Exponential function Definition and 166 Threads

I know that for 1 variable, one can write ##e^{f(x)} = \sum_{n = 0}^{\infty}\frac{(f(x))^n}{n!}##. In the case of 2-variables ##f(x,y)##, I assume we cannot write ##e^{f(x,y)} = \sum_{n = 0}^{\infty}\frac{(f(x,y))^n}{n!}## right (because of how the Taylor series is defined for multiple...

Hi PF The logarithm is the inverse function to exponentiation. But, focusing on exponentiation, here comes the graph: And, next, the quote An exponential function is a function of the form ##f(x)=a^x##, where the base ##a## is a positive constant and the exponent ##x## is the variable...

I wonder if it's ##f(x)=2^{x}-1## considered an exponential function because in my textbook it's stated that the set of values of an exponential function is a set of positive real numbers, while when graphing this function I get values(y line) that are not positive(graph in attachments), so I am...

Preface: I have not done serious math in years. Today I tried to do something fancy for a game mechanic I'm designing. I've got an item with a variable power level. It uses x amount of ammo to produce f(x) amount of kaboom. Initially it was linear, e.g. fL(x) = x, but I didn't like the scaling...

My trial : I think ## \int ~ dy ~ e^{-2 \alpha(y)} ## dose not simply equal: ## - \frac{1}{2}e^{-2 \alpha(y)} ## cause ##\alpha## is a function in ##y ##. So any help about the right answer is appreciated!

Hi everybody We can't differentiate ##x^x## neither like a power function nor an exponential function. But ##x^x=e^{x\mbox{ln}x}##. So ##\dfrac{d}{dx}x^x=\dfrac{d}{dx}e^{x\mbox{ln}x}=x^x(\mbox{ln}x+1)## And here comes the doubt: prove the domain of ##x^x## is ##(0, +\infty)## Why is only...

I want to know the frequency domain spectrum of an exponential which is modulated with a sine function that is changing with time. The time-domain form is, s(t) = e^{j \frac{4\pi}{\lambda} \mu \frac{\sin(\Omega t)}{\Omega}} Here, \mu , \Omega and \lambda are constants. A quick...

Ok, first I tried to show that ##A = \left \{a^{r}|r\in\mathbb{Q},r<x \right \}## does not have a maximum value. Assume ##\left\{ a^{r}\right\}## has a maximum, ##a^{r_m}##. By this hypothesis, ##r_{m}<x## and ##r_{m}>r\forall r\neq r_{m}\in\mathbb{Q}##. Consider now that ## q\in\mathbb{Q}|q>0##...

I try to proof it but i got stuck right here, i want your opinions Can I get a solution if i continue by this way? or Do I have to take another? and if it is so, what would yo do?

the integral is: and according to mathematica, it should evaluate to be: . So it looks like some sort of Gaussian integral, but I'm not sure how to get there. I tried turning the cos function into an exponential as well: however, I don't think this helps the issue much.

Hello everyone, can anybody solve this limit? This is really tough one for me, thank you in advance.

Homework Statement -2^x = y Homework EquationsThe Attempt at a Solution When I plug this function in my graphing calculator, it appears to be 2^x reflected across the x axis. This doesn't make sense to me. For example, for x values of 1 and 2, the value of y is not on the same half of the...

For example, in linear differential equations, there might be these questions where we'd directly use e∫pdx as the integrating factor and then substitute it in this really cliche formula but I never really understood where it came from. Help ?

Homework Statement given the formula m=n*e^(-nt) show that the maximum of this curve is at m=1/(t*e^(1)). 2. The attempt at a solution I can show this graphically but I am curious if it is possible to do it by hand?

Homework Statement Show ##\int_{0}^{1}e_n(x)\overline e_k(x) dx = 1## if ##n=k## and ##0## otherwise. Homework Equations ##e_n(x) = e^{2\pi inx}##. The Attempt at a Solution Consider 2 cases: case 1: ##n=k##. Then ##\int_{0}^{1} e_n(x) \bar e_k(x) dx = \int_{0}^{1} e_n(x)e_{-k}(x) dx =...

I was wondering if anyone could point me to a set of rules for finding the base of an exponential function? I can figure out that the base of f(x)=7^x is 7 and the base of f(x)=3^{2x} is 9 but even though I know f(x)=8^{\frac{4}{3}x} has a base of 16, I don't know how that answer was reached.

How to Integrate it::: ∫e^(ax²+bx+c)dx Or in general e raised to quadratic or any polynomial. I am trying hard to recall but I couldn't recall this integration. I tried using By-parts but the integration goes on and on.

