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

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12 views

Financial Mathematics--Finding Compounding Period given Annual and Effective Interest Rates

I'm trying to find a compounding period C when given an annual interest rate r and effective annual yield i. I'm working with the following equation: $i=(1+r/C)^C-1$ I'm having trouble re-writing ...
2
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0answers
49 views

Solving exp integral in closed form?

I am trying to solve the following integrals: 1) $\int \int e^{-(\frac{x^2}{2 m^2} +\frac{y^2}{2 m^2})} dxdy $ 2) $\int \int e^{-(\frac{x^2}{2 m^2} +\frac{y^2}{2 n^2})} dxdy $ 3) $\int \int ...
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1answer
34 views

Solving Equation for $x$

Solve $(a + \sqrt {a^2 - 1})^{x^2 - 2x} + (a - \sqrt {a^2 - 1})^{x^2 - 2x} - a = 0$ for $x$ , where $a>1$ . My approach is as follows : $(a + \sqrt {a^2 - 1}) (a - \sqrt {a^2 - 1})=1 $ Let $(a + ...
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4answers
137 views

Dubious “proof” of $e^x$ derivative?

The proof to which I am referring is amply discussed here: Derivative of exponential function proof, but I remain unconvinced by the answers that pertain to the specific proof discovered by ...
2
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4answers
124 views

What is $\lim\limits_{n\to {\infty}} (\frac{n}{1+n})^n$? [duplicate]

What is $\lim\limits_{n\to {\infty}} (\frac{n}{1+n})^n$. Is it possible to write the function $f(x)=x^n$ and since we know $\frac{n}{1+n}\to 1$, so $f(\frac{n}{1+n})\to 1^n=1$. So the limit it $1$. ...
2
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4answers
46 views

Proving uniform convergence of $(1+\frac{x}{n})^n$ to $e^x$ on compact intervals in the real numbers

My goal is to prove that if $b> a > 0$ are real numbers, then: $\lim_{n \rightarrow \infty} \int_a^b (1 + x/n)^n e^{-x} dx = b-a$. I think the best way to do this is to show that $(1+x/n)^n$ ...
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1answer
70 views

Show by series definition of exponential function that $\exp(-x) \rightarrow 0 $ for $x \rightarrow \infty.$

There are many arguments I have seen using $\ln-$ arguments and other properties of the exponential function to show the existence of this limit $\exp(-x) \rightarrow 0 $ for $x \rightarrow \infty$. ...
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1answer
28 views

Is it possible to represent pieces of two functions with one equation?

I'm trying to create a rudimentary weighting system for evaluating how close two numbers are to each other. (This corresponds to string lengths - coding project for work... happy to explain in more ...
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3answers
44 views

Calculating exponential limit

I've been breaking my mind over this one. Find the limit. $\lim\limits_{n \to \infty} (\frac{n^2+3}{n^2+5 n-4})^{2n} $ I know it equals $\frac{1}{e^{10}} $ but can't figure out how to find it. Help? ...
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5answers
108 views

Maclaurin Expansion for $e^{e^{z}}$ at $z=0$

I need to find terms up to degree $5$ of $e^{e^{z}}$ at $z=0$. I tried letting $\omega = e^{z} \approx 1+z+\frac{z^{2}}{2!}+\frac{z^{3}}{3!}+\cdots$, and then substituting these first few terms ...
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1answer
35 views

Acceleration: If I know distance, time, and initial velocity, what's acceleration and final velocity?

So I know the Initial Velocity ($V_i$), Time ($t$), and Distance ($d$). I know that $$d = V_it + \frac{1}{2} at^2$$ If I rearrange this, would acceleration $a = \dfrac{2(d - V_it)}{t^2}$ ? Then ...
0
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1answer
39 views

Finding probability a particle will appear after t seconds (exponential r.v)

Suppose you are watching a radioactive source that emits particles at a rate described by the exponential density with $\lambda=1$ The probability $P(0,T)$ that a particle will appear in the next T ...
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2answers
52 views

Approximate $\log(1-e^x)$ where $x<0$

The title is pretty self-explanatory, I need to calculate the logit function ($x=\log(p)$): $$x-\log(1-e^x)$$ Where $x<0$, And my problem is to approximate $$\log(1-e^x)$$ I was thinking of ...
3
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1answer
99 views

Find monotonic functions going from $0$ to $+\infty$ for $x \in (-\infty,+\infty)$ (similar to $e^x$)

How can we find functions on $\mathbb{R}$ with exponential-like properties, namely: $f(x)$ is infinitely differentiable; $f(x)$ and all its derivatives are monotonic; $f(x)$ and all its derivatives ...
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2answers
30 views

Solving exponential equation (quadratic type)

I fail trying to solve the following equation: $9^x-6^x-2^{2x+1}=0$ Trying to write it as a quadratic equation makes my constant term exponential $(3^x)^2-2^x3^x-2^{2x+1}=0$ How can I solve this ...
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2answers
24 views

Clarification Needed Regarding $\sinh^{-1}(-3)$

As the definition of $\sinh^{-1}(x)$ goes : $\sinh^{-1}(x)=\ln\left(x+\sqrt{x^{2}+1}\right)$ So what I expect to get is $\sinh^{-1}(-3)=\ln\left(-3+\sqrt{10}\right)$ The value inside of the ...
0
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1answer
25 views

Using exponential decay function to predict outcome

Let's say I have a graph that follows the function $y= ae^{-bx}$ , and I'm trying to predict the chlorine residue left in a pool after a certain amount of time. So for $2$ hours, the chlorine residue ...
2
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0answers
49 views

Is there a way to solve the exponential equation $a^x + b^x + c^x = d$ analytically?

