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

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1answer
22 views

Determining the gradient of a decaying function

I have data that can be fitted using an exponentially decaying function: $y = e^{-t/\tau}$ and I want to determine the value of $\tau$. I see that if I make the t-axis logarithmic I get a straight ...
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2answers
424 views

Concentration of a drug after repeated injections

After injection of a dose $D$ of insulin, the concentration of insulin in a patient's system decays exponentially and so it can be written as $D\exp^{-at}$ where $t$ represents time in hours and $a$ ...
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0answers
75 views

Expected waiting time of the process in the queue given that the process is served

Consider the following situation: There is one server with exponential service time with parameter $\lambda$. One process is waiting in the queue. The waiting time is exponential with parameter ...
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1answer
26 views

solving exponential functions

$(27^{x - 1})(3^x) = 9^{2x-3}$. I apologize if you do not understand the equation. I was unsure on how exactly to represent it correctly. I have gotten to the step in the equation where it is ...
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1answer
47 views

derivative of a definite integral with base e

$$\frac{d}{dx} \int_3^{x^2} e^{t^3} dt$$ I can sorta figure out how to solve problems like this, if it was an indefinite integral...
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1answer
26 views

Sketching Logs with Quadratic Terms

$\log(x^2+1) = y$ asymptote at $x^2+1 > 0$ and so there is no asymptote $x$ and $y$ intercept at $(0,0)$ How do you know that the function goes both directions, and has a dip in the middle? ...
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1answer
40 views

Can the variables of $y = A + B \mathrm{e}^{C t}$ be solved analytically given 3 sets of points?

Given the non-linear equation $y = A + B \mathrm{e}^{C t}$ and 3 sets of points: ($y_1$, $t_1$), ($y_2$, $t_2$), ($y_3$, $t_3$), can the variables $A$, $B$, and $C$ be calculated analytically? ...
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3answers
101 views

Continuity proof for exponential

Show that $f(x) = e^x$ is continuous using the epsilon-delta definition. I can't seem to write down anything meaningful...
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1answer
36 views

Find the inverse function about a exponential related function

Here is the function:$$y = 4x + {x^m},where{\text{ 0 < m}} \leqslant {\text{1;}}$$ Approximately results is acceptable.
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4answers
741 views

Absolute value of complex exponential

Can something explain to me why the absolute value of a complex exponential is 1? (Or at least that's what my textbook is telling me.) For example: $$|e^{-2i}|=1, i=\sqrt {-1}$$
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3answers
156 views

Show that $\,\,n!<\mathrm{e}\left(\frac{n}{2}\right)^n$

I'd like to prove that $\,\,n!<\mathrm{e}\left(\frac{n}{2}\right)^n$. What I have so far: $\sqrt[n]{n!} = \sqrt[n]{1\cdot 2 \cdot \ldots \cdot n} \leq \frac{1+\ldots ...
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1answer
62 views

Product of exponential distributions

Suppose $X_1$ is $\mathrm{Exp}(\lambda_1)$ and $X_2$ is $\mathrm{Exp}(\lambda_2)$. $X_1$ and $X_2$ are independent. Let $Y = \min (X_1, X_2)$ and $Z = \max (X_1, X_2)$ and $W = ZY$ . Compute the ...
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2answers
47 views

Solving an equation with both linear and exponential terms

Can I find an algebraic solution for the equation below? Thank you. $$ x+e^{x}(x+a)=b $$
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1answer
28 views

n events of one process occuring before m events of another process

Assume that you have two independent Poisson processes, N1( t ) with rate λ1 and N2( t ) with rate λ2 . What is the probability that n events occur in the first process before m events occur in the ...
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1answer
36 views

Problems with another characterization of exponential functions

As in other two of my questions, which are already answered by myself, I am treating exponential function again. Now, from the perspective of continuity only. These means, I can not use any single ...
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5answers
140 views

If $\frac{d}{dx}e{^x} = e{^x}$, then why does $\frac{d}{dx}e^{-14}$ = 0?

If $\frac{d}{dx}e{^x} = e{^x}$, then why does $\frac{d}{dx}e^{-14}$ = 0? Why doesn't $\frac{d}{dx}e^{-14}$ = $e^{-14}$? I don't understand.
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3answers
369 views

Find a closed form from the given power series

I have the power series $\sum_{n=0}^{\infty} {z^{2n}\over{n!}}$, how do I find the closed form for this power series. I am aware that $e^z=\sum_{n=0}^{\infty} {z^{n}\over{n!}}$, so I tried to ...
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2answers
301 views

Exponential pop. growth when only given population at two instances of time.