Hello,I was wondering. Is the exponential function, the only function where ##y'=y##. I know we can write an infinite amount of functions just by multiplying ##e^{x}## by a constant. This is not my point. Lets say in general, is there another function other than ##y(x)=ae^{x}## (##a## is a...

I have a model where the probability is spherically symmetric and follows an exponential law. Now I need the probability density function of this model. The problem is the singularity at the origin. How can I handle this? P(r) = ∫p(r) dr = exp(-μr) p(r) = dP(r)/(4πr²dr) One way I tried to...

1. The problem statement, all variables, and given/known data Task begins by giving sample function and a corresponding Fourier transform $$f(t) = e^{-t^2 / 2} \quad \Longleftrightarrow \quad F(\omega) = \sqrt{2 \pi} e^{-\omega^2 / 2}$$ and then asks to find the Fourier transform of $$f_a(t) =...

Let f(z) = $e^{e^{z}}$ . Find Re(f) and Im(f). I don't know how to deal with the exponential within an exponential. Does anybody know how to deal with this?

Hey! :o I am looking at the following: Show that $\displaystyle{\text{exp}(1)=\sum_{k=0}^{\infty}\frac{1}{k!}=e}$ with $\displaystyle{e:=\lim_{n\rightarrow \infty}\left (1+\frac{1}{n}\right )^n}$. Hint: Use the binomial theorem and compare with the partial sum $s_n$ of the series...

Homework Statement Determine the Fourier-transfroms of the functions \begin{equation*} a) f : f(t) = H(t+3) - H(t-3) \text{ and } g : g(t) = \cos(5t) f(t) \end{equation*} and \begin{equation*} b) f : f(t) = e^{-2|t|} \text{ and } g : g(t) = \cos(3t) f(t) \end{equation*}Homework Equations The...

Hello all, I have a complicated function: \[f(x)=\left ( e^{x}+x \right )^{^{\frac{1}{x}}}\] I need to find it's derivative and it's limit when x goes to infinity. As for the derivative, I thought maybe to use LN, so that I can get rid of the exponent, am I correct? How should I approach...

Homework Statement The question is given just like this: ##\frac{d}{dx}(exp\int_1^x P(s)\ ds)## = ? I assume they want me to find the derivative of the whole thing. Homework EquationsThe Attempt at a Solution I'm thinking the first step is: ##\frac{d}{dx}(exp\int_1^x P(s)\ ds) = (exp\int_1^x...

Homework Statement a= eD/R*T*G make a linear equations and calculate the value for D and G R=8,3 and constant D,G=constant T= variable Homework Equations y=ax+b y=numberax*bThe Attempt at a Solution ax= E/(R*T) x= 1/T a= E/R y= (E/R)*x+G I don't know how to move on and if this is even correct/

Homework Statement [/B] Solve the equation ## e^{2x}+2=e^{3x-4}##Homework EquationsThe Attempt at a Solution I know by using Newton-Raphson method the problem can be solved, i however tried solving it as follows ##e^{3x-4}-e^{2x}-2=0, e^{3x}-e^{2x}.e^{4}-2e^{4}=0, p^3-p^2.e^4-2e^4=0...

1. Homework Statement Find derivative of y=e^(cos(t)+lnt) Homework EquationsThe Attempt at a Solution So just using the chain rule: y'=e^(cos(t)+lnt)*(-sin(t)+1/t) The answer in the back of the book is y'=e^(cos(t))*(1-tsin(t))

The wave function is an exponential function, if I plot the real part of it, I don't get a wave graph like sine or cosine function, Why the wave function is not represented by a trigonometric ratio instead. Also, the wave function cannot be plotted since it is imaginary, why is it imaginary? Thanks

Hey guys, I need your help. I am not sure if this is the right part of the forum to ask this question. So I started reading papers about the Lyapunov Exponent, but there is something I do not understand in the formula. Why ? It seems logical that we want because we want to get the Exponent...

So I'm trying out various practice problems and for some reason I can't get the same answer when it comes to problems involving natural exponentials. Here's the problem A type of lightbulb is labeled as having an average lifetime of 1000 hours. It's reasonable to model the probability of...

Mod note: Changed title from "Differential Euler's Number" 1. Homework Statement Find the derivative. f(t)=etsin2t The Attempt at a Solution f'(t)=etsin2t(sin2t)(cos2t)(2) However the book seems to say that there should be an extra "t" in the solution. Some help?

