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

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0
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2answers
25 views

Value of $\lambda$

If $y=e^{2\cos^{-1}x}$ also $$(1-x^2)y_2-xy_1-\lambda y=0$$ then the value of $\lambda$ is. I see that the question is incomplete but answer is given as $2$. Am I missing on anything.
1
vote
3answers
43 views

How to determine Laurent series associated to $f(z)$ [on hold]

The function is $$f(z)= \frac{1}{(e^z -1)},$$ $z$ belong to $\mathbb{C}$ and $0<|z|<1$. I need a general expression in term of a sum from 0 to infinity
2
votes
1answer
15 views

Exponential limit convergence for each $x$

I have $f_n(x)=\left( 1+\frac{-e^{-x}}{n} \right)^n$, what about the convergence to $f(x)=e^{-e^{-x}}$? is it true $\forall x?$ I say yes, but how can I show this? Is continuity of $f_n$ enough?
3
votes
4answers
96 views

Solve $2^x+4^x=2$

This is the equation, but the result is different from wolframalpha: $$2^x+4^x=2$$ $$2^x+2^{2x}=2^1$$ $$x+2x=1$$ $$x=\frac{1}{3}$$ WolframAlpha: $x=0$ Where is the error?
0
votes
1answer
33 views

Limits involving $e$ [on hold]

I am looking to solve the following limits. $\displaystyle\lim_{x\to-\infty}5e^{-x}$ and $\displaystyle\lim_{x\to2}\frac{1}{2e-ex}$ Any help would be appreciated.
8
votes
1answer
90 views

Fake proof that $\frac{e^x-1}{e^x+1}=e^x$, via integrating $\operatorname{sech} x$ in two ways

We start with the integral: $$\int \text{sech}(x)dx$$ Method 1 \begin{align} \int \text{sech}(x)dx & = \int\frac{2}{e^x+e^{-x}}dx \\ &= \int\frac{2e^x}{e^{2x}+1}dx \end{align} Using the ...
0
votes
2answers
44 views

Prove that $\frac{1}{2} (e^R - e^{-R}) \geqslant \frac{1}{4} e^R $ with $R > 0$

Sorry to bother you with silly question, but I can't figure out how to prove: $$\frac{1}{2} (e^R - e^{-R}) \geqslant \frac{1}{4} e^R $$ with $R > 0$. I tried different ways but that didn't lead ...
2
votes
3answers
42 views

Identity with exponential function: $\lim_{n\to\infty}\frac{n^{2n}}{(n+1)^{2n}} = \frac{1}{e^2}$

Could you please explain me how we got this identity $\lim_{n\rightarrow \infty}\frac{n^{2n}}{(n+1)^{2n}} = \frac{1}{e^2}$ when we know $\lim_{n\rightarrow \infty}(1+\frac{1}{n})^n = e$ Thanks!
0
votes
1answer
21 views

applying exponents in an absolute value brackets

So part of the problem I'm trying to solve is this: $|2-9|^{3{^3}}$ being the exponent, to the power of 3. Do I have to apply the exponent to everything in the bracket? 2*2*2 and 9*9*9 or does the ...
0
votes
3answers
41 views

Does this sequence have a closed form representation?

We know that $$ \sum_{s=0}^\infty \frac{\lambda^{s}}{s!} = e^\lambda$$ Relatedly, $$ \sum_{s=1}^\infty \frac{\lambda^{s}}{s!}s = \lambda \sum_{s=1}^\infty \frac{\lambda^{s-1}}{(s-1)!}$$ For which ...
3
votes
0answers
67 views

Does this equation have no solutions?

The question is this : The source from where I got this question was devoid of any answers to it, so I came here, this is how I proceeded : LHS : $((((({(x)^x})^{2x})^{3x})^{....x^2})^2 = (((((x)^...
-1
votes
0answers
21 views

Add a factor to an exponential function from 0 to n, so that the result for x = 0 and x = n stays thr same [closed]

I am trying to figure out how to add a factor to my function y=EXP(x/255)-1)*255/(EXP(1)-1) (x=0...255) (Wolfram Alpha plot), so that the conditions ...
0
votes
2answers
100 views

Proving that $\pi$ and $e$ are rational numbers [duplicate]

Maybe this question is too dumb to be asked, but it's really bugging me so I decide to ask it anyway. I hope you bear with me. Okay, it's known that both sides of the following series equal. $$\pi=...
4
votes
1answer
69 views

Ways to squeeze $e$ by hand

Let $a$ and $b$ be the lower and upper bound of $e$, respectively. Both $a$ and $b$ are rational numbers. Without using a calculator and without knowing the value of $e$, find $a$ and $b$ where $b-a&...
7
votes
3answers
138 views

Proving $\pi \gt e+\frac{1}{e} \gt \pi-\frac{1}{\pi} \gt e$

I created this problem for myself as a fun exercise. I want to prove the following statement: $$\pi \gt e+\dfrac{1}{e} \gt \pi-\dfrac{1}{\pi} \gt e$$ I found that the following upper/lower bounds ...
2
votes
1answer
75 views

Is it valid to write $1 = \lim_{x \rightarrow 0} \frac{e^x-1}{x} = \frac{\lim_{x \rightarrow 0} (e^x) -1}{\lim_{x \rightarrow 0} x}$?

