Questions tagged [exponential-function]

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

The exponential function is one of the most important functions in mathematics (though it would have to admit that the linear function ranks even higher in importance).

Definition: If $~b~$ be any number such that $~b\gt 0~$ and $~b\neq 1~$ then an exponential function is a function in the form,$$f(x)=a~b^x$$ where $~b~$ is called the base , the exponent,$~x~$ can be any real number and $~a\neq0~$.

${}$

Properties:

  • The graph of $~f(x)~$ will always contain the point $~(0,1)~$. Or put another way, $~f(0)=1~$ regardless of the value of $~b~$.
  • For every possible $~b~$we have $~b^x\gt 0~$. Note that this implies that $~b^x\neq 0~$.
  • If $~0\lt b\lt 1~$then the graph of $~b^x~$ will decrease as we move from left to right. Check out the graph of $~\left(\frac{1}{2}\right)^x~$ above for verification of this property.
  • If $~b\gt 1~$ then the graph of $~b^x~$ will increase as we move from left to right. Check out the graph of $~2^x~$ above for verification of this property.
  • If $~b^x=b^y~$, then $~x=y~$.

${}$ The Natural Exponential Function: In mathematics, the natural exponential function is $$f(x)=e^x~,$$ where $e$ is Euler's number.

Note: $f(x)=e^x~$ is a special exponential function. In fact this is so special that for many people this is THE exponential function.

Applications:

Exponential functions are solutions to the simplest types of dynamical systems. It is used to model a relationship in which a constant change in the independent variable gives the same proportional change (i.e. percentage increase or decrease) in the dependent variable. Exponential growth occurs when the growth rate of the value of a mathematical function is proportional to the function's current value. Exponential decay occurs in the same way when the growth rate is negative.

References:

https://en.wikipedia.org/wiki/Exponential_function

http://www.montereyinstitute.org/courses/DevelopmentalMath/COURSE_TEXT2_RESOURCE/U18_L1_T1_text_final.html

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Simplifying the derivative of $f(x)= \frac{e^x - e^{-x}}{e^x+e^{-x}}$

I was having some trouble on simplifying the derivative because I didn't know if it's correct. The original function is $$f(x)= \frac{e^x - e^{-x}}{e^x+e^{-x}}$$ What would the simplified derivative be with no negative exponents? So far I…
Jadon
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How do we 'know' that $2^x$ is continuous?

It is intuitive for $2^n$, if $n$ is an integer, to exist. How do we know that less intuitive values such as $2^\frac{1}{2}$, $2^\sqrt{2}$, $2^\pi$ etc exist? I'd like to accept that $2^x$ is continuous, but how can we be sure of the existence of…
Trogdor
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How to invert a simple exponential growth formula

I think this is simple but my math skills are limited. I have a basic exponential growth formula: $$y=x \cdot (1-p)^n$$ and I have $y$ and $x$ and $n$ values and I need value of $p$. Then when I solve for $p$, I have to calculate $y$ with different…
bacuto d.
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Exponential extension of $\mathbb{Q}$

A non-trivial exponential function $E:\mathbb{K} \rightarrow \mathbb{K}$ in a field $\mathbb{K}$ is a function such that \begin{split} E(x+y)=E(x)E(y) \quad \forall x,y \in \mathbb{K} \\ E(x)=1 \iff x=0 \end{split} For the exponential function such…
Emilio Novati
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This three-variable system of equations seems impossible to solve

$$g = af^b + c$$ $$i = ah^b + c$$ $$k = aj^b + c$$ I want to solve for $a$, $b$, and $c$. $f$, $g$, $h$, $i$, $j$, and $k$ are inputs to the equations, so they don't have to be solved for. Just imagine that these variables (that are not $a$, $b$, or…
sOvr9000
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Estimating the behavior for large $n$

I want to find how these coefficients increase/decrease as $n$ increases: $$ C_n = \frac{1}{n!} \left[(n+\alpha)^{n-\alpha-\frac{1}{2}}\right]$$ with $\alpha=\frac{1}{br-1}$ and $0\leq b,r \leq 1$. I used the Stirling's Approximation factorial …
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Intiutive argument that $\exp' = \exp$

Is there any intuitive argument or visual "proof" that $\exp' = \exp$? Suppose you have defined the Euler number $\mathrm{e}$ as limit of the sequence $(a_n)$ where $a_n = \left (1 + \frac{1}{n} \right)^n \quad \forall n > 0$, and that the $\exp(x)$…
Julia
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l'Hôpital and it's use in derivation

In for example $$\lim_{x\rightarrow 0} \frac{e^{ax} - 1 - ax}{1 - \cos x}$$ We would use l'Hôpital rule and derive it twice to get $a^2$ How do you see this when just looking at the given function, when do you know you should use l'Hôpital and can…
user184106
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Fnd $ \int\frac{e^{-2x-x^2}}{( x+1)^2}\,dx$

Find $\displaystyle\int \dfrac{e^{-2x-x^2}}{\left( x+1\right)^2}\hspace{1mm}dx$. If I do Integration by parts, I end up with $\displaystyle\int e^{-2x-x^2}\hspace{1mm}dx$ Which I believe cannot be integrated
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Find a particular solution of $\,\,y''+3y'+2y=\exp(\mathrm{e}^x)$

I already solved for the homogeneous one, but I'm still looking for the particular solution of the differential equation: $$y''+3y'+2y=\exp(\mathrm{e}^x)$$ The homogeneous solutions of this system are $\mathrm{e}^{-x}$ and $\mathrm{e}^{-2x}$. I've…
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How to solve this exponential equation? $2^{2x}3^x=4^{3x+1}$.

I haven't been able to find the correct answer to this exponential equation: $$\eqalign{ 2^{2x}3^x&=4^{3x+1}\\ 2^{2x} 3^x &= 2^2 \times 2^x \times 3^x\\ 4^{3x+1} &= 4^3 \times 4^x \times 4\\ 6^x \times 4 &= 4^x \times 256\\ x\log_6 6 + \log_6 4 &=…
jn025
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Exponent laws with Negative bases

$-49 =7^x$ is the question. Here I m supposed to solve for what power of $7$ will give me $-49$. Or in other words, I have to solve for $x$. This looks fairly simply when thinking about the exponent rules for it looks as if you could make the…
Ashton
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If $A$ is a $2\times 2$ matrix with a repeated eigenvalue $r$, then $\mathrm{e}^{At}=\mathrm{e}^{rt}\left[I+(A-rI)t\right]$

If $A$ is a $2\times 2$ matrix with a repeated eigenvalue $r$, show that $\mathrm{e}^{At}=\mathrm{e}^{rt}\left[I+(A-rI)t\right]$. I have already been able to show that if $A$ is an arbitrary $2\times 2$ matrix $$ \left( \begin{array}\\ a & b…
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Does n power of e grow much more faster than its Maclaurin polynomial?

I wonder how to calculate the following limit: $$ \lim_{n\rightarrow\infty}\frac{1+n+\frac{{}n^{2}}{2!}+\cdots +\frac{n^{n}}{n!}}{e^{n}} $$ In the first sight, I think it should be zero, because exponential function is much faster than polynomial.…
jintok
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Rational Exponent

Is $a^{p/q}$ equal to $a^{2p/2q}$? Do we need to simplify $p/q$ to its lowest terms? I need a strict mathematical definition which proves one or another statement. For example: $(-8)^{1/3} = -2$ Is $(-8)^{2/6}$ equal to $\sqrt[6]{(-8)^2} =…
Ivan Ehreshi
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