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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Why does integrating a complex exponential give the delta function?

How come, when we integrate a complex exponential from $ -\infty $ to $ \infty $, we get a scaled delta function? $$ \begin{align} \int_{-\infty}^{\infty} e^{i k x} \; dk & = 2 \pi \delta \left ( x \right ) \end{align} $$ Specifically, why do we say…
QuantumFool
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$\pi^4 + \pi^5 \approx e^6$ is anything special going on here?

Saw it in the news: $$(\pi^4 + \pi^5)^{\Large\frac16} \approx 2.71828180861$$ Is this just pigeon-hole? DISCUSSION: counterfeit $e$ using $\pi$'s Given enough integers and $\pi$'s we can approximate just about any number. In formal…
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Proving $\mathrm e <3$

Well I am just asking myself if there's a more elegant way of proving $$2<\exp(1)=\mathrm e<3$$ than doing it by induction and using the fact of $\lim\limits_{n\rightarrow\infty}\left(1+\frac1n\right)^n=\mathrm e$, is there one (or some)…
sheldoor
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Why does $e$ have multiple definitions?

The number $e$ seems to have multiple definitions: $$\lim \limits_{n \to \infty} \left(1 + \frac{1}{n}\right)^n$$ The unique number $a$ such that $\int_1^a\frac{1}{x} \, dx = 1$ The unique number $a$ such that $\frac{d}{dx}a^x=a^x$ The base of the…
Jay
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Unifying the connections between the trigonometric and hyperbolic functions

There are many, many connections between the trigonometric and hyperbolic functions, some of which are listed here. It is probably too optimistic to expect that a single insight could explain all of these connections, but is there a holistic way of…
Joe
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A question comparing $\pi^e$ to $e^\pi$

I was doing an algebra problem set following a chapter on logarithms and exponentiation, and it presented this "bonus question": Without using your calculator, determine which is larger: $e^\pi$ or $\pi^e$. I wasn't able to come up with anything,…
ivan
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Can hyperbolic functions be defined in terms of trigonometric functions?

For example, can $\sinh x$ be written as a function of $\sin x$? Another question, are hyperbolic functions dependent of their trigonometric correspondence in any way?
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Is this formula for $\frac{e^2-3}{e^2+1}$ known? How to prove it?

I found an interesting infinite sequence recently in the form of a 'two storey continued fraction' with natural number…
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How to verify if a curve is exponential by eyeballing?

A plane curve is printed on a piece of paper with the directions of both axes specified. How can I (roughly) verify if the curve is of the form $y=a e^{bx}+c$ without fitting or doing any quantitative calculation? For example, for linear curves, I…
arax
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If there are entire $G_k$s such that $f=\exp\circ\exp\circ\cdots \circ\exp\circ G_k$ ($k$ times), must $f$ be constant?

I am a French guest and I hope that my English isn't too bad... So here is my issue: I consider an entire function $f$ which satisfies the following property for each complex number $z\in \mathbb{C}$: $\forall ~ k \in \mathbb{N}^*$, there exists an…
Stabilo
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Prove $|e^{i\theta} -1| \leq |\theta|$

Could you help me to prove $$ |e^{i\theta} -1| \leq |\theta| $$ I am studying the proof of differentiability of Fourier Series, and my book used this lemma. How does it work?
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Is the Fibonacci sequence exponential?

I could not find any information on this online so I thought I'd make a question about this. If we take the Fibonacci sequence $F_n = F_{n-1} + F_{n-2}$, is this growing exponentially? Or perhaps if we consider it as a function $F(x) = F(x-1) +…
Stijn
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Absolute value of complex exponential

Can somebody explain to me why the absolute value of a complex exponential is 1? (Or at least that's what my textbook says.) For example: $$|e^{-2i}|=1, i=\sqrt {-1}$$
codedude
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Is $e$ a coincidence?

$e$ has many definitions and properties. The one I'm most used to is $$\lim_{n\to \infty}\left(1+\frac{1}{n}\right)^n $$ If someone asked me (and I didn't know about $e$): Is there a constant $c$ such that the equation $\frac{d}{dx}c^x=c^x $ is…
Stewie Griffin
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"What if" math joke: the derivative of $\ln(x)^e$

Randall Munroe, the creator of xkcd in his latest book What if writes (p. 175) that the mathematical analog of the phrase "knock me over with a feather" is seeing the expression $ \ln( x )^{e}$. And he writes regarding this expression: "it's not…