Tagged Questions

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

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0
votes
1answer
22 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 ...
1
vote
1answer
23 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
votes
1answer
23 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 ...
1
vote
1answer
43 views

Natural Logarithm - solve the equation

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

Implicit differentiation involving exponential functions [on hold]

How do I find $dy/dx$ for $xe^{-y/2} + ye^{-x/2} - 2 = 0$?
1
vote
5answers
34 views

$y=ab^x $solve for $x$. How can I accomplish this? [on hold]

I have the question $900=1500(0.95)^x$ as an exponential decay question. How do I solve for $x$?
0
votes
1answer
23 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 ...
4
votes
0answers
27 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 ...
-3
votes
0answers
30 views

how can i prove property of exponential function

Let A,B be nxn matrices with B nonsingular. How can I prove that $e^{(B^{-1}ABt)}$=$B^{-1}$$e^{{At}B}$ $\forall$ t ?
0
votes
1answer
32 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 $A\exp(Bt)=\exp(Bt)A$?
0
votes
0answers
32 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
votes
1answer
54 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: ...
2
votes
2answers
55 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
votes
1answer
19 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 ...
1
vote
0answers
59 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
votes
1answer
42 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?
1
vote
1answer
61 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 ...
0
votes
0answers
17 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 ...
1
vote
0answers
21 views

I can't plot this function (y = (8*(1-exp^(-800000000*x))) on Scilab. Error 144 [closed]

I'm a beginner using Scilab, so probably it is a stupid question. Also, I don't know how to change Scilab console messages to English, I'm sorry about that. Here is my code: function [y] = f(x) y ...
2
votes
1answer
51 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
votes
0answers
20 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
23 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 ...
0
votes
1answer
21 views

polynomial solution of second order differential equation

Find the polynomial solution $$u_n(x) = x^n + a_1x^{n-1}+...+a_n$$ of the differential equation $$u_n'' + xu_n' - nu_n = 0$$ satisfied by u_n(x). Note that this is entry-level calculus, so in my ...
1
vote
1answer
24 views

Constant raised to the power of an even or odd function

Suppose that $a$ is a positive real number, that $f(x)$ is an even function and that $g(x)$ is an odd function. Would $a^{f(x)}$ be an even or odd function? And would $a^{g(x)}$ be an even or odd ...
2
votes
5answers
54 views

Prove that $e^x \ge$ its Maclaurin polynomial with n terms [closed]

a) show that $e^x \geq 1+x$ for all $x\geq 0$ b) deduce that $e^x \geq 1+x+\frac{1}{2}x^2$ for $x\geq0$ c) use induction to prove that for $x\geq 0, n\in \mathbb{N}$ $$e^x\ge ...
1
vote
3answers
30 views

Derivative of $e^\sqrt{4x+4}$

$$f(x)=e^\sqrt{4x+4}$$ $f(x)=e^u$ $u=\sqrt{4x+4}=(4x+4)^{1/2}$ $u\;'=\dfrac{1}{2}(4x+4)^{-1/2}=\dfrac{1}{2\sqrt{4x+4}}$ I don't know how to proceed from here. Thanks.
0
votes
1answer
57 views

Why isn't Euler's formula multivalued?

So it seems that all complex exponential functions are multivalued except for ones with base $e$. Why? Shouldn't all exponentials be multivalued?
-1
votes
1answer
28 views

An exponential inequality

Assume $a(t)\geq 0$ and $b(t)\geq 0$. i can show the following inequality $\mid e^{-\int_0^ta(s)ds}-e^{-\int_0^tb(s)ds} \mid\leq T\max_{0\leq t \leq T}\mid a(t)-b(t)\mid$ by writing $\mid ...
4
votes
5answers
129 views

Showing that $e^{-2} < \ln 2$

I have to prove the following inequality: $e^{-2} < \ln2.$ Using Bernoulli's inequality, I showed that $2 \leq e$, and using this result I tried to simplify the inequality by using an upper ...
0
votes
0answers
11 views

Exponential decay and logarithmic functions

How do you use experiential decay functions and logarithmic to create a mathematical model to compare the ages of two bones (Bone A and Bone B). When Bone A contains $3$ times the amount of ...
0
votes
1answer
16 views

Re-arrange the following equation to express $t$ in terms of $V$. [closed]

$$V = 2e^{5t-9}$$ I'm not sure how to solve this.
-1
votes
0answers
20 views

Integral vanishes as $r \rightarrow \infty$, exponent

I'm integrating $e^{-x^2}$ over the boundary of a triangle and in order to finish my solution I need to show that $$\int_0^r (e^{y^2-r^2-2iry})i dy$$ vanishes as $r$ approaches infinity. Could you ...
0
votes
2answers
37 views

Let $A$ be a single $p\times p$ Jordan block. Find general solution to $\dfrac{dx}{dt} = Ax$

Let $A$ be a single $p\times p$ Jordan block. Find the general solution to $\,\dfrac{dx}{dt} = Ax$. What should I approach first? Please help!
0
votes
0answers
7 views

Second derivative of Bregman divergence

Suppose I define an exponential family distribution: $$ f(x; \theta) = \exp \left( \langle x, \theta \rangle - h(x) - \psi(\theta)\right) $$ where the log-partition function is: $$ \psi(\theta) = ...
1
vote
0answers
73 views

how to solve this limit with $e^{x}$

I was trying to solve the derivative of $e^{x}$ the traditional way with the definition of the derivative: $$ \lim_{h\rightarrow 0}\frac{e^{x+h}-e^{x}}{h} $$ so I solved like this: ...
1
vote
2answers
44 views

Estimating the behavior for large $n$

I want to find how these coefficients increase/decrease as $n$ increases: $$ C_n = \frac{1}{n!} \left[(n+\alpha)^{n-\alpha-\frac{1}{2}}\right]$$ with $\alpha=\frac{1}{br-1}$ and $0\leq b,r \leq 1$. ...
-2
votes
1answer
41 views

How do I show that $-2^{n+1} + 2^{n} = -2^{n}$ [closed]

How do I show that $$ -2^{n+1} + 2^{n} = -2^{n} $$
0
votes
1answer
26 views

Limit of a sequence, power of one minus an exponential

What is the limit of the following quantity $L \rightarrow \infty$, $$ (1 - \exp(-cL))^{\delta L} $$ for any $c$ and $\delta$ positive constants?
2
votes
4answers
174 views

Intiutive argument that $\exp' = \exp$

Is there any intuitive argument or visual "proof" that $\exp' = \exp