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

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35 views

Derivative of an exponential function

I am trying to solve $$\frac{1}{e^{x}}$$ I first tried using the quotient rule, and ended up with: $$\frac{e^{x}}{(e^{x})^2}$$ That was not the right answer, so I took a look at wolfram, and ...
0
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1answer
45 views

An integral involves Gamma function

Thanks for your attention, I meet an integral involves Gamma function and exponential function as follows:$$\int_a^\infty {{x^\alpha }} {e^{cx}}\Gamma \left( {s,bx} \right)dx$$ where $a > 0,s ...
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0answers
93 views

Using the central limit theorem to prove a statement regarding normal distribution, from a population with exponential distribution

X1, . . . , Xn are a random sample from a population having an exponential distribution with rate parameter λ. Use the Central Limit Theorem to show that, for large values of n, sqrt(n)*(λx − 1) ∼ ...
3
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1answer
118 views

Prove that the exponential function is differentiable

Imagine that you are writing a book on the foundations of analysis. You have already proved that for each $a > 1$ there is a unique function $f_a(x) = a^x$ satisfying the following: $f_a$ is an ...
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3answers
36 views

How to solve $n$ from $c \leq 1.618^{n+1} -(-0.618)^{n+1}$

I need to solve the bound for $n$ from this inequality: $$c \leq 1.618^{n+1} -(-0.618)^{n+1},$$ where $c$ is some known constant value. How can I solve this? At first I was going to take the ...
0
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1answer
45 views

Challenging Question: for Expected Value of a particular probability density function

I've been stuck on this for a while and it's been driving me crazy. Any help would be greatly appreciated. I am trying to find the Expected Value of the following Probability Density Functions (where ...
0
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1answer
32 views

Differential equation: the law of natural growth and the law of natural decay

I understand that $\frac{dy}{dx} = k*y$ and when $k>0$ this is the law of natural growth and when $k<0%$ this is the law of natural decay, but my textbook gives an example of radioactive decay ...
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4answers
88 views

Simplifying the derivative of $f(x)= \frac{e^x - e^{-x}}{e^x+e^{-x}}$

I was having some trouble on simplifying the derivative because I didn't know if it's correct. The original function is $$f(x)= \frac{e^x - e^{-x}}{e^x+e^{-x}}$$ What would the simplified derivative ...
-1
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2answers
72 views

How to integrate $e^{-x^2 -y^2}$?

How do you compute the following integral:$e^{-x^2 -y^2}$? Wolfram already gives the answer as: $\frac{1}{2} \sqrt{\pi} e^{-y^2} \text{erf}(x) + C$, but I have no idea how to get there. I tried ...
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3answers
68 views

How to solve for x: $e*x + e^{-x} = 0$

I already know the answer is supposed to be $x=-1$, but I have no idea how to get to that. What I've done so far is: $ln(e^{-x}) = ln(e)+ln(-x)$ $-x = 1 + ln(-x)$ $ln(-x) + x = -1$ or $ln(x * -e^x) = ...
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1answer
33 views

What is doubling time of tumor? (Using exponential growth)

So here is the question i received: The rate of growth of a tumor is directly proportional to the size of the tumor. If the tumor is $5mm$ across at the time $t=0$ and is $8mm$ across $3$ months ...
2
votes
1answer
42 views

Evaluate Infinite Sums

I'm having trouble with this question. $$E=\sum_0^\infty \frac{nhve^{-nx}}{\sum_0^\infty e^{-nx}}$$ where x=(hv)/(kT) Evaluate both sums and show that E=hv(e^x -1)^-1 I've tried comparing ...
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vote
2answers
56 views

How do we 'know' that $2^x$ is continuous?

It is intuitive for $2^n$, if $n$ is an integer, to exist. How do we know that less intuitive values such as $2^\frac{1}{2}$, $2^\sqrt{2}$, $2^\pi$ etc exist? I'd like to accept that $2^x$ is ...
3
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1answer
35 views

How to invert a simple exponential growth formula

I think this is simple but my math skills are limited. I have a basic exponential growth formula: $$y=x \cdot (1-p)^n$$ and I have $y$ and $x$ and $n$ values and I need value of $p$. Then when I solve ...
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2answers
17 views

How do you define a function to get the output you need for a particular parameter?

Suppose we have a function $y(x)$ such that $y (\frac{-e^{-2\lambda} + e^{-\lambda}}{1-e^{-2\lambda}}) = \lambda$ How can I determine $y(x)$? Are there steps that outline how to solve such a ...
18
votes
3answers
275 views

What combinatorial quantity the tetration of two natural numbers represents?