I just want to clarify one thing I was never really sure on. First the question: $$1 = \lim_{x \rightarrow 0} \frac{e^x-1}{x} = \frac{\lim_{x \rightarrow 0} (e^x) -1}{\lim_{x \rightarrow 0} x}$$ is ...
0
votes
3answers
51 views

how so simplify this exponential equations

((a^3/2)/(b^3))/((a^-1)/(b^2)) I tried to solve this problem many times, however I tend to get the wrong answer. Here is the method I tried (((a^3)^1/2)/(b^3))*... sorry I get confused i got (...
0
votes
2answers
70 views

How can I get the expression of x?

If there is $$x^2e^{A\sqrt x}=B$$ then what is the expression of $x$? If this cannot be solved, is there any approximation?
3
votes
2answers
71 views

Proofing that the exponential function is continuous in every $x_{0}$

Given: $$\exp: \mathbb{R} \ni x \mapsto \sum_{k=0}^{\infty } \frac{1}{k!} x^{k} \in \mathbb{R}$$ also $e = \exp(1)$. For all $x \in \mathbb{R}$ with $\left | x \right | \leq 1$: $$\left | \exp(x) -...
2
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1answer
60 views

Proving that the exponential function is continuous

We aren't allowed to use many tricks such as difference quotient / integral calculus... Prove that $\exp$ is continuous at $x_{0}=0$ .....................................................................
0
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0answers
27 views

Integrate this expression containing exponential

I am trying to integrate $$ \int x^a f'(x)\lambda \exp(f(x)\lambda) dx$$ I'm not that great at these things, but I noticed that $f'(x)\lambda \exp(f(x)\lambda = \frac{d}{dx} \exp(f(x)\lambda)$. I ...
0
votes
1answer
29 views

How can I prove this inequality involving the exponential function?

Given $$\exp: \mathbb{R} \ni x \mapsto \sum_{k=0}^{\infty } \frac{1}{k!} x^{k} \in \mathbb{R}$$ also $e = \exp(1)$. For all $x \in \mathbb{R}$ with $\left | x \right | \leq 1$: $$\left | \exp(x) - ...
3
votes
1answer
57 views

Calculate: $f(a)=\int\limits_{-\infty}^{\infty} \exp(-|x|^a)\mathrm{d}x$

Given the following function: $$ f(a)=\int_{-\infty}^{\infty} \exp(-|x|^a)\mathrm{d}x $$ For which values of $a$ is it possible to give an exact value for this function? I only know $f(2)=\sqrt{\pi}$...
0
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1answer
29 views

Please explain how to do this, base e [closed]

Express $ 3^X$, $x^\pi$, $x^{\sin x} $ using base $e$.
0
votes
0answers
7 views

How for continuous compounding of interest, the difference in balance over difference in time is equal to interest rate times balance at the instant?

I was reading the chapter on exponential growth and decay in Morris Klein's Calculus book. He says $\delta A=A(0.04)\delta t$, 0.04 the interest is in unit percent per year. A is balance. t is time. ...
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votes
0answers
20 views

Determining bounds for a sum with nested infinite series

I am computing the inner product of the characters of the trivial and the $k$-th irreducible two dimensional representations of the dihedral group $D_n$ of order $2 n$ when $n$ is even. The ...
-1
votes
1answer
89 views

Prove $e^x - e^y \leq e |x-y|$ for $x$ belonging to $[0,1]$ [closed]

I'm not sure how to go about this. Does it involve using MVT? I got as far as saying $e = \frac{e^x - e^y}{x-y}$.
3
votes
2answers
86 views

I can't complete the integration of $e^{\sqrt{x}}$

Compute $\displaystyle\int_0^1e^{\sqrt{x}}\,dx$ That's a picture of how far I could get while trying to integrate $e^{\sqrt{x}}$. I tried the substitution method first, (boxed part) and then went ...
0
votes
1answer
22 views

Exponential decay + a recurrence relation

I'm not sure if I get this right, some pointers could be helpful. Say you have to take 60m of some sort of medication at midnight. It has a blood half-life of 6 hours. Meaning that after 24 hours 3....
0
votes
1answer
92 views

What is the value of $e^{-10000}$?

