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

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Finding the CDF for $Y = e^X$ when $X \sim N(0,1)$

Problem: Let $X \sim N(0,1)$ and let $Y = e^X$. Find the CDF for $Y$. Attempted Solution: Let $y = e^x$ so that $x = \ln(y)$. Then $$ F(y) = P(Y \le y) = P(Y \le e^x) = P(X \le \ln(y)) = F_X(\ln(...
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2answers
66 views

Is this proof that $e$ is irrational correct?

I should mention that I still haven't taken Calculus or even Pre-Calculus, which is why I want to ask this. I've seen proofs $e$ is irrational, but not this one. Is this correct, and if it isn't, why ...
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1answer
29 views

Exponential Function with start and end point

I have the following situation. I have an start point of 40 degrees temperature and endpoint of 69 degrees. Now i want to normalize all values in this range into an skala from 1-15. This should ...
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2answers
46 views

Integrating exponent with polynomial

http://i.stack.imgur.com/4tXNr.jpg $e^{x^2/2}\int e^{-x^2/2}(-x^3+x)\ dx$ turns out to be equal to $e^{x^2/2}[e^{-x^2/2}(x^2+1)] $ Is there a easier method of integrating such functions? I can't ...
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5answers
151 views

The integral $\int_0^\infty e^{-t^2}dt$ [duplicate]

Me and my highschool teacher have argued about the limit for quite a long time. We have easily reached the conclusion that integral from $0$ to $x$ of $e^{-t^2}dt$ has a limit somewhere between $0$ ...
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1answer
54 views

Exponential growth and decay question [closed]

A city has a growing population at a rate proportional to the current population, that is: $$\frac{dP}{dx}=kP.$$ Verify that $P(t)=P_0e^{kt}$, $t>0$ is a solution of the equation. If the ...
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4answers
388 views

Exponential Equations with Fractions

I have had some issues with the following two equations: $$ \frac{3^{n-2}}{9^{1-n}}=9$$ $$\frac{5^{3n-3}}{25^{n-3}}=125$$ If anyone could work them out step by step that would be awesome. I ...
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3answers
49 views

Prove $\sum_{k = 2}^\infty \ln(1+\frac{1}{k^2})$ converges using $\exp(x) \geq 1+x$.

All I've got so far is $$\exp(x) \geq 1+x \Rightarrow x \geq \ln(1+x) \Rightarrow \frac{1}{k^2} \geq \ln\left(1+\frac{1}{k^2}\right)$$ which (since $\ln(1+\frac{1}{k^2})$ is larger than zero) means ...
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4answers
58 views

What does $e^{a*ln(x)}$ equal in terms of $a$ and $x$, and how is this found?

I saw somewhere that it would be $x^a$, but I'm not sure why.
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2answers
20 views

Why does the exponential distribution's pdf integrate to 1? [duplicate]

From All of Statistics pg. 29: EXPONENTIAL DISTRIBUTION. $X$ has an Exponential distribution with paramater $\beta$, denoted by $X \sim \text{Exp}(\beta)$, if $$ f(x) = \frac{1}{\beta}e^{-x/\...
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3answers
139 views

Why doesn't continuously compounded interest make me a zillionaire?

It would seem that if I have some money and I get an interest on it every second, I'd be a zillionaire in no time. However, as the formula for the continuously compounded interest is: $A(t) = P(1 + \...
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2answers
21 views

Derivative that includes several functions of time

I'd like to compute the following derivative (i.e., solve for $v$): \begin{align} \frac{dv}{dt} = \frac{-v(t) + I_{rec}(t) + I_{ext}(t)}{\tau_m}. \end{align} I know that if I had $\frac{dv}{dt}=\...
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1answer
33 views

How sum of exponential variables is a gamma variable [duplicate]

I have the task to calculate $P(S_{100}\geq 200)$ where $S_{100}=\sum^{100}_{i=1} X_i$ and $X_i$, $i=1,2, \cdots, 100$ are independent $exp(\lambda)$ random variables. One method is to use the fact ...
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1answer
46 views

Is $x^x$ in the same asymptotic growth class as an exponential function?

