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

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0
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0answers
13 views

find angular velocity for so that: $\exp(jt) = \exp( j(3t+\pi/3) )$ [on hold]

I have a fourier series in which there are two different arguments on the exponential function: $jt$ and $j(3t+\pi/3)$ and I have to "choose" a fitting angular velocity. It it probably easy yet it ...
0
votes
1answer
20 views

How do I write this complex number in exponential form?

$$ -4 - i 16\sqrt{5}$$ Example: I know we can write $-8-i8\sqrt{3}$ as $16e^{i(-2\pi/3 + 2k\pi)}$ where $k = 0,\pm1, \pm2,....$
1
vote
1answer
21 views

How to prove the inequality? [on hold]

Set $f(x)=1-(1-\lambda)^x$, where $\lambda \in (0,1)$, show that $f(x)/x \ge f(x)-f(x-1)$ holds for any $x\ge 1$.
0
votes
1answer
21 views

Is $Y=aX^b\cdot\exp(X)$ a rational or exponential function?

Is $Y=aX^b\cdot\exp(X)$ a rational or exponential function? $Y$ and $X$ are real variables, $a$ and $b$ are parameters. Someone said this is a product of polynomial and exponential function. Do we ...
2
votes
2answers
79 views

How to calculate $\exp(-x)$ using Taylor series

We know that the Taylor series expansion of $e^x$ is \begin{equation} e^x = \sum\limits_{i=1}^{\infty}\frac{x^{i-1}}{(i-1)!}. \end{equation} If I have to use this formula to evaluate $e^{-20}$, how ...
2
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0answers
24 views

How to estimate parameters of exponential functions?

I am a new bird to math! I have an expression as follows, $$e^{-ux}+e^{-uy}=z$$ I have at least ten pairs of $(x,y,z)$, so how can I estimate the value of $u$? And how to evaluate my results? Hand ...
1
vote
2answers
67 views

Solve $e = xe^x$

I know it it seems trivial that $x = 1$, but I would like to know a more rigorous solution involving algebra. I tried solving for it, but could not come up with a proper solution. My attempt: $e = ...
1
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0answers
15 views

Solve $b_1 e^{-a_1x^2}-b_2 e^{-a_2x^2}-b_3 e^{-a_3x^2}=0, \forall x$

Suppose, \begin{align*} b_1 e^{-a_1x^2}-b_2 e^{-a_2x^2}-b_3 e^{-a_3x^2}=0, \forall x \end{align*} Assume $a_1,a_2,a_3, b_1,b_2, b_3>0$ What are the possible values of $a_1,a_2,a_3, b_1,b_2, ...
3
votes
2answers
64 views

the value of $e$ and the method of getting it

We define e to be a number which satisfies the following condition $$\lim _{a \to 0} \frac{e^a-1}{a}=1. $$ How did we arrive to the following from above equation $$e=\lim _{n \to \infty} ...
1
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2answers
27 views

Find the general expression from the antiderivative

I am having trouble computing the original function. Question states: Let $f$ be a differentiable, positive function, such that $$f'(x)=x*f(x)$$ for all real numbers x. A) Find the general ...
6
votes
4answers
990 views

Slick proof of exponential inequality

Today I saw that using taylor series, one can show that $e^x+e^{-x}\leq 2e^{x^2/2}$. Is there a slick proof using some sort of Jensen-type inequality or integral bound?
3
votes
4answers
155 views

How to solve current exponential equation? [on hold]

There is an equation: $$3^x + 7^x = 21^x$$ How to solve this?
1
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0answers
28 views

exponential integration with fractional powers

I am trying to solve the following integral $$\int_{-\infty}^a \frac{\beta_1 \beta_2}{y^2(c-y)^2} e^{-\beta_1/(c-y)} e^{-\beta_2/y} \, dy$$ where $a<0$, $c>0$, $\beta_1>0$, $\beta_2>0$ I ...
1
vote
3answers
41 views

Need help with an inverse function

$$g(x) = \frac{100}{1+2^{-x}}$$ Ok, i have this expression and my task is to find the inverse. My answer to that is -ln2((100-x)/x). Which is wrong when i test it. Can someone help me with this?
0
votes
1answer
48 views

Physical Proof of Euler's Formula

I would like to construct a geometrical or physical proof of Euler's Formula $e^{ix}=\cos x +i\sin x $. If anyone has constructed such a proof before I would love to see it, if not, I would like some ...
-1
votes
0answers
18 views

Exponential estimate/inequality

I have a vector $x=(x_1,\dots, x_n)\in \mathbb{R}^n$ and some variance $\sigma^2 >0$. I know that the following inequality is wrong (but I present it because it would make world nicer in my view) ...
0
votes
1answer
30 views

Number of solutions to an equation

Hello guys I have a simple question to ask. For example I have the equation : $$x^n + x^{n-1} + x^{n-2} + ... + 1 = 0$$ I read somewhere that the number of solutions to an equation is given by the ...
2
votes
0answers
36 views

