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

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2answers
40 views

Implicit Differentiation problem (Exponential Derivatives) Please help!

Use the process of implicit differentiation to find $dy/dx$ given that: $$x^2e^y − y^2e^x=0 $$ I am trying first to find $y$, $$y^2e^x = x^2e^y$$ $$y^2 = (x^2e^y)/e^x$$ $$y = ...
14
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0answers
186 views

Peculiar locations of the root and the maximum of $(x+1)^{x+1}-x^{x+2}$

Related to some other problems, I got interested in this function: $$(x+1)^{x+1}-x^{x+2}$$ Its root is very close to $\pi$: (Mathematica code) ...
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3answers
68 views

How to prove $A^{n\times n}=I_n\Rightarrow A^n=A^{f(n\times n)}$?

Let $A\in M_2(\mathbb{Z})$ s.t. there is a positive integer $n$ satisfying $A^n=I_2$. Show that $A^{12}=I_2$. I have no idea where to start. Suggestions?
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0answers
35 views

Growth rate of bacteria involving logarithmic functions

I was trying to solve the following question but I keep getting the wrong answer, could anyone help me out and see why? A bacteria culture starts with 900 bacteria and grows at a rate proportional to ...
1
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1answer
64 views

easy exponential population growth problem help?

The question is: Let $C(t)$ be the number of cougars on an island at time t years (where $t > 0$). The number of cougars is increasing at a rate directly proportional to $3500 * C(t)$. Also, $C(0) ...
4
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4answers
108 views

Prove that $a^x$ is continuous

I'm having trouble with proving the following: Let $a > 0$ be a positive real number. Show that the function $f : \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x) := a^x$ is continuous. I'm a ...
6
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4answers
199 views

Solving an exponential equation involving e: $e^x-e^{-x}=\frac{3}{2}$

In a previous exam, my professor had the question \begin{equation*} e^x-e^{-x}=\frac{3}{2}. \end{equation*} I attempted to take the natural log of both side to solve it, but evidently that was ...
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1answer
31 views

Operator - Exponential form

It is well known that for every unitary operator $\hat U$ an exponential of the form $$ \hat U = e^{i\hat H} $$ exists ($\hat H$ is hermitian). But I can only prove it the other way round: $$ ...
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2answers
59 views

least squares using exponential model

I'm trying to fit values from this model $$y(x)=ae^{−bx}+c$$ where a, b and c are 3 different parameters that I want to find with least squares. So using least squares I want to find the value of a, b ...
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1answer
32 views

spherical wave expansion

In the paper, Sheen, David M., Douglas L. McMakin, and Thomas E. Hall. "Three-dimensional millimeter-wave imaging for concealed weapon detection." Microwave Theory and Techniques, IEEE Transactions ...
0
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1answer
27 views

How to derive sigmoid function from e by scaling & translating?

The Sigmoid function is like this: $\frac{1}{1 + e^{-x}}$ Can it be derived by simply scaling and translating the graph of $e^{-x}$ ? It looks to me as thought you could: 1). Translate it up, by 1 ...
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1answer
38 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
47 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
26 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 ...
-2
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1answer
56 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 ...
3
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2answers
90 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 $$e^x=\int_{-x}^{+\infty}e^{-t}dt. $$ ...
0
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0answers
12 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
votes
1answer
46 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 ...
0
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1answer
32 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
30 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 ...
0
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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
17 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
29 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
30 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
48 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} ...
0
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2answers
29 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
54 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 ...
0
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3answers
61 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 $\frac{dy}{dx}$.
1
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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 ...
1
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0answers
24 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 ...
1
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1answer
32 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
1
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1answer
29 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 ...
0
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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 ...
0
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0answers
25 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
43 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
402 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
75 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?
2
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4answers
47 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
28 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 ...
3
votes
3answers
72 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) * ...
0
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0answers
16 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
31 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 ...
0
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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.)
6
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0answers
48 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 ...