I see that for any natural number $a$, $\lim_{x\to\infty} \tfrac{x^x}{a^x}$ approaches $\infty$, so the limit does not exist. So is this function have a different big-O than $O(a^x)$, for example? So ...
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2answers
52 views

Sequence of functions that converges to Gaussian

Define $f_n:[0,\infty)\to\mathbb{R}$ as follows: $$ f_n(x) = \begin{cases} \left(1-\frac{x^2}{n}\right)^n & 0\leq x\leq \sqrt{n} \\ 0 & \text{otherwise}\end{cases} $$ I need to show that $f_n$ ...
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1answer
34 views

Prove the inequality involving exponential function in form of $\exp( \frac{1}{x} )$

For $\nu > 0$, $0 < x \leq \nu $, and a positive integer $S$, (we think) following an inequality always holds $1- \left( \frac{1}{x+1} \right)^S \geq \exp \left( -\frac{1}{Sx} \right) $ Does ...
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1answer
92 views

Solve the system of equations $x^y=y^x$

Solve the system of equations $$ x^y=y^x \\ a^x=b^y $$ I could not solve this despite many tries
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3answers
40 views

Solving an exponential function with a sum

I have to solve equations of this kind to $x$: $3^{2x-1} + 1 = 28 \cdot 3^{x-2}$ I don't get the trick to eliminate the $+1$ in the equation. Can someone show me how I can solve this? Thanks!
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5answers
416 views

Can someone help me with this question of finding x as exponent?

The equation is: $$6^{x+1} - 6^x = 3^{x+4} - 3^x$$ I need to find x. I forgot how to use logarithm laws. Help would be appreciated. Thanks.
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2answers
49 views

How to verify $(1+\frac{1}{n})^2(1-\frac{1}{n^2})^{n-1}\geq \exp(\frac{1}{n})$

How to verify this inequality? Assuming that $n\in \mathbb{N}^+$, we have: $$\left(1+\frac{1}{n}\right)^2\left(1-\frac{1}{n^2}\right)^{n-1}\geq \exp\left(\frac{1}{n}\right).$$
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0answers
50 views

Solve $x = \sqrt[x]2$

How does one solve $x = \sqrt[x]2$ for $x$? This can be otherwise stated as $x = 2^{1/x}$ Raising both sides to the power of $x$: $x^x = (2^{1/x})^x$ $x^x = 2$ But I don't know where I can go from ...
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0answers
63 views

What's the worst sequence that still leads to a converging series?

As a background, I'm initially interested in sequences $a_n$ giving rise to functions $\sum_{n=0}^\infty a_nx^n$ for $x\in(0,1)$ and their diverging behavior for $x$ to $1$. E.g. the geometric series $...
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0answers
40 views

Solving an exponential inequality problem

How do I prove the following inequality : $$\Bigg(\frac{2}{\alpha^2} \, \big( e^{\alpha x} - e^{\alpha y} \big) \, + \, e^{\alpha y} (y^2 - x^2) \; \Bigg) > 0 $$ given, $x, y > 0$ ? Can ...
3
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2answers
119 views

How does one solve the equation $3^x+5\cdot3^x\cdot2^x-2^x=0$?

I've tried to solve equations $$3^x+5\cdot3^x\cdot2^x-2^x=0$$ $$81^x−2·54^x−36^x−2·24^x+16^x=0$$ but I failed. I don't know where to start. Any help is welcome
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2answers
122 views

How can $e^a = 0$ when integrating?