Comprehensive summary of where the function $\pi^{-\frac x\pi}$ can be encountered

I am studying the special functions, including the Riemann Xi and Zeta, and everywhere a function $\pi^{-\frac x\pi}$ pops up, usually as multiplier to the Gamma function. But yet I am not sure this ...
0
votes
0answers
19 views

Equation involving a modulus and variable in an exponent

How would I solve for the first positive non-zero integer value for $x$ in this equation? Equation: $1 \equiv 4^x \pmod{199}$
2
votes
5answers
127 views

$e^{x} > 1$ and $0 < e^{x} < 1$

So $$\exp(x) := \sum_{n=0}^{\infty} \frac {x^n} {n!}$$ How to prove that $\exp(x) > 1$ when $x > 0$ and moreover $\exp(x) < 1$ when $x<0$ Is it possible with induction? Or must I use ...
0
votes
2answers
34 views

Solve for $m$ in $d^m = n$ [duplicate]

I believe the answer is $m = \lceil \sqrt[d]n \rceil$ or $\lfloor \sqrt[d]n \rfloor$. Can anyone help me?
8
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0answers
198 views
+100

Approximation of the exponential

Let $c>1,k\in\mathbb{N}$. Let's consider two approximations of the exponential function : The first one is the most common one $f_k(x)=\left(1+\frac{x}{c^k}\right)^{c^k}$ and the second one is ...
1
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2answers
22 views

How to find the x intercepts

$\frac{4}{3} e^{3x} + 2 e^{2x} - 8 e^x$ I have some confusion especially because of the e how can I approach the solution? The solution of the x-intercept is 0.838 Many thanks
-4
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1answer
32 views

Exponential Math Functions [closed]

Iodine — 131 is a radioactive substance used in nuclear medicine. Suppose a patient was given a dose of 6mL. The half-life of Iodine-131 is 8 days. Determine the amount of iodine-131 in the patient ...
6
votes
0answers
96 views

Integrate this monster

Can you please help me? I've been trying for some time now to integrate this: $$\int_0^\infty g^{-(a+1)} \; \exp\left\{-\left(\frac{b}{g} + \frac{1}{2} \sum_{i=1}^{n} ...
1
vote
1answer
83 views

Equation $e^{\frac{1}{x}} - x =0$

Can someone solve this equations with steps $$e^{\frac{1}{x}} - x =0$$ I dont know how to start. I tried adding logarithms but that doesn't help.
2
votes
0answers
28 views

What are conditions for an infinite sum with a complex parameter not to be analyitically extendable?

I'm looking for a sequence $f(n)$, so that $g(z):=\lim_{N\to\infty}\sum_{n=0}^N\exp\left(-z\cdot f(n)\right),$ with $z$ so that this converges classically, defines a function which can not be ...
0
votes
0answers
28 views

math in medicine

Use the following parameter values for the model of the periodic arterial pulse that we considered in class: Rs = 17:5 mmHg/(liter/min), Csa = 0:00175 liters/mmHg, V0=0.07 liters, T = (1/80) min. ...
1
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1answer
45 views

Let $f(x) = \exp (x^2 − x + 6)$. Choose Dom(f) so that $f^{−1}$ exists. What is $f^{−1}$ and Dom($f^{−1}$) in your case?

I have already got $$y=\exp(x^2-x+16)$$ $$\ln y = x^2-x+6$$ $$\ln x=y^2-y+6$$ I know for getting inverse function we need to solve for $x$, but what should i do in this case?
-2
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2answers
46 views

Solving exponential equations like $6^{3x}=4^{2x-3}$ using logarithms

I'm trying to solve these using logarithms: $a$) $9^{x+1} = 27^{2x-3}$ $b$) $6^{3x} =4^{2x-3}$ $c$) $210=40(1.5)^x.$ I'm trying to practice logarithms by doing various questions. It's been a ...
0
votes
1answer
26 views

Logarithms and exponential decay

The table describes the cooling of a cup of coffee as it sits on your teacher’s desk in the math office. Time (min) $0, 4, 8, 12, 16, 20$ Temperature (celsius) $55, 47, 40, 34, 29, 25$ a) Calculate ...
7
votes
1answer
64 views

Reason for LCM of all numbers from 1 .. n equals roughly $e^n$

I computed the LCM for all natural numbers from 1 up to a limit $n$ and plotted the result over $n$. Due to the fast-raising numbers, I plotted the logarithm of the result and was surprised to find a ...
0
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0answers
10 views

Exponential decay of the temperature of coffee

The table describes the cooling of a cup of coffee as it sits on your teacher’s desk in the math office. Time (min) 0 4 8 12 16 20 Temperature (celsius) 55 47 40 34 29 25 a) Calculate a, the ...
0
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2answers
36 views

Forming equations for exponential growth/decay questions

Problem Dry cleaners use a cleaning fluid that is purified by evaporation and condensation after each cleaning cycle. Every time the fluid is purified, 2.1% of it is lost. The fluid has to be topped ...
1
vote
1answer
27 views

difference between poisson and exponential distributions in the context of client server systems?

