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

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Function Growth Question

Which is greater as $n$ gets larger, $f(n) = 2^{2^{2^n}}$ or $g(n) = 100^{100^n}$? I tried differentiating the terms but it didn't really reveal anything. Can anyone come up with a solution? Thanks ...
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1answer
17 views

Isolating $t$ in doubling time formula: $500000 = 120 \cdot 2^{\frac t 2}$

I am a having trouble figuring out a way to rearrange the formula $500000 = 120 \cdot 2^{\frac t 2}$ in order to isolate t and get the time.
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0answers
10 views

Deriving equation for sequential decay?

The differential equation describing the decay of a particle (p1) into another particle (p2), which then decays into a further particle (p3) is: where is the number of p2 particles, and is the ...
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0answers
7 views

Exponential Distribution Unbiased Estimate of Coefficient of Variation?

Through simulation, I've noticed that estimates of the coefficient of variation (CV) of exponentially distributed variables are biased at low sample sizes (as seen in the plot I made). I've seen an ...
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1answer
22 views

Derivative Word Problem about Virus Spreading

I had this question on a practice sheet for our calculus unit, and I am kind of confused by the following question. At lunch one day, the flu rapidly starts infecting the students at the school. ...
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1answer
45 views

Limit representation of the exponential function

A well known fact is that $$\lim_{n\to\infty} \left(1+\frac{a}{n}\right)^n=e^a$$ Now I was wondering what if $a$ also depends on $n$? In particular take $$\lim_{n\to\infty} ...
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1answer
34 views

The function $f(x)$ is defined on the real numbers has the property that $f(f(x)) ( 1+f(x)) = - f(x)$ for all $x$ in the domain of $f$ . [on hold]

The function $f(x)$ is defined on the real numbers has the property that $f(f(x)) ( 1+f(x)) = - f(x)$ for all $x$ in the domain of $f$. If the number $3$ is in the domain and range of $f$ then the ...
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2answers
16 views

Convert a value in a logarithmic sequence to a linear equivalent

Sorry if im asking a silly question. Its been a while since varsity maths. I have a logarithmic sequence ranging from [1 to 32]. It is a signal strength value from a modem. I need to display this ...
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1answer
34 views

Compounded Interest with Exponentially Increasing Periodic Payments

Given the formula $$v_a = p\left(\frac{\left(1+\frac{r}{n}\right) ^{nt}-1}{\frac{r}{n}}\right)$$ for the value $v_a$ of an account growing at a periodic rate $r$ with a regular deposit $p$ compounded ...
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2answers
56 views

Exponential Integration [duplicate]

I don't know how to solve this equation: $$\int_0^\infty e^{-x} (x-a)^m dx$$ where $a$ is a constant and $m$ = $n+1$ Thanks in advance for your help.
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3answers
55 views

Finding the dy/dx of a complicated function

I need urgent help on this question. I have no clue how to solve it as it's very complicated to me. The question is the following: Given $y=\frac{2xy}{x^2 + y}$ find $\frac{dy}{dx}$.
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2answers
53 views

Manipulating the definition of $e$

I know that $\lim\limits_{n\rightarrow \infty}(1+\frac{1}{n})^n=e$ I'm trying to show $\lim\limits_{t\rightarrow \infty}(1+\frac{1}{t^2})^{t^2}=e$ If I write $n=t^2$ then ...
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1answer
20 views

Hypercubes, exponential functions [on hold]

Kyle is trying to write an interactive computer program that draws cubes in any number of dimensions. From his research, he found that the number of the form n^4 with an an positive integer is called ...
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0answers
22 views

Speed of the usual approximation of the exponential

Let's consider the usual approximation of the exponential function $f_n(x)=(1+\frac{x}n)^n$. What do we know about its speed of convergence to the exponential? That is to say, how can we characterize ...
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1answer
24 views

Draw the graphs of $y=x-1$, $y=x$, $y=x+1$, & $y=xe^{\frac{-1}{|x|}}$ for , $-\infty< x< \infty$ using the same $X$ and $Y$ axes.

In the above question, I could easily plot the linear equations. But I don't know how to plot $y=xe^{\frac{-1}{|x|}}$. Can you please explain me, how to draw this exponential curve? Thank you
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0answers
20 views

How to determine contours by looking at the exponential integrands?

