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

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1answer
17 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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2answers
38 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
46 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
48 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
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
22 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
18 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
13 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
39 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
361 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
66 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
37 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
98 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) ...
1
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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 $a_1 < a_0$ ...
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
63 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
24 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
43 views

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

$$2^{2x}-2^{x+2}=5$$ 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
36 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.)
3
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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
50 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? ...
4
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2answers
62 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 $|a|$ ...
1
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1answer
28 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% ...
2
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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?
3
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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 ...
0
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1answer
36 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 ...
2
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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 ...
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1answer
37 views

How to prove the limit of “the exponential of a sequence”

So given a convergent sequence $\{a_n\}_{n=1}^\infty$ with limit $a$, I'd like to prove that $$\lim_{n\to\infty} \left(1+\frac{a_n}{n}\right)^n=e^a.\quad(1)$$ Knowing that $e$ is defined by ...
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1answer
34 views

Is the derivative of a exponential function a^x always greater than the derivative of a polynomial x^n as x approaches infinity

with n and a being any constants > than 1. I have tried taking the $\lim\limits_{x \to \infty} a^x / x^n$, and l'hopitals is telling me than $x^n$ can always be reduced to 1 with multiple iterations, ...
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1answer
53 views

How do I calculate these limits? [duplicate]

How would I go about calculating $$\lim_{n\to\infty}\frac{\left(1 + \frac11\right)^1 + \left(1 + \frac12\right)^2 + \left(1 + \frac13\right)^3 + \cdots + \left(1 + \frac1n\right)^n}n$$ and ...
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2answers
22 views

Determining half life without logs, given only reduction undergone and total time taken

I have a half-life question that I can't solve. There's very limited information given. Even the half-life formula has not been taught yet. The mass of a radioactive substance in a certain sample ...
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0answers
34 views

exponential and linear nature in one equation

Please accept my apology in advance as i am not very good in math. I am looking for equation for my simulation that gives the exponential behavior in the initial x-axis points and turned to linear ...
0
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1answer
40 views

The existence of anti-derivatives

The only thing I can think of is that the function is continuous hence the anti derivative exists. I was wondering if there is anything else that needs to be done/said?
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0answers
30 views

strong convex implies exp-concave

Prove that if f is strong convex (for some m>0) $\mbox(\nabla f(\mathbf{x})-\nabla f(\mathbf{y}))^{T}(\mathbf{x}-\mathbf{y})\geq m||\mathbf{x}-\mathbf{y}||_{2}^{2} $ then f is also ...
0
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1answer
15 views

Reasoning behind method of steepest descent

I am considering the method of steepest descent from my notes. I have written that $$\int_a^b dx e^{g(x)} \sim e^{g(x_0)} \int_{\infty}^{\infty}dx \exp \left[-\frac{1}{2}(x-x_0)^2|g^"(x_0)|\right] ...
4
votes
1answer
91 views

If $\frac{x-1}{e^x-1} = y$ then $x=?$

I have following equation: $$\frac{x-1}{e^x-1} = y$$ I want to solve this equation such that I have the value of $x$ in the term of $y.$ i.e. inverse of the equation
6
votes
4answers
514 views

How do pocket calculators calculate exponents?

I'd like to know specifically how a pocket calculator (TI calculators also apply) calculates $e^{0.1}$, and what methods or algorithms pocket calculators use in order to produce their answer.