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

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1answer
40 views

SAT2 Level 2 Book Answer Error

I am currently studying for my SAT2 Subject Test in Mathematics Level 2 and was check my answers to a practice test when I can across this (below) question. Problem: George invests $\$1000$ into ...
1
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1answer
48 views

modulo RSA decrypt question

Given the following RSA generated public key: $P(3, 55)$. Which integer value should be chosen for $d$ to decrypt messages encrypted with $P$? Check your answer with $M = 8$ and $C = 17$. ...
0
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1answer
27 views

Fit exponential distribution with noise

I'm trying to fit an exponential with noise (which in this case is a constant $c$) like this one $$y(x)=αe^{−αx}+c,$$ having $(x_i, y_i)$ values (So $α$ and $c$ are unknown and are the ones that I ...
2
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1answer
43 views

Integration of $\int \frac{e^x}{e^{2x} + 1}dx$ [duplicate]

I came across this question and I was unable to solve it. I know a bit about integrating linear functions, but I don't know how to integrate when two functions are divided. Please explain. I'm new to ...
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1answer
75 views

Finite power series [duplicate]

I'm a student and I'm looking for a solution for the following finite power series: $$ \sum_{n=0}^m \frac{1}{n!} x^n $$ By "solution" I meant expansion of the series and finding a closed form ...
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2answers
91 views

Is $\exp$ the only function satisfying $f(x)=\displaystyle \int_{-x}^{+\infty} f(-t) dt$?

Today in class we first dealt with improper integrals, and as an example we found $ \displaystyle \int_0 ^{+\infty} e^{-x}dx=1$. Soon, I noticed that in fact $$e^x=\int_{-x}^{+\infty}e^{-t}dt. $$ ...
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0answers
13 views

Minimum of function involving exponentials

I am trying to prove that this function involving exponentials: $g(x)=\frac{\sqrt{2 \pi } \left(1-2 e^{-2 \pi ^2 x ^2}\right) x }{2 e^{-\frac{1}{8 x ^2}}+\sqrt{2 \pi } x -1}$, when $x\geq1/2$ Is ...
2
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1answer
49 views

Maclaurin series for a function

Provided I have the function \begin{equation*} f(x)=(1+x)^{1/x}, \end{equation*} and I want to calculate a 3rd order Maclaurin series, how can that be done without taking direct derivatives (as ...
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2answers
100 views

Does $y = (-1)^x$ where $x∈ℝ$, change exponentially?

Is $y = (-1)^x$ an exponential curve, or just a sinusoidal one, can it be said to change exponentially as with positive exponents? I'm sure W/A showed this as being sinusoidal with an integer ...
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1answer
34 views

Solve for $N$: $2000N=(0.9025)^{\log_2 N}$

I want to find the value of $N$ while $2000N=(0.9025)^{\log_2 N}$ (This is sample value not actual) How to solve it? The Whole Question which i am solving is $Pe=(Pt/N)(1-δ)^{\log_2 N}$ ...
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1answer
32 views

Find the range of the function $f(x)$ if $f(x) = 2^x + \frac{4}{2^x}$

I tried this by a logical approach as the sum of two positive numbers is constant will be minimum if they are equal , i.e. $\frac{4}{2^x}$ each should be equal to $2.$ Hence minimum value will be $4.$ ...
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2answers
29 views

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
23 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
20 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 ...
1
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1answer
49 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 ...
0
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1answer
33 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. ...
0
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1answer
50 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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2answers
40 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 ...
1
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1answer
71 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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3answers
62 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
54 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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0answers
26 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
35 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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1answer
32 views

Characteristic Function limit to 0

When calculating the limit of the following characteristic function $$ \frac{1}{n+1}\left[ \frac{1-\exp\left( \left(n+2 \right)it \right)}{ 1-\exp(it) } \right]$$ and taking its limit when ...
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0answers
25 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
28 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 ...
2
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1answer
45 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= ...
3
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7answers
489 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
95 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
48 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
33 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 $a_1 < a_0$ ...
1
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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 ...
3
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3answers
80 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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0answers
39 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
17 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
34 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
61 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
53 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
140 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
64 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
12 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
45 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
31 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
22 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% ...
3
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1answer
48 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
125 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
38 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
37 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
89 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 ...