I have a problem where I'm only given the population of a "bacteria culture" at two instances in time: 2 hours and 4 hours. The problem says the population of bacteria is 125 after 2 hours, and 350 ...
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5answers
250 views

Does the series $\,\displaystyle\sum_{n = 1}^{\infty}\left(2^{1/n} - 1\right)\,$ converge?

I'm trying to determine if the following sum converges or diverges (this is question 38 in section 11.7 of Stewart's Early Transcendentals): $$\sum_{n = 1}^{\infty}(2^{1/n} - 1)$$ I've considered ...
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2answers
104 views

Show that $\sum\limits_{n=1}^\infty \frac{1}{n^z}$ converges absolutely

I want to show that $\large\sum\limits_{n=1}^\infty \frac{1}{n^z}$ converges absolutely for $\Re(z) > 1$, so I want to show that $\large\sum\limits_{n=1}^\infty |\frac{1}{n^z}|$ converges, or ...
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1answer
37 views

variance of minimum of squared exponential random variable

Given $Y_1 $ to $Y_n$ are exponential r.v's with mean $\theta$ find $\operatorname{var}[\min(Y^2 )]$ with the help of gamma distribution. attempt: $\min(Y) $ is exponential with $(\text{mean} = ...
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4answers
630 views

Prove that e is irrational

Prove that e is an irrational number. Recall that e $=\displaystyle\sum_{n=0}^\infty\frac{1}{n!},\,\,$ and assume $\mathrm{e}$ is rational. Then $$\sum\limits_{k=0}^\infty \frac{1}{k!} = ...
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1answer
46 views

A definite integral

$$\int_0^1\sqrt{\left(3-3t^2\right)^2+\left(6t\right)^2}\,dt$$ I am trying to take this integral. I know the answer is 4. But I am having trouble taking the integral itself. I've tried foiling and ...
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2answers
426 views

Find Log equation from data points

I have the following data points, (left hand column goes from 0-127, right hand column goes from 30-22000 hz. Is there any calculator I can use to find a "log" function of this data, so that it comes ...
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1answer
102 views

Steps to Graph Exponential Equations & Absolute Value

how to sketch: $-e^{|-x-1|} + 2$ Can someone clarify: $|f(x)|:$ we draw $f(x)$ and then reflect the ($-y$ parts) in the $x$-axis $f|(x)|:$ we draw $f(x)$ and then reflect the ($-x$ parts) in the ...
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1answer
121 views

An Application of Rouche's Theorem to Two Cases

Here is my question - it is an example sheet question, completely non-examinable: [I have managed this first part, but am including it to help give a sense of where the question is going.] $(i)$ ...
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3answers
41 views

Complex derivative involving exponents and natural log

Find: $\frac{d}{dx} a^{x\ln x}$ I have tried several methods involving u-substitution etc, but can't figure it out.
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1answer
39 views

Lifetime of exponential variable of a battery

Suppose that the operating lifetime of a certain type of battery is an exponential random variable with parameter $\theta=2$ $($measured in years$)$. Find the probability that a battery of this type ...
0
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1answer
34 views

Proving properties about complex exponential

I defined $a^z$ for $z \in \mathbb{C}$ as $a^z = \exp(z\log(a))$ and I proved it is continous, now I want to show that $a^n = a \cdot a \cdot a \cdot \ldots \cdot a$ for $n \in \mathbb{N}$ so ...
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1answer
47 views

Continuity and other properties of complex exponential

So I think I can do the others, but part (i) about showing the continuity of $a^z$ has me stumped. I always get really stuck when it comes to proving continuity (I am using the metric spaces ...
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1answer
95 views

Problem of the Week! [duplicate]

This week in Algebra II we are studying the Hanoi tower's. Our assignment was to find what type of formula would give the number of moves it would take to solve the puzzle. After using a T-chart ...
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1answer
69 views

Derivative of exponential functions

Can anyone present an intuitive reason for why the derivatives of exponential functions, lets say, as apposed to polynomials, grow more rapidly than the functions themselves? i.e. $$ y = e^{x^2}\\ ...
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4answers
98 views

Limit of $\frac{e^{x}}{\ln(x)}$

I don't know how to find the limit $$\lim_{x\to\infty}\frac{e^x}{\log x}.$$ How can I do this ? Thank you in advance.
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1answer
32 views

Show that $g(x)=x\ln{x}$ and $g(x)=e^x$ are bounded below.

