} {z^{n}\over{n!}}$, so I tried to ...
1
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2answers
104 views

Exponential pop. growth when only given population at two instances of time.

I have a problem where I'm only given the population of a "bacteria culture" at two instances in time: 2 hours and 4 hours. The problem says the population of bacteria is 125 after 2 hours, and 350 ...
2
votes
4answers
203 views

Does the series $\,\displaystyle\sum_{n = 1}^{\infty}\left(2^{1/n} - 1\right)\,$ converge?

I'm trying to determine if the following sum converges or diverges (this is question 38 in section 11.7 of Stewart's Early Transcendentals): $$\sum_{n = 1}^{\infty}(2^{1/n} - 1)$$ I've considered ...