What is the value of $e^{-10000}$? We know that the function $e$ does not attain value $0$ anymore. But in R and Matlab the value of $e^{-10000}$ is given as $0$ which is not correct anymore. I ...
0
votes
1answer
19 views

Bounding an exponential integral

I'm having trouble seeing this bound I've seen on a proof. Let $f$ be a polynomial, and $F$ the polynomial obtained from $f$ by replacing each coefficient by its absolute value. Then: $$\bigg{|}\...
0
votes
0answers
39 views

Prove Exponential series from Binomial Expansion

I try to prove the Exponential series : $$\exp(x) = \sum_{k=0}^{\infty} \dfrac{x^k}{k!}$$ From the definition of the exponential function $$\exp(x) \stackrel{\mathrm{def}}{=} \lim_{n\to\infty} \left(...
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votes
2answers
64 views

Exponential of a number [closed]

What is exponential of a number? $E^{10}=22026.4657948$ What is the mathematical calculation behind the above calculation? Regards, Philip
0
votes
1answer
40 views

Proof that $a^x$ goes towards infinity as x goes towards infinity

I'm tasked to prove that $a^x \rightarrow \infty $ when $x \rightarrow \infty$ provided that (a > 1). I've found a very rigorous proof for this. But my question is, why can't it be logically realized ...
0
votes
0answers
16 views

How do I determine and sketch the images $g(\mathbb{R})^2$ as sets and as geometric objects?

It's given function $g(x, y) = \begin{pmatrix}e^x \cos y\\ e^x \sin y\end{pmatrix}$. How do I determine and sketch the images $g(\mathbb{R})^2$ as sets and as geometric objects?
1
vote
4answers
80 views

Prove that for all $a > 0$: $\int_0^{\pi/2}e^{-a\cos x}\cos(a\sin x)dx = \frac{\pi}{2} - \int_0^a\frac{\sin x}{x}dx$

Prove that for all $a > 0$: $$\int\limits_0^{\pi/2}e^{-a\cos x}\cos(a\sin x)dx = \cfrac{\pi}{2} - \int\limits_0^a\cfrac{\sin x}{x}dx$$ I have no idea how to solve it. But the task looks very ...
3
votes
2answers
48 views

How do I simplify this Log with a Fraction in it?

So I have: $$ \log_2(5x) + \log_2 3 + \frac{\log_2 10}{2} $$ I understand that when there is addition, and the bases are the same, I can simply multiply what is in the parenthesis. So for the first ...
0
votes
1answer
34 views

Find how many years must elapse before the proportions of red kangaroos and grey kangaroos are reversed, assuming the same rates continue to apply.

I have this question (sorry I'm not able to embed it): Q.7. There are approximately ten times as many red kangaroos as grey kangaroos in a certain area. If the population of grey kangaroos ...
4
votes
2answers
68 views

For which classes of matrix can the matrix exponential be easily computed?

We have diagonal matrices $A = \mbox{diag} (\lambda_1, \ldots, \lambda_n)$ for which matrix exponential has simple form $e^A = \mbox{diag} (e^{\lambda_1}, \ldots, e^{\lambda_n})$, and it can be ...
0
votes
0answers
14 views

Finding the CDF for $Y = e^X$ when $X \sim N(0,1)$

Problem: Let $X \sim N(0,1)$ and let $Y = e^X$. Find the CDF for $Y$. Attempted Solution: Let $y = e^x$ so that $x = \ln(y)$. Then $$ F(y) = P(Y \le y) = P(Y \le e^x) = P(X \le \ln(y)) = F_X(\ln(...
0
votes
2answers
65 views

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 ...
-1
votes
1answer
27 views

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 ...
0
votes
2answers
46 views

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 ...
2
votes
5answers
149 views

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 $0$ ...
0
votes
1answer
53 views

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 ...
3
votes
4answers
387 views

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 ...
1
vote
3answers
49 views

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 ...
0
votes
4answers
58 views

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.
0
votes
2answers
20 views

Why does the exponential distribution's pdf integrate to 1? [duplicate]

From All of Statistics pg. 29: EXPONENTIAL DISTRIBUTION. $X$ has an Exponential distribution with paramater $\beta$, denoted by $X \sim \text{Exp}(\beta)$, if $$ f(x) = \frac{1}{\beta}e^{-x/\...
2
votes
3answers
139 views

Why doesn't continuously compounded interest make me a zillionaire?

It would seem that if I have some money and I get an interest on it every second, I'd be a zillionaire in no time. However, as the formula for the continuously compounded interest is: $A(t) = P(1 + \...