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

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1
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2answers
31 views

Find when the population is growing the fastest, under the logistic model

The population $P$ of an island $y$ years after colonization is given by the function: $\displaystyle P = \frac{250}{1 + 4e^{-0.01y}}$. After how many years was the population growing the fastest? ...
4
votes
2answers
60 views

Proving that $3^n<n!$ when $n\geq 7$

It's been 10 years since my last math class so I'm very rusty. How would I go about proving $$3^n < n!$$ where $n \geq 7$? I understand that factorials grow faster than set values with a variable ...
0
votes
0answers
11 views

Shifting a series of functions whilst maintaining symmetry

I have a function y = a*Exp[-(x - b)^2/2*c^2] + d*Exp[-Abs[-e*x]] + f Which is symmetrical when the coefficient of b is equal to 0 however it loses symmetry as ...
0
votes
1answer
38 views

Integral - complex exp. term

Does anyone know a suitable method to integrate and/or know the answer to: $\int\limits_{-\pi}^{\pi}$ $\log\Big[\tfrac{2 - a\exp({-it})}{1 - a\exp({-it})}\Big] $ ${\mathrm{d}t}$, for constant $|a|$ ...
1
vote
1answer
25 views

Linearizing an expression involving exponentials

How can I linearize $f(x) = A(1-\text{exp}(Bx))$? I tried to take the natural logaritm, but could not find something that looks like linear. I am trying to find a fitting curve for this by hand. $A$ ...
0
votes
0answers
20 views

How are Compound Interest and Infinite Series related?

The mathematical constant e happens to be both $\lim_{n\to\infty}(1 + \frac{1}{n})^n$ and $1 + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + ..$. The 1st formula is of compound interest with 100% ...
2
votes
1answer
37 views

Taylor series of f(x + a) becomes exponential

In my symmetries of classical mechanics course we have looked at taylor expansions. Our notes claim that; $$ f(x + a) = \sum_{n=0}^\infty \frac{1}{n!} f^{(n)}(x)a^n ≡ \exp{\left( a ...
5
votes
9answers
111 views

If $a_n = \frac{e^{n}}{e^{2n}-1}$ how do I show that $a_{n+1} \leq a_n$?

Let $$a_n = \frac{e^{n}}{e^{2n}-1}$$ How do I show that $a_{n+1} \leq a_n$? I don't know how to deal with the $-1$ in the denominator.
-5
votes
1answer
35 views

Oil decay at 13%, how long until it is less than 21% of original?

My teacher gave me this problem, and it is very wordy, I don't really even understand what it is asking. First I took 100 and multiplied it by 0.13 subtracting that number from 100 and completing the ...
-1
votes
1answer
34 views

Exponential Function Equation and inverse Pre-Cal

I am not completely sure if I wrote the equation correctly. For A I wrote: $m(t)=100(b^x)$ Not sure it is correct...but how do I find the inverse? That doesn't make sense to me. Do I use log?
3
votes
2answers
37 views

Derivative of Matrix Exponential as Integral

I saw this "standard" identity in a physics paper and I was wondering how to prove it \begin{align*} \frac{d}{dx} e^{A+xB}\bigg|_{x = 0} = e^A\int_0^1 e^{A\tau}B e^{-A\tau}\,d\tau \end{align*} I tried ...
0
votes
0answers
30 views

Help understanding exponential formula

I am reading a paper in which a group is approximating data that fits an exponentially declining curve. They use the following formula to fit the data, where τ is the y-axis variable and v is the x ...
0
votes
1answer
34 views

Derivative of an expodential function

Let the function: \begin{equation} \phi(x)=\begin{cases} e^{-\frac{1}{1-x^2}}, & x\in (-1,1)\\ 0, &x \not \in (-1,1)\end{cases} \end{equation} How can I show that $\phi(x)$ is differentiable ...
0
votes
0answers
14 views

Characteristic function of an asymmetric Laplace distributed random variable

What is the characteristic function of a random variable with density $$f_X(x) = \frac{1}{2} [ 1_{x>0} \, a e^{-a x} + 1_{x<0} \, b e^{b x} ], \; \; \; \quad a,b > 0 \quad \quad ? $$ My ...
1
vote
1answer
52 views

Proving that the matrix exponential map is surjective onto the general linear group

Let $M_n(\mathbb{F})$ be the set of all $n\times n$ with entries in $\mathbb{F}$ and let $\exp:M_n(\mathbb{C})\to M_n(\mathbb{C})$ be defined by $$ \exp(A)=\sum_{k=0}^{\infty}\frac{A^k}{k!},$$ for ...
2
votes
1answer
37 views

How to prove the limit of “the exponential of a sequence”

So given a convergent sequence $\{a_n\}_{n=1}^\infty$ with limit $a$, I'd like to prove that $$\lim_{n\to\infty} \left(1+\frac{a_n}{n}\right)^n=e^a.\quad(1)$$ Knowing that $e$ is defined by ...
0
votes
1answer
34 views