I was doing a question which asked me to turn the following into polar equation:$$(y + x − x(x^2 + y^2))\frac{dy}{dx} = y − x − y(x^2 + y^2)$$ With a lot of messy algebra I can get it down to: $$\...
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4answers
65 views

How to prove that $\frac {e^{b^2-1}}{b^2}$ ≥ 1

How to prove that $$\frac {e^{b^2-1}}{b^2} \ge 1?$$ Use logarithm or limit or what? Or do we have to use it as a conclusion to prove it backwards? And how to prove it forwards, that is, without ...
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2answers
53 views

Calculate $\int_{-\pi}^{\pi} \frac{xe^{ix}} {1+\cos^2 {x}} dx$

So I'm trying to calculate $$ \int_{-\pi}^{\pi} \frac{xe^{ix}} {1+\cos^2 {x}} dx $$ knowing that if $f(a+b-x)=f(x)$ then $$ \int_{a}^{b} xf(x)dx=\frac{a+b}{2} \int_{a}^{b} f(x)dx, $$ but it doesn't ...
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1answer
112 views

How to evaluate or approximate this kind of recursion: $a(n+1) = m \cdot \exp\left(\frac{-K \cdot (m - a(n))}{m}\right),\ n \geq 1$?

Edit: In the original post, I put the function $$a(n+1) = m \cdot \exp(-K \cdot a(n) / m),\ n \geq 2$$ which is not the function I wanted to study. The correct one is the one given below I came up ...
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1answer
33 views
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22 views

Exponential-ish function from 0,0 to 1,1: how to push the turning point of the curve

I am trying to find a weighting function to map $x$ values $0 < x < 1$ to a $y$ values $0 < y < 1$, following something similar to an exponential curve. So far, I have been using the ...
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1answer
28 views

Problems Calculating Fractional Derivative

I have been trying to calculate the fractional derivative of $e^{ax}$ using the Liouville Left-Sided derivative, which states that, for $x>0$ and $0<n<1$, $D^n f(x) = \frac{1}{1-n} \frac{d}{...
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2answers
36 views

Infinite differentiability of a function with a removable discontinuity

How would I prove that $\frac x{e^x-1}$ is infinitely differentiable? (This question came up since the No 1 answer in Maclaurin series for $\frac{x}{e^x-1}$ states that the function is infinitely ...
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1answer
20 views

Limit of trig functions

We have to evaluate $$\lim_{x\to 2} \frac{\cos^x a +\sin^x a -1}{x-2}.$$ I am working on it for hours I tried using series , replacing $\cos a$ by $t$ and $\sin a$ by $\sqrt{1-t^2}$ but not got any ...
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2answers
32 views

Graphing log with number in front of “log”

When I have something like $y = log_2(x)$ I know that I have to turn it into exponential form and get: $2^y = x$. Next, I make a table for $X,Y$ and choose about 5 values for $y$, typically $-1, 0, 1, ...
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2answers
39 views

What will the value of an account be after $12$ years if the account earns $4.91\%$ a year and if someone invests $\$20,000$?

Second National Bank offers an account that earns $4.91\%$ per year, compounded continuously. If a person invests $\$20,000$ in this account, what will be the value of the account at the end of $12$ ...
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1answer
80 views

Entire function $f$ such that $\lim\limits_{z\rightarrow \infty}f(z)=0$ and $f(0)=1$?

The question is this: Does there exist an entire function $f$ such that $\lim_{z\rightarrow \infty}f(z)=0$ and $f(0)=1$. I immediately would point to $f(z)=e^{-z}$. It is entire and satisfies the ...
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3answers
19 views

isolating x with two variables and negative exponents

I have: $$ 4^y = x^{-2} $$ Can someone hint to me what I need to do to isolate $x$? I'm not sure what to do.
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1answer
38 views

Convergence of $\sum_{k \geq 1} e^{-tk} \cos kz$

I would like to find the convergence of the series $\sum_{k \geq 1} e^{-tk} \cos kz$. Clearly, this series converge in using the comparison test or the integral. How could I get an explicit function ...
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2answers
36 views