Tetration is a generalization of exponentiation in arithmetic and a part of a series of other generalized notions, Hyperoperators. Consider $m\uparrow n$ denotes the tetration of $m$ and $n$. i.e. ...
8
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1answer
244 views

Evaluate $\int_{-1}^{1} \exp(x+e^{x})\,dx$

Evaluate $$\int_{-1}^{1} \exp({x+e^{x}})\,dx$$ where $\exp(x)=e^x$. Can anyone give me any tips on where to start with this? I've tried doing it be substitution, with $ u=e^x$ and ended up needing ...
1
vote
1answer
32 views

How to calculate the probability of two events happening within a certain time period using exponential distribution

I know how to calculate the probability of one event taking place within a set time period with exponential distribution but I'm having difficulty figuring out how to calculate what would happen if ...
2
votes
1answer
22 views

Exponential distribution lightbulb

The time it takes for a lightbulb to burn out is exponentially distributed with mean $u$ which is a random variable. Asssume that $u$ is distributed with density $$f(x)=\frac{8}{x^3}$$ for $x \in ...
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2answers
33 views

Problem in exponential/log calculus question

I have no idea how to approach this question, $\frac{dQ}{dt} = Q$ and $Q = e$ when $t = 0$, find $Q$ in terms of $t$. I can approach it logically, and the only way $y' = e$ when $t = 0$ is $y= ...
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2answers
11 views

Multiplying Fractional Exponent with a whole

Studying for a math exam and I can't understand the working for this one section of a question. $x^{\frac{1}{2}}2x =2x^{\frac{3}{2}}$ but I'm not sure how it's done, would someone kindly explain?
2
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1answer
34 views

Equality of exponential functions from geometric series

I'm currently trying to understand why the first and second line of this equation are in fact equal. This is taken from "Introduction to the Physics of Waves" by Tim Freegarde from a chapter about ...
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0answers
28 views

Bouncing back and forth between anti-difference and finite difference in finite calculus at exponential functions.

I've been reading 'concrete mathematics(knuth)' and just don't get how I'm supposed to bounce back and forth between anti-difference and finite difference in finite calculus, specifically at ...
1
vote
2answers
19 views

Problem with quadratics and exponentiation

I was bored and started solving the following equation: $$2^x = x^2$$ I can see two solutions: $x = 2$ and $x = 4$. WolframAlpha tells me there is one more, but I can hardly get the two I mentioned ...
2
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1answer
74 views

The exponential extension of $\mathbb{Q}$ is a proper subset of $\mathbb{C}$?

This question come from a recent post Exponential extension of $\mathbb{Q}$. An exponential field is a field $\mathbb{K}$ where it's well defined a function $E:\mathbb{K} \rightarrow \mathbb{K}$ ...
2
votes
3answers
74 views

Finding the limit $\lim_{n\to\infty} \frac{n\left(\sqrt[n]{n}-1\right)}{\log n}$

I try to calculate the following limit: $$\lim_{n\to\infty}\frac{n\left(\sqrt[n]{n}-1\right)}{\log n}$$ I think it should equal 1, because: $$\exp(x)=\lim_{n\to\infty}\left(1+\frac{x}{n}\right)^{n}$$ ...
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1answer
39 views

Teaching non-integer exponents

Say I'm teaching a younger student the concept of exponents. Now the basic example to begin with is say $2^3$ where it can be 'visualised' as having two blocks, then doubling the number of blocks, ...
2
votes
1answer
64 views

Are $\sin(\alpha\beta)$ and $\sin(\alpha^{\beta})$ expressible in terms of $\sin(\alpha)$ and $\sin(\beta)$?

There is a well known formula for expressing $\sin(\alpha+\beta)$ just using $\sin(\alpha)$ and $\sin(\beta)$. It is enough to replace $\cos$ in the formula ...
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2answers
36 views

Inequality involving exponential function

It is trivial to show that $$\sqrt {1+2x} < e^x, \quad \text{for}\, x>0$$ Is it true the stronger inequality $$\sqrt {1+\frac{8x}{4-x}} < e^x, \quad \text{for}\, x>0?$$
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2answers
93 views

Independent Exponential Random Variables

I am currently trying to figure out a problem and it is using notation that I have never seen before so I am pretty stuck, any suggestions would be greatly appreciated! Let $X, Y, Z$ be independent ...
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2answers
90 views

generate random number from exponential distribution with random right truncation

I need to draw a random number from an exponential distribution (rate $mu$) that is right-truncated with the truncation value coming from a gamma distribution (shape $k$, rate $lambda$). My naive ...
0
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1answer
45 views

Relationship between constants in an equation

I have the following equation: $e^{ax} + e ^{bx} = e ^{cx}$ Is it possible to find a relationship between constants $c$ and $a$, $b$ that holds for all $x$'s? Thanks in advance.
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1answer
38 views

Solving for $\lambda$ in an exponential distribution given an average

Studying for a mid-term, and not sure how to go about the following problem. Given $t = 700$ as an average, I have to solve for lambda. I'm thinking since t is determined, I don't need any integrals ...
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1answer
54 views

