# Tagged Questions

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

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### If $(\sqrt{x^2-5x+6} + \sqrt{x^2-5x+4})^{x/2} + (\sqrt{x^2-5x+6} - \sqrt{x^2-5x+4})^{x/2}=2^{\frac{x+4}{4}}$, find $x$.

The main question is : If $(\sqrt{x^2-5x+6} + \sqrt{x^2-5x+4})^{x/2} + (\sqrt{x^2-5x+6} - \sqrt{x^2-5x+4})^{x/2}=2^{\frac{x+4}{4}}$, find $x$. My method : I first began by substituting $x^2-5x+5$ as ...
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### $\int_{- \infty}^{\infty} \frac{f(x)}{1+\exp{g(x)}}dx=\int_{0}^{\infty} f(x) dx$ for $f(x)=f(-x),~g(x)=-g(-x)$ - are there other formulas like that?

If $f(x)$ any even function, integrable on $(0,\infty)$ and $g(x)$ any odd function, then we have: $$\int_{- \infty}^{\infty} \frac{f(x)}{1+e^{g(x)}}dx=\int_{0}^{\infty} f(x) dx \tag{1}$$ The ...
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### What is the determinant of exp(matrix)? [duplicate]

Given a square matrix $A$, form the Lie series of it, which is defined by: $$e^A = I + A + \frac{1}{2} A^2 + \frac{1}{3!} A^3 + \cdots + \frac{1}{n!} A^n = \sum_{k=0}^\infty \frac{1}{k!} A^k$$ Is ...
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### Show that for all $z \in \overline{D}(0;1)$, $(3-e)|z| \leq |e^z - 1|\leq |z|(e-1)$

Show that for all $z \in \overline{D}(0;1)$, $(3-e)|z| \leq |e^z - 1|\leq |z|(e-1)$ I think I'm supposed to use the following chain of inequalities $$|e^z -1|\leq e^{|z|}-1 \leq |z|e^{|z|}$$ But ...
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### Why does $a\cdot r^{-1}$ equate to $\frac {a}{r} = 1$?

Why is $a\cdot r^{-1}=1$ equivalent to $\frac {a}{r} = 1$? I am trying to write exponential functions from graphs; two points were given: $(-1,1)$ & $(-2,5)$. I am trying to find an equation ...
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### The value of an investment in Canada Savings Bonds is modelled by $A(t) = A_0 e^{0.0255t}$… Rest of question below.

The value of an investment in Canada Savings Bonds is modeled by $$A(t) = A_0 e^{0.0255t}$$, where A is the amount the investment is worth after $t$ years, and $A_0$ is the initial amount invested. At ...