I am studying client's request arrival patterns on web and application servers. About web server's request arrival pattern I read that "The request arrival rate on web server follows Poisson ...
0
votes
3answers
58 views

Complex number problem- separating into real and imaginary parts!

Please help with a question that I am working on just now...:) If $z=2e^{i\theta}$ where $0<\theta<\pi$, how can I find the real and imaginary parts of $w=(z-2)/(z+2)$? Hence, how can I ...
0
votes
1answer
59 views

Prove $\lim_{n \to \infty}$ $(1+\frac xn-o(\frac 1n))^n=e^x$ [duplicate]

We know that $\lim_{n \to \infty}$ $(1+\frac xn)^n=e^x$. How to prove that $\lim_{n \to \infty}$ $(1+\frac xn-o(\frac 1n))^n=e^x$? Attempt of the proof: Let $\epsilon>0$ $\exists n_0$ such that ...
0
votes
3answers
67 views

Why is $ \overline{e^z} = e^\overline{z} $?

How can you conjugate an entire function? $ \overline{exp(z)} $ I need an equivalent. I thought this is only possible with complex numbers. What is the proof for $ \overline{e^z} = e^\overline{z} $ ...
1
vote
1answer
49 views

Uniform Convergence to the Exponential Function over a Compact Interval

I'm trying to show that the sequence of functions $f_n(x)=(1+(x/n))^n$ converges uniformly to $f(x)=e^x$ over any compact interval of the real line. We're assuming that it converges pointwise. Here is ...
2
votes
2answers
44 views

yet another simple Laplace transform

what is $ℒ(t^2e^{3t})$ I have got this far so far: $=\int_{0}^\infty (t^2e^{t(3-s)})$ Integration by parts using: $u = t^2$ and $du = 2t$ $v = \frac{e^{t(3-2)}}{3-s}$ and $dv = e^{t(3-s)}$ Which ...
0
votes
1answer
71 views

Solve exponential equation $6\times3^{2x}-13\times 6^x +6\times 2^{2x}=0$

I have tried solving the following equation by using exponential properties and logarithms, but can not find some link between all of the terms: $$6\times3^{2x}-13 \times6^x +6\times 2^{2x}=0$$ ...
1
vote
1answer
24 views

Exponential Growth and Decay / compound interest

This is the question: "If you want to have $\$75,000$ after $35$ years in your account that pays $12\%$ annual interest compounded quarterly, how much should you put in as your original investment?" ...
1
vote
2answers
58 views

Integral $\int_0^b \frac{1-\exp\{-x\}}{x}\text{d}x\qquad 0<b<\infty$

Is there any closed form expression for the definite integral $$\int_0^b \frac{1-\exp\{-x\}}{x}\text{d}x\qquad 0<b<\infty$$ as I could not find one in Gradshteyn and Ryzhik Table of Integrals?
0
votes
1answer
31 views

Point of intersection between two exponentials with a constant term

Is there any way to solve algebraically for $x$: $a^x - b^x = C$ If not, is there a commonly used function that can be used to represent its solution? e.g., the Lambert W function for $a^x - bx = C$ ...
4
votes
1answer
24 views

Exponential Growth and Decay : $y = a (1+r)^t$

I know this is a really basic question for this website, but I can't find it anywhere else. This is the question: "If you deposit $\$3,750$ in an account that pays $6\%$ annual interest compounded ...
1
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1answer
28 views

Absolute value in exponential, signal energy?

How can this give this result? Isn't the absolute of $(e^(-2*t))$ always 1?
1
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7answers
151 views

How to estimate the value of $e$. [closed]

I am currently studying how to estimate $e$. To solve this problem I use these methods discuss below: Method 1: We know that $e^x = 1 + \dfrac{x}{1!} + \dfrac{x}{2!}+ \cdots $ So if we consider a ...
2
votes
0answers
71 views

Solve for $x: \ln(x+4)+\ln(x-2)=5$

Solve for x: $\ln(x+4)+\ln(x-2)=5$ Where do I go from here? If there weren't four terms in the equation I would use the quadratic formula. How can I solve for x? EDIT 1: Is this correct? ...
4
votes
1answer
71 views

Solve the integral [closed]

Can anyone solve these two integrals . $$ \int_{0}^{ \infty } \frac{x^2 e^{-x^2/2 \sigma ^2}}{(x-a)^2+b^2} dx $$ and $$ \int_{0}^{ \infty } \frac{e^{-(\ln x - \mu )^2/2 \sigma ...
-1
votes
3answers
91 views

Which is greater $e^{\pi}$ or $\pi^e$? [duplicate]

Recently I asked a question on Maths SE Proof that at most one of $e\pi$ and $e+\pi$ can be rational after solving this one one I was thinking whether $e^\pi$ is greater or $\pi^e$ ? On calculating ...