Homework Statement Hi, I have come across this equation in modelling exponential growth and decay. I am wondering if it is possible to solve it algebraically or not? Homework Equations 8000-1.2031 * e^ (0.763x)=(0.5992×e^0.7895x) The Attempt at a Solution Brought all e^(ax) values to one...

Homework Statement Find the first and second derivative of the following function: F(x)=e4ex Homework Equations d/dx ex = ex d/dx ax = axln(a) The Attempt at a Solution I know the derivative of ex is just ex, but I'm not sure how to go about starting this one. I'm near certain I need to use...

I am multiplying a lognormal distribution by an function to scale it larger. While I know that scaling a lognormal distribution by a constant multiplier yields a lognormal distribution, in this case the multiplier is not a constant. Instead, smaller values from the lognormal distribution are...

Please delete, got mixed up, apologies.

How do I write taylor expansion of a function of x,y,z (not at origin) as an exponential function? Please see the attached image. I need help with the cross terms. I don't know how to include them in the exponential function?

Homework Statement In writing the definition of ##e## i.e. ##e=\displaystyle\lim_{n\rightarrow\infty}(1+\frac{1}{n})^n##, why do we denote the variable by 'n' despite the fact that the formula holds for n∈(-∞,∞)? Is there any specific reason behind this notation i.e. does the variable have...

Homework Statement superman has been disabled by a nearby amount of kryptonite, which decays exponentially. If Superman cannot regain his power until 90% og the kryptonite disintegrates, then how long will it be before he regain his powers? Use r=-0.138629. Round to the nearest day. a. 4days...

Homework Statement [/B] I need to differentiate the exponential function i = 12.5 (1-e^-t/CR) and I need to plot a table so that I can do a graph of i against t but I'm not sure how. (CR is the equivelant of Capacitance 20 Micro Fards and Resistance 300 Kilo Ohms)Homework Equations [/B] How do...

I'm modelling a variable output Y which has a value of 1 at x=0. I've noticed that in the system I'm modelling, as x increases, y increases at an exponentially decreasing rate, up until a limit of around 1.53. I view this as changes in x causing the Y value to increase by a max of 53%. The...

Demonstrate that ##|e^{z^2}| \le e^{|z|^2}## We have at our disposal the theorem which states ##Re(z) \le |z|##. Here is my work: ##e^{|z|^2} \ge e^{(Re(z))^2} \iff## By the theorem stated above. ##e^{|z|^2} \ge e^x## We note that ##y^2 \ge 0##, and that multiplying by ##-1## will give us...

Homework Statement The graph goes through the points (-2, 13) and (0, 5) and has the horizontal asymptote y = 4. f(−2) = ____ therefore: ____(B^____ ) = ____ b = The Attempt at a Solution f(−2) = 13 therefore: 1 (B^-2 ) = 13 b = ? not sure Thank you

When I was first introduced to a derivation of the taylor series representation of the exponential function here (pg 25): http://paginas.fisica.uson.mx/horacio.munguia/Personal/Documentos/Libros/Euler%20The_Master%20of%20Us.pdf I noted the author, Dunham mentioning that the argument was non...

So this question is from Khan Academy. I understood the first part and chose the correct function, but the second question(from 40 degrees to 30 degrees change) explanation confused me. _____________________________________________________________________________ QUESTION: Ajay made a...

This is part of a differential equations group project problem where I solve a set of differential equations to obtain the solution to a function. The part that I am stuck at involves taking the log of an exponential function, though there may be a mistake on the book's part, but I'm not sure...

Homework Statement Prove that y(x,t)=De^{-(Bx-Ct)^{2}} obeys the wave equation Homework Equations The wave equation: \frac{d^{2}y(x,t)}{dx^{2}}=\frac{1}{v^{2}}\frac{d^{2}y(x,t)}{dt^{2}} The Attempt at a Solution 1: y(x,t)=De^{-u^{2}}; \frac{du}{dx}=B; \frac{du}{dt}=-C 2...

OK, I'm new to multi-variable calculus and I got this question in my exercises that asks me to integrate e^{-2(x+y)} over a diamond that is centered around the origin: \int\int_D e^{-2x-2y} dA where D=\{ (x,y): |x|+|y| \leq 1 \} I know that the region I'm integrating over is symmetric...

Hello everyone, I was solving this problem: Justify that ln6= ln2+ln3 So: exp(ln2+ln3)=exp(ln2)*exp(ln3)= 2*3= 6 = exp(ln6) Till here, my work was okay. What I didn't understand is : why should we say that the exponential function is strictly increasing over R before being able to simplify the...