Rate of tumour increasing, exponential growth

Suppose the volume, $V$, of a spherical tumour with a radius of $r = 2\,\textrm{cm}$ uniformly grows at a rate of $dV/dt=0.3\,\textrm{cm}^3/\textrm{day}$, where $t$ is the time in days. At what rate ...
0
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1answer
99 views

exponential growth, e coli

Suppose an E coli culture is growing exponentially at 37 degrees celsius. After 20 minutes at that temperature, there are 1.28x10^7 E. coli cells. After 60 minutes, there are 2.4 x 10^7 cells. How ...
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1answer
48 views

Natural Logarithm - solve the equation

I am having problems understanding how to solve $e^{4x}+4e^{2x}-21 = 0$.
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1answer
30 views

Calculus help for viruses

The center for disease control has found that a virus is spreading at a rate of $4.3$% per year. That is $\frac {dV}{dt}=.043V$. If there are currently $12,000$ people infected by the virus, how long ...
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0answers
38 views

Write $\sum_{k=1}^nk\sin(kx)^2$ in closed form

$\underline{Given:}$ Write in closed form $$\sum_{k=1}^nk\sin(kx)^2$$ using the fact that $$\sum_{k=1}^nku^k=\frac u{(1-u)^2}[(n)u^{n+1}(n+1)u^n+1]$$ $\underline{My\ Work:}$ I substituted ...
0
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1answer
36 views

Commuting matrices and exponential function

Let $A,B$ be $n\times n$ commuting matrices, that is $AB=BA$. I also know that $\exp(Bt)=X(t)X(0)^{-1}$ where $X$ is the fundamental matrix function. How can I show that $A\exp(Bt)=\exp(Bt)A$?
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0answers
34 views

fundamental matrix function

I consider the homogeneous system $x'(t)=Ax(t)$ where $A$ is a $3x3$ matrix $$A=\pmatrix{-2 &0 &0 \\ 4 &-2& 0 \\ 1& 0& -2} $$ I need to determine the fundamental matrix ...
0
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1answer
72 views

how to visulaize Euler formula

What is $\theta$ significance in Euler equation $$e^{i\theta}=\cos(\theta) +i\sin(\theta)$$ Does $\theta$ have any impact on unit circle construction? Reference: ...
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2answers
117 views

Exponential extension of $\mathbb{Q}$

A non-trivial exponential function $E:\mathbb{K} \rightarrow \mathbb{K}$ in a field $\mathbb{K}$ is a function such that \begin{split} E(x+y)=E(x)E(y) \quad \forall x,y \in \mathbb{K} \\ E(x)=1 \iff ...
0
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1answer
33 views

Probability of choosing at least one correct envelope for n letters as $n \rightarrow \infty$

There are n letters for which each has a specific envelope. If each letter has randomly been put into an envelope, what is the probability of choosing at least one correct envelope as $n \rightarrow ...
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0answers
61 views

Are exponents with a base very close to $1$ (such as $1.0001$) useful in Mathematics?

I was curious if exponents with a base very close to $1$ are ever used in Mathematics and for what applications. For example, when I was in college, my Calculus professor told me that logarithms are ...
0
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1answer
47 views

Nature of the range of $e^x$

Apart from the trivial cases, $x=\log a$ where $a\in\mathbb{Q}$, are all values of $e^x$ irrational? Are some transcendental?
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1answer
88 views

This three-variable system of equations seems impossible to solve

$$g = af^b + c$$ $$i = ah^b + c$$ $$k = aj^b + c$$ I want to solve for $a$, $b$, and $c$. $f$, $g$, $h$, $i$, $j$, and $k$ are inputs to the equations, so they don't have to be solved for. Just ...
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0answers
41 views

Proving that any continuous homomorphism of $\mathbb{R}/(2\pi\mathbb{Z})$ int0 $T$* is neccesarily an exponential function

This is an exercise form Katznelson's book on Harmonic Analysis, so I want to solve it using his hint. T* here denotes the multiplicative group of units of complex numbers of unit norm. That is to ...
2
votes
1answer
74 views

A hard exercise on endomorphisms and determinants

The following exercise has been bugging me for some days, could someone help me with it ? Let $E$ be a $\mathbb{C}$-vector space with dimension $n$ and $f\in\mathcal{L}(E)$ ($\mathcal{L}(E)$ denotes ...
0
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0answers
38 views

Triple integral containing definite integral and exponentials with trigonometric functions

I am attempting to solve the following integral analytically: $$ \int_{z=5i}^{z=1} \int_{t=\csc^{-12}(z)}^{t=2} \int_{\theta=\sin^{t}(z)}^{\theta=t^2} {[\mathrm{e}^{t\cos(\mathrm{e}^{i \theta})} + ...
0
votes
1answer
27 views

What are the rules being used to compute $\lim\limits_{x\rightarrow \frac{\pi}{2}} (1-\cos x)^{\tan x}$?

I am given $\lim\limits_{x\rightarrow \frac{\pi}{2}} \frac{\ln(1-\cos x)}{\cos x} = -1$ So, $(1-\cos x)^{\tan x} = e^{(\tan x) \ln(1-\cos x)}$ and as $x\rightarrow \frac{\pi}{2}$, we have: $(\tan ...