Show that $g(x)$ is bounded below, for $0\leq x$: a) $g(x) = \left\{ \begin{array}{ll} 0 & \mbox{if } x=0 \\ x\ln{x} & \mbox{if } x>0 \end{array} \right.$ b) $g(x)=e^x$ For (a), ...
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1answer
88 views

Prove logarithm rules using definition as the inverse exponential

Problem $3$. Show that $\operatorname{exp} : \Bbb R \to (0, \infty)$ is bijective. Its inverse function is called the (natural) logarithm $\log : (0, \infty) \to \Bbb R$. Verify the logarithm ...
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3answers
2k views

“Linearize” an exponential-looking graph with log function

This may be a beginner question, but I can't quite wrap my head around logs... I have a set of data (from an experiment) which gives me an exponential-looking graph (Fig 1). I'd like to "linearize" ...
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1answer
122 views

Proving that the exponential function is bijective

Prove that $\exp: \mathbb{R} \mapsto (0,\infty)$ is a bijection. Okay, so the first part is really easy: injectivity follows directly from writing the exponential function as a series. ...
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1answer
73 views

Relation/connection between $n!$ or $e$ and $2^n$

What is the relation/connection between $n!$ or $e$ and $2^n$ ? Is the there a relation/connection between $n!$ or $e$ and $2^n$?
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0answers
16 views

How would I design a formula that increases the length of pauses exponentially based on current speed?

I'm writing a program that presents users words in a flash-card fashion, at a speed they define (say, 500 cards/min). When a "section" of cards is done, I want there to be a pause before the next one ...
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2answers
166 views

Prove $\sum_{n=1}^\infty(e-\sum_{k=0}^n\frac1{k!})=1$

This comes from the comments section of this question here, credits Lucian. The statement is $$\sum_{n=1}^\infty\left(e-\sum_{k=0}^n\frac1{k!}\right)=1$$ This looks really interesting, so I was ...
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3answers
81 views

Analytical aptitude - Division of exponentials.

What is the remainder when 6^17 + 117^6 is divided by 7? How to approach these type of questions?
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1answer
94 views

Rational and trascendental numbers: $\pi$, $e$ and $\pi+e$ [duplicate]

The numbers $\pi,e$ are trascendentals, but if consider: $\pi+e$ then is rational, trascendental? Thanks
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1answer
77 views

When does this non linear 2 equation system have solutions? What is the solution?

I need to solve the following system: $$ \begin{cases} a x_0^2 = \exp{ \left( -\dfrac{x_0^2}{4 \sigma^2} \right) } +a r^2 \\ \exp{ \left( -\dfrac{x_0^2}{4 \sigma^2} \right) } + 4 a \sigma^2 = 0 ...
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2answers
28 views

Calculus Exponential Functions Again

This one wants us to evaluate the following limits of this exponential function. $$\lim_{x \to \infty} \frac6{e^x-6}$$ I'm not sure how to approach this problem. I did easily figure out this version ...
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1answer
120 views

(Basic High School Mathematics) Graphing the inverse square law

I did an experiment measuring the intensity of light in relation to the distance away from a source. How would I graph the avg intensity over 1/distance squared? Note that T1 = trial 1 etc.. It's ...
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1answer
56 views

$|x|^{|x|}$ is continuous in $\mathbb{R}$

I'm trying to show this now my self, but still no go. There isn't really a concrete attempt that I can show.. Help?
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7answers
130 views

for $n$ an integer, why is $n^0=1$ ??

This is so going to cost me.... I was wondering why for any integer $n$: $n^0 =1$. Perhaps It's because $n$ is a round number and if $m$ is a non negative integer, also round then: $$n^m = 1 \cdot ...
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3answers
119 views

$0^0$ — indeterminate, or $1$? [duplicate]

One of my teachers argued today that 0^0 = 1. However, WolframAlpha, intuition(?) and various other sources say otherwise... 0^0 doesn't really "mean" anything.. can anyone clear this up with some ...
0
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1answer
94 views

How does the complex exponential function transform the unit circle?

I know you can write every complex number on the unit circle as $e^{i\theta} = \cos(\theta)+i\sin(\theta).$ But what does it look like when you raise $e$ to the values? You get ...
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1answer
32 views

Multi part problem to prove functional relation of the exponential function

I'm worried about part (i) right now mostly. Part 3 is easy, and part 2 I can probably get after some work. I know that $\exp(-z) = \large\sum\limits_{n=0}^\infty \frac{-z^n}{n!} = ...