Limit of indeteminate form $1^{(∞)}$

If we consider the function $f(x)=[(ax+1)/(bx+2)]^{x}$ where $a$,$b$ >$0$ and a I tried as follows]1 But at end i got stuck .
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0answers
24 views

Integral computation with Mathematica and Sympy differ

To compute the integral: $I = \int_{0}^{+oo} ue^{Au^{2}+Bu}du$ where $A<0$ and $B>0$ I have tried both Mathematica and Sympy but they yield different results: Mathematica yields: $ I = \frac{\...
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1answer
48 views

Express $y = KC^x$ as a linear function

Consider an exponential relationship of the form $y = KC^x$ where $K$ and $C$ are constants. Express the exponential function $y = KC^x$ as a linear function and describe how you would obtain the ...
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2answers
20 views

How to show that: $\log_a (x^{a}-x)-\log_a \Big(\dfrac{x^{a}-x}{a}\Big)=1$, where $a$ and $x$ are positive integers.

I was studying Fermat's Little Theorem and Logarithm to see if there is any interesting result or correlation exist between the two. So I came up with this equation. I know few basic logarithmic ...
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2answers
80 views

Check that $\lim\limits_{n\to\infty}\sum\limits_{i=1}^{n}\left(\frac{i+x}{n}\right)^n=\frac{e^{x+1}}{e-1}$

Show that $$\lim_{n\to\infty}\sum_{i=1}^{n}\left(\frac{i+x}{n}\right)^n=\frac{e^{x+1}}{e-1}$$ Any hints how I can tackle this problem? Although I checked on a sum calculator that it converges ...
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1answer
28 views

Prove, for every $l \geq 3$ , the $\Big( 1- \dfrac{1}{2 \cdot l}\Big)^{2 \cdot l} < \dfrac{1}{e}$ holds

I need to prove that for every $l \geq 3$, the $\Big( 1- \dfrac{1}{2 \cdot l}\Big)^{2 \cdot l} < \dfrac{1}{e}$ holds. ($l$ is integer) This is what I tried so far. $$ \begin{align} x &= \...
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1answer
54 views

$(-1)^0$ , calculating zeroth power of a negative number

I wish to calculate the zeroth power of a negative number $(-1)^0 = (-1)^{2-2}$ =$\frac{(-1)^2}{(-1)^2} = 1$ But when I put it in a calculator, it comes out to be $-1$.
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2answers
33 views

Equations with variable Exponents

I am struggling to find a solution to $x^{x-5}=5$, although clearly from plotting the graph of $f(x)=x^{x-5}-5$ I can see that there are two real solutions, but I have no idea how to evaluate them, or ...
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3answers
61 views

exponential function and mathematical induction

May I ask how to solve the problem? Use mathematical induction to prove that for $x\geq0$ and positive integer $n$, $$e^x \geq 1 + x + \frac{x^2}{2!} + \cdots + \frac{x^n}{n!}$$
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1answer
28 views

A basic question about the decay rate of $te^{-t}$ as $t$ tends to infinity

It is well-known that $te^{-t}$ tends to $0$ as $t$ tends to infinity. But I want to know the decay rate of $te^{-t}$ as $t$ tends to infinity. Using Taylor expansion of $e^{t}$ we have: $${t /e^{t}}=...
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3answers
33 views

Subtracting powers with variable in exponent

I am having some troubles with a question that subtracts powers. Solve for unknown: $$3^{x+4} - 5(3^x) = 684$$ I have a hunch that I should apply factorization somehow. Do I multiply 5 and 3 to ...
2
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1answer
25 views

What is the solution for $y(t)=e^{-\frac{t}{\tau y(t)}}$?

A simple quadratic flow model leads to the following apparently simple equation $$y(t)=e^{-\frac{t}{\tau y(t)}}$$ where the flow, $y$ is a function of time, $t$ and $\tau $ is a constant. But is ...