Is the derivative of a exponential function a^x always greater than the derivative of a polynomial x^n as x approaches infinity

with n and a being any constants > than 1. I have tried taking the $\lim\limits_{x \to \infty} a^x / x^n$, and l'hopitals is telling me than $x^n$ can always be reduced to 1 with multiple iterations, ...
-2
votes
1answer
53 views

How do I calculate these limits? [duplicate]

How would I go about calculating $$\lim_{n\to\infty}\frac{\left(1 + \frac11\right)^1 + \left(1 + \frac12\right)^2 + \left(1 + \frac13\right)^3 + \cdots + \left(1 + \frac1n\right)^n}n$$ and ...
1
vote
2answers
21 views

Determining half life without logs, given only reduction undergone and total time taken

I have a half-life question that I can't solve. There's very limited information given. Even the half-life formula has not been taught yet. The mass of a radioactive substance in a certain sample ...
0
votes
0answers
34 views

exponential and linear nature in one equation

Please accept my apology in advance as i am not very good in math. I am looking for equation for my simulation that gives the exponential behavior in the initial x-axis points and turned to linear ...
0
votes
1answer
38 views

The existence of anti-derivatives

The only thing I can think of is that the function is continuous hence the anti derivative exists. I was wondering if there is anything else that needs to be done/said?
0
votes
0answers
28 views

strong convex implies exp-concave

Prove that if f is strong convex (for some m>0) $\mbox(\nabla f(\mathbf{x})-\nabla f(\mathbf{y}))^{T}(\mathbf{x}-\mathbf{y})\geq m||\mathbf{x}-\mathbf{y}||_{2}^{2} $ then f is also ...
0
votes
1answer
12 views

Reasoning behind method of steepest descent

I am considering the method of steepest descent from my notes. I have written that $$\int_a^b dx e^{g(x)} \sim e^{g(x_0)} \int_{\infty}^{\infty}dx \exp \left[-\frac{1}{2}(x-x_0)^2|g^"(x_0)|\right] ...
4
votes
1answer
87 views

If $\frac{x-1}{e^x-1} = y$ then $x=?$

I have following equation: $$\frac{x-1}{e^x-1} = y$$ I want to solve this equation such that I have the value of $x$ in the term of $y.$ i.e. inverse of the equation
0
votes
1answer
30 views

Transpose exponential equation [closed]

Could somebody please help with transposing the following equation to isolate x to the left side of the equation to solve for x? $$ y = 10^{1.830 \log(x)} + 2.686 $$
6
votes
4answers
507 views

How do pocket calculators calculate exponents?

I'd like to know specifically how a pocket calculator (TI calculators also apply) calculates $e^{0.1}$, and what methods or algorithms pocket calculators use in order to produce their answer.
2
votes
2answers
62 views

Solving equations of the form $ae^x + bx +c =0$

In a recent piece of homework I needed to solve an equation of the form $ae^x + bx +c = 0$ where $a,b$ and $c$ are constants. I could not do it; no matter how I tried I either went in circles or hit a ...
1
vote
2answers
24 views

Find the distribution function of bivariate distribution

Find the distribution function of $$f_{X,Y}(x,y)=\begin{cases} e^{-y}, & \text{if $0< x<y < \infty$} \\ 0, & \text{ otherwise} \end{cases}$$ Trial : According to my calculation ...
0
votes
2answers
18 views

Partial Derivative of Exponential Quotient

How would I go about finding the partial derivative with respect to $y$ of $z = (x^2/(1-y^3))^{0.5}$ The way I thought to do it was to get rid off the brackets and square root, making ...
0
votes
1answer
38 views

How to find out b's value in e^(bx)?

It is killing me......I can not get this right! The answer is A. Can anyone please help!
2
votes
3answers
20 views

Geometric Series: Calculating the drug level prior to a maximum dosage.

A dose of $D$ milligrams of a drug is taken every 12 hours. Assume that the drug's half-life is such that every $12$ hours a fraction $r$, with $0<r<1$ of the drug remains in the blood. Let ...
1
vote
2answers
36 views

How do you find the general expression for the k^{th} derivative of an exponetial function with a function in the exponent?

