# Tagged Questions

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

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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### How to calculate this integral $\frac{1}{2} \int_0^1\ 1.5 e^{-ik\pi \ t} \ \ dt, \, k \in \mathbb{Z}$ [closed]

$$\frac{1}{2} \int_0^1\ 1.5 e^{-ik\pi\ t} \ \ dt, \, k \in \mathbb{Z}$$
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### 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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### 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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### 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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### 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 ...
### 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 .