So I came across this equation. $$a^x + b^x + c^x = d$$ where $a, b, c$ and $d$ are all constants. And I just wondered, is there any way to solve for x analytically?
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3answers
32 views

Is there a non-exponential function whose limit at infinity is a real, irrational number?

$e$, for example, can be calculated through a non-polynomial function $(1+1/x)^x$, but I cant think of an example for a non-exponential function (or rational function) where the limit to infinity ...
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1answer
63 views

Prove that $n! = O(n^n)$

I thought $n^n$ was greater than $n!$. How would I go about proving this? I have this so far: Assume that $P$($n$) is true $n!$ = O($n^n$) Assume that $P$($n+1$) is also true $(n+1)! ...
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1answer
64 views

Two problems with Exponents

How to solve following problems on exponents: $$\frac1{1+p^{a-b}+p^{a-c}}+\frac1{1+p^{b-c}+p^{b-a}}+\frac1{1+p^{c-a}+p^{c-b}}=?$$ and If $a^2bc^2=5^3$ and $ab^2=5^6$, what is $abc$? Please ...
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1answer
47 views

How to solve this difficult one variable equation analytically?

Would anybody like to explain me clearly how to solve analytically this equation? $$5.56=\frac{1-e^{-5.5x}}{1-e^{-x}}$$ I have already solved it with Mathematica and it gives $x=-0.004809$. However, ...
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0answers
27 views

calculating growth rate with exponentials vs. non-exponentials?

i'd like to calculate the growth rate of growing bacterial population. at $t_1 = 0 hr$, the population size (based on density) is: $N_1 = 0.17$ and at $t_2 = 0.5hr$ it is $N_2 = 0.25$. using a simple ...
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2answers
35 views

How can I solve this trig exponential equation ??

I tried taking $\log_2$ of both sides and I got: $(\sin(x))^2\cdot(1-\cos(x))=1$ Is it correct? If it is, how should I continue ?
2
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2answers
50 views

$\lim_{n\to \infty} (1+n+\cos n) ^\frac{1}{2n+n \sin n}$

While in class, we were proving a limit problem using the Squeeze Theorem, but when I was reviewing my notes, I came up with a problem,, The first question was to prove that $$\lim_{n\to ...
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2answers
64 views

Does this definition of $e$ even make sense?

This sprung from a conversation here. In Stewart's Calculus textbook, he defined $e$ as the unique solution to $\lim\limits_{h\to 0}\frac{x^h-1}{h}=1$. Ahmed asked how do you define $x^h$ is not by ...
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2answers
44 views

What is $\frac{(-2)^{x}}{2^{x-1}}$

The title says it all: $$\frac{(-2)^{x}}{2^{x-1}}$$ How is this computed? I'm reviewing the finer points of exponents so a thorough explanation would be most appreciated!
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1answer
27 views

Can anyone walk me through calculating the differential equation…

I need help solving this equation, i've attempted using numerous methods. But I'm give choices with square roots as an exponent of e, and I haven't been able to match any of them. $$ ...
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1answer
31 views

When will the population of a sample double (using dif-eq)?

I have the initial equation $$\frac{dP}{dt}=kp$$ where P is the population, t is time, and k is some positive constant. The rest of what I'm given is that P(0) = A, what is the time for the population ...
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2answers
14 views

Solution to initial condition problem

$y=-ln(1-e^{(t+c)})$ I'm trying to find the solution to the initial condition $y(0)=-ln2$ Isolate c $0=ln(2)-ln(1-e^c)$ $0=ln({2\over1-e^c})$ $-e^c=2-1$ $e^c=-1$ $c=0$ I can't figure out ...
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1answer
34 views

Probability of exponential growth event

Under the assumption of exponential growth of a population of cells, the population size at time $t$, $N(t)$, is: $$N(t) = N_0\exp(rt)$$ where $r$ is the rate of division and $t$ is time. What is ...
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1answer
14 views

Is it possible to clear the x using the Lambert function?