I know that we determine the contours in contour integrals by looking at the exponential integrand (assuming there is indeed an exponential integrand in the given integral) but I don't know how. For ...
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0answers
16 views

when cdf=( i-0.5)/n and you have a negative

I am stuck with when you set your cdf to equal $\frac{i-0.5}{n}$ for when you are plotting QQ plots. I have: $$-e^{\frac{-x^2}{2\sigma^2}}= \frac{i-0.5}{n}$$ Then I got stuck because I need to take ...
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1answer
41 views

integration by parts exponential

How do you integrate $$\frac{x}{\sigma^2} \exp \left( \frac{-x^2}{2\sigma^2}\right)$$ I have so far tried integration by parts and have gotten stuck. $$u= \frac{x}{\sigma^2}$$ $$du= ...
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7answers
373 views

How to evaluate the limit $\lim_{h \to 0} \frac{e^{2+h}-e^2}h$?

$$ {\lim \limits_{h \to 0}} { {e^{2+h}-e^2 } \over {h} } $$ Due to time constraints, evaluating limits with e in them wasn't covered and I have this on the AP exam review. How do I proceed?
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5answers
67 views

Prove that $ex \leq e^x$ for all $x \in \mathbb{R}$

This is easy to prove for negative $x$ but what about positive $x$? Should I use MVT?
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4answers
41 views

Troubles understanding this limit

I have troubles understanding this limit: $$\lim_{x\to0} \frac{a^x -1}{x}=ln( a)$$ I have the following proof: $$\frac{a^x -1}{x}=\frac{e^{xlna}-1}{x}=\frac{e^{xlna}-1}{x ln(a)}ln(a) ...
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2answers
121 views

Exponential and logarithmic functions [on hold]

The temperature, $T (C^◦)$, of a quenched steel plate at time t is given by $$\large{T = f(t) = 10 + \frac{85}{1 + e^\frac{\large{\textbf{t}}−60}{10}}}$$ where $0 ≤ t ≤ 100$ measured in seconds. (a) ...
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2answers
22 views

Population ratio of Birth control to no birth control

A country currently has a population of $N_0$ and growth rate of $a_0$. The country introduces, at $t = 0$, a birth control scheme which hopes to gradually reduced the growth rate to $a_1 < a_0$ ...
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1answer
30 views

Find range of the given function : $ f(x) = \frac{e^x}{1+ [x] } $ when $ x \ge 0 $

Find the range for $ f(x) = \cfrac{e^x}{1+[x] } $ when $x\ge 0$ . Where $ [.] $ denotes greatest integer function. My book answers it in a very straight forward manner - Here f(x) is ...
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3answers
64 views

Zeroes of sin(x)

Consider the function f = $\sin(x)$ defined as $$ \sin(x) = \frac{e^{ix}- e^{-ix}}{2i} $$ How to prove that the only zeroes of this function lie on the line $i = 0$ in the complex plane and ...
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2answers
29 views

Solve this: $log_3{a}=log_{10}{a}$

Solve this: $log_3{a}=log_{10}{a}$ Please don't use this $$\log_b (x) = \frac{\log_{10} (x)}{\log_{10} (b)} = \frac{\log_{e} (x)}{\log_{e} (b)}. \,$$ We haven't learnt it yet.
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0answers
25 views

Prove that for the function $u=c a^x$ there is a number $T$ for which $f(x+T)=\frac 1 2 f(x)$.

Prove that for the function $u=ca^x$ there is a number $T$ for which $f(x+T)=\frac 1 2 f(x)$. Here's what I did: $$ca^x a^T= \frac 1 2 x a^x \Longrightarrow a^T=\frac 1 2 $$ Is it OK or should I ...
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4answers
44 views

Solve for $x$ given $2^{2x} - 2^{x+2} = 5$ [closed]

$$2^{2x}-2^{x+2}=5$$ I know I am being dumb, but I can't figure out how to factor this one. I need to solve for $x$.
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0answers
37 views

Exponent - Solving for an unknown within an expectation

I have reached a stage where I need to solve for an unknown number, $\theta$ . However, I stuck and don't know how to proceed further. The equation to be solved is: $E\left[ \exp(\theta a^i) * ...
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0answers
15 views

Computing efficiently a small base to the power a large number

Is there a fast algorithm to compute an exponential with a small base, (namely , close to 1) For example, computing 1.01 to the power 100?
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0answers
27 views

Minimizing Unintegrable Exponential Function

I am trying to develop an algorithm which minimizes an unintegrable function. I don't have a strong mathematics background and am unaware of such strategies. My integral is of the following form: ...
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0answers
16 views

Find the set of points on which the maps of $e^z$ and $\log(z-1)$ are expanding and contracting.

I understand that $e^z$ is has a domain $\Omega$ such that $\Omega = \Bbb {C}$ and is analytic on the whole complex plane, but I have never been tasked with understanding the map of a function that is ...
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2answers
59 views

Is it possible to find the value of $x$ where $e^x$ exceeds $x^{10}$ by hand?