I'm looking for a general expression for the function $\frac{\delta^k}{\delta \mu^k}[e^{n\mu + \mu^2}]_{\mu=0}$ I was thinking I could use the taylor expansion coefficients, but the function in the ...
2
votes
1answer
25 views

Complex modulus Inequality using $|exp(z)-1|$

I think I am almost there: Prove $\left|z\right|/4 < \left|\exp(z)-1\right|<7\left|z\right|/4$ for all $0<|z|<1$. MY ADVANCES First we note that $$ \left|\exp(z)-1\right| = ...
1
vote
1answer
21 views

Integral using complex numbers shortcut

I want to compute the following integral $$- \frac{1}{M(\lambda_1-\lambda_2)}\int\limits_{-\infty}^t(e^{\lambda_1(t-t')}-e^{\lambda_2(t-t')})(\beta\omega A\sin\omega t' +g)\;dt'$$ Here the integral ...
2
votes
0answers
34 views

Interpreting and understanding the identity $e^{iz} = \cos(z) \pm \sqrt{\cos^2(z) - 1}$

A question in my complex analysis book (Gamelin's "Complex Analysis", question I.8.7) asks me to prove that $e^{iz} = \cos(z) \pm \sqrt{\cos^2(z) - 1}$. Using the identity $\cos(z) = \frac{e^{iz} + ...
1
vote
1answer
17 views

Limit at $\infty$ of a polynomial multiplied by a negative exponential

I am trying to show $\int_0^{\infty} x^2 e^{-2 x} dx = 1/4 $ Integration by parts gets the indefinite integral $$\int x^2 e^{-2 x} dx = \frac{-1}{4} e^{-2 x} (2 x^2+2 x+1)+constant$$ In order to ...
1
vote
1answer
27 views

Help finding the inverse of an exponential function

$$f(x)=6^{3x+9}-2.$$ I got to one step, but I became lost. I understand that I'm converting it into logarithmic form, but I don't understand what the next steps are. \begin{align*} ...
4
votes
2answers
282 views

How is this definite integral solved: $\int_{-\sqrt3}^{\sqrt3}{e^x\over(e^x+1)(x^2+1)}dx $?

$$\int_{-\sqrt3}^{\sqrt3}{e^x\over(e^x+1)(x^2+1)}dx $$ Tried partially integrating, had no luck.. Any thoughts?
3
votes
1answer
32 views

Definite integral of exponential function

I'd like to get some feedback on an integral calculation, if anyone might spot something wrong with my work: $$\int_4^9{\frac{2e^{\sqrt{x}}}{\sqrt{x}}}\,\,dx$$ Using substitution, let $$u = ...
0
votes
2answers
72 views

What is the Derivative of x^x [duplicate]

I was browsing through my old textbook and I found this problem: Find Derivative of $ x^x$ My work : Haven't got a clue yet, where to start?
2
votes
3answers
100 views

How could this be true $n=\log(e^n)$?

I am learning elementary logarithms. How could this be true $n=\log(e^n)$? I searched online and couldn't find any info on this, could anyone give me some clue?
2
votes
2answers
43 views

Differentiating exponential functions - is base e the only situation?

My maths book gives the example of; Where $$ f = e^x $$ $$ f` = e^x $$ It only uses the example of base e in all of the questions so does that mean this is the only situation where the differential ...
0
votes
1answer
27 views

Issue with Compound Interest

Here is the question: Noha is investing ${$}2517$ in an account compounded monthly. She wants to have ${$}3000$ in $3$ years for a trip to Europe. What interest rate, to the nearest hundredth of a ...
-3
votes
3answers
63 views

Solve for X in a difficult exponential function [closed]

Solve for $X$ when $3^{x^x}=1000$ By hand please (without evaluating the intersection on the graph). How is it done?
1
vote
1answer
32 views

integral formulas for sinus and cosinus

Consider the formulas $$\int_0^{2π} \cos(nx) \cos(mx)\, dx = 0$$ and $$\int_0^{2π} \sin(nx) \sin(mx) \, dx = 0$$ for $m, n \in \mathbb{N_0}, m ≠ n$. These formulas can be proven when given the ...
22
votes
7answers
3k views

Is $\exp(x)$ the same as $e^x$?

For homework I have to find the derivative of $\text {exp}(6x^5+4x^3)$ but I am not sure if this is equivalent to $e^{6x^5+4x^3}$ If there is a difference, what do I do to calculate the derivative of ...
2
votes
5answers
84 views

How to properly solve this inequality $2^x < \frac{3}{4}$?

How to properly solve this inequality? $$2^x < \frac{3}{4}$$ I know that it will be something like that: $$ x \stackrel{?}{\ldots} \log_2\frac{3}{4} $$ But I don't know how to decide if it should ...
1
vote
2answers
78 views

Minimum of $x^x$

So the title says almost everything. Why is the minimum of the function $x\mapsto x^x$ exactly at $x= e^{-1}$? Don't get me wrong. I am able to write $x^x$ as $\exp(x \ln(x))$ myself and can ...
4
votes
3answers
112 views

$F(x)+G(y)= e^{x+y}?$

Are there functions $F(x)$, $G(y)$, such that $F(x)+G(y)=e^{x+y}$ , where $x,y$ are real numbers? I have been trying all elementary functions, and have no clues on what else I could do.
0
votes
1answer
57 views

Integration of $\int\frac{e^{x}}{\sqrt{x}} dx $

How do I integrate $\int\frac{e^{x}}{\sqrt{x}} dx $ I can't think of a suitable substitution. Is it even possible?