$ y = \frac{x^2}{4} - \frac{ln(x)}{2} $ Solving, I get to: $ e^{4y} = \frac{e^{x^2}}{x^2} $ But I don't know how to continue.
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14 views

Rearranging summation terms including a complex exponential expression

I'm reading a paper on signal processing and having a hard time wrapping my head around a step the author takes. The signal of interest is defined as $r_k = e^{j(2\pi\Delta f k T_s + \theta)} + v_k$ ...
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1answer
46 views

Stuck solving $\ln(e^y-1)-y=t+c$ for $y$

I'm trying to solve for $y$ $\ln(e^y-1)-y=t+c$ $e^y-1=e^{(t+c+y)}$ $e^y=e^{(t+c+y)}+1$ $y=t+c+y+1$ Where am I going wrong?
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0answers
43 views

Question about the connection between exponential and logarithmic functions

Does this make sense to anyone? What advice would you give me to clarify my reasoning and explanation? One of the really "neat" features of the exponential function: $$f(x)=e^x$$ is the fact that ...
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0answers
19 views

Proving Exponential Convergence

Consider the function $\dot{x} = f(x,t)$. I want to show that if there exists a function $V(x,t)$ and some positive constants $h,\delta,k_1,k_2,$ and $k_3$ such that for all $x \in B(0,h)$ and for all ...
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0answers
22 views

Closed form roots of sum of exponential functions

Do anyone know a way to solve an equation like the following (over the complex numbers)? $1+2^z+3^z=0$ I certainly cannot. I've tried by hand, and by mathematica, but I can't figure it out. Thanks in ...
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1answer
29 views

Integration of complex exponential function over $\mathbb C$

Find the limit $$\lim_{z \to \infty}\int_{\mathbb C}|w|e^{-|z-w|^2}dA(w) $$ where A is area measure such that dA=rdrd$\theta$ Please help me, I did four page computation by changing to polar ...
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3answers
54 views

Solve for $x$ for the following exponential equation $2^{2x+1} = 3^{2x+1}$. What am I doing wrong?

$2^{2x+1} = 3^{2x+1}$ $2^1=3$? Why can't I take $\log_2$ of both sides ?
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1answer
44 views

Exponential equation on the set of real numbers

Solve the following equation on the set of real numbers: $8^x+27^x+2·30^x+54^x+60^x=12^x+18^x+20^x+24^x+45^x+90^x$ $x=1; x=0; x=-1$ are trivial solutions, but I'm stuck with proving that there are ...
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0answers
14 views

Bounds for infinite series involving exponentials

Let $$ S(a,b):=\sum_{j=1}^\infty \exp(-a j^b ) , \quad a,b > 0 $$ which (due to monotonicity) can be bounded by $$ S(a,b) \leq \int_0^\infty \exp(-a x^b ) \, \mathrm{d} x = ...
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2answers
66 views

At what point does exponential growth dominate polynomial growth?

It's well-known that exponential growth eventually overtakes polynomial growth (link, link). So for any non-negative integer $d$ and positive $\epsilon$, there exists $t^* \ge 0$ for which $$ 1 + ...
3
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2answers
97 views

Solve $\sqrt x = 1 + \ln(3 + x)$ algebraically

I am having trouble with this homework problem. I am able to graph and find the solution, but I am curious as to how one would do this algebraically. The way I began, was subtracting $1$ on both ...
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4answers
114 views

Solving $e^x - 3 = 0$ [closed]

I want to solve this equation for $x$: $$e^x - 3 = 0$$ Can somebody give me some hints? Thanks.
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3answers
37 views

How do i solve these exponential equations? [closed]

Is there a way to solve these exponential equations without using logarithms? I tried to get the same base for all the terms, but I could not make it. Is there any other general procedure that I can ...
2
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0answers
50 views

Does the continued fraction for $e^{3/n}$ have a pattern?

Wikipedia has patterns for the simple continued fractions $e^{1/n},e^{2/n}$, which made me wonder whether there is one known for $e^{3/n}?$ (by pattern, I mean that the partial quotients $a_n$ can ...
2
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4answers
75 views

How to solve an equation like $2{^x} + x = 2 $?

I encountered an equation similar to this in an old math exam. $2{^x} + x = 2 $ I couldn't figure it out and ended up with a mess of logarithmic functions. The answer sheet indicated it should be ...
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1answer
19 views

Differing graphs for simple inverse exponential problem

In class, we are learning exponential functions. The following inverse exponential problem is bothering me: $y=x^{-\frac{1}{9}}$. When graphed, I feel that it should look like it does on Desmos: ...
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1answer
31 views

How to solve the following limit using mathematics Stirling $\lim\limits_{n\to \infty}\frac{n!}{n^ne^{-n}\sqrt{2\pi n}}=1$

How to solve the following limit using mathematics Stirling $\lim\limits_{n\to \infty}\frac{n!}{n^ne^{-n}\sqrt{2\pi n}}$. a) $\lim\limits_{n\to \infty}\frac{n!e^n}{n^{n+1/2}}$. b) $\lim\limits_{n\to ...
7
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1answer
1k views

A curious property of $\operatorname{frac}(e\cdot k)$

Let $\alpha > 0$ be a real number and let us consider the set $S(\alpha)$ of those natural numbers $n$ such that the fractional part of $\alpha \cdot n$ "begins" with the representation of $n$ (in ...