All I managed is to "simplify" the equation $e^x=x^{10}$ to $\frac{x}{\ln{x}}=10$. Is there some way or trick to make the equation look like $x=\dots$? (Solve the equation, in other words.)
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0answers
39 views

$\pi$ base $e$ or $\pi=\sum\limits_{n=-1}^{\infty} a_ne^{-n}$ where $a_n\in\{0,1,-1\}$

I was "playing with $\pi$" trying to look at it in different numeral systems and it's not so hard to obtain $\pi$ base $2$ or $3$ or even $\varphi=\frac{\sqrt{5}+1}{2}$, using Maclaurin series of ...
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2answers
53 views

Find when the population is growing the fastest, under the logistic model

The population $P$ of an island $y$ years after colonization is given by the function: $\displaystyle P = \frac{250}{1 + 4e^{-0.01y}}$. After how many years was the population growing the fastest? ...
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2answers
63 views

Proving that $3^n<n!$ when $n\geq 7$

It's been 10 years since my last math class so I'm very rusty. How would I go about proving $$3^n < n!$$ where $n \geq 7$? I understand that factorials grow faster than set values with a variable ...
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0answers
11 views

Shifting a series of functions whilst maintaining symmetry

I have a function y = a*Exp[-(x - b)^2/2*c^2] + d*Exp[-Abs[-e*x]] + f Which is symmetrical when the coefficient of b is equal to 0 however it loses symmetry as ...
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1answer
39 views

Integral - complex exp. term

Does anyone know a suitable method to integrate and/or know the answer to: $\int\limits_{-\pi}^{\pi}$ $\log\Big[\tfrac{2 - a\exp({-it})}{1 - a\exp({-it})}\Big] $ ${\mathrm{d}t}$, for constant $|a|$ ...
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1answer
29 views

Linearizing an expression involving exponentials

How can I linearize $f(x) = A(1-\text{exp}(Bx))$? I tried to take the natural logaritm, but could not find something that looks like linear. I am trying to find a fitting curve for this by hand. $A$ ...
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0answers
20 views

How are Compound Interest and Infinite Series related?

The mathematical constant e happens to be both $\lim_{n\to\infty}(1 + \frac{1}{n})^n$ and $1 + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + ..$. The 1st formula is of compound interest with 100% ...
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1answer
40 views

Taylor series of f(x + a) becomes exponential

In my symmetries of classical mechanics course we have looked at taylor expansions. Our notes claim that; $$ f(x + a) = \sum_{n=0}^\infty \frac{1}{n!} f^{(n)}(x)a^n ≡ \exp{\left( a ...
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9answers
118 views

If $a_n = \frac{e^{n}}{e^{2n}-1}$ how do I show that $a_{n+1} \leq a_n$?

Let $$a_n = \frac{e^{n}}{e^{2n}-1}$$ How do I show that $a_{n+1} \leq a_n$? I don't know how to deal with the $-1$ in the denominator.
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1answer
36 views

Oil decay at 13%, how long until it is less than 21% of original?

My teacher gave me this problem, and it is very wordy, I don't really even understand what it is asking. First I took 100 and multiplied it by 0.13 subtracting that number from 100 and completing the ...
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1answer
36 views

Exponential Function Equation and inverse Pre-Cal

I am not completely sure if I wrote the equation correctly. For A I wrote: $m(t)=100(b^x)$ Not sure it is correct...but how do I find the inverse? That doesn't make sense to me. Do I use log?
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2answers
84 views

Evaluate $\lim_{x\rightarrow \infty}(1+\frac{1}{\sqrt{x}})^{\sqrt{x}}$. Euler's Limit

Evaluate $\lim\limits_{x\rightarrow \infty}(1+\frac{1}{\sqrt{x}})^{\sqrt{x}}$. Can I get some help? I am thinking that the limit does not exist. If you approach it from the left and then from the ...
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2answers
38 views

Derivative of Matrix Exponential as Integral

I saw this "standard" identity in a physics paper and I was wondering how to prove it \begin{align*} \frac{d}{dx} e^{A+xB}\bigg|_{x = 0} = e^A\int_0^1 e^{A\tau}B e^{-A\tau}\,d\tau \end{align*} I tried ...
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0answers
31 views

Help understanding exponential formula

I am reading a paper in which a group is approximating data that fits an exponentially declining curve. They use the following formula to fit the data, where τ is the y-axis variable and v is the x ...
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1answer
37 views

Derivative of an expodential function

Let the function: \begin{equation} \phi(x)=\begin{cases} e^{-\frac{1}{1-x^2}}, & x\in (-1,1)\\ 0, &x \not \in (-1,1)\end{cases} \end{equation} How can I show that $\phi(x)$ is differentiable ...
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0answers
14 views

Characteristic function of an asymmetric Laplace distributed random variable

What is the characteristic function of a random variable with density $$f_X(x) = \frac{1}{2} [ 1_{x>0} \, a e^{-a x} + 1_{x<0} \, b e^{b x} ], \; \; \; \quad a,b > 0 \quad \quad ? $$ My ...
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1answer
59 views

Proving that the matrix exponential map is surjective onto the general linear group

Let $M_n(\mathbb{F})$ be the set of all $n\times n$ with entries in $\mathbb{F}$ and let $\exp:M_n(\mathbb{C})\to M_n(\mathbb{C})$ be defined by $$ \exp(A)=\sum_{k=0}^{\infty}\frac{A^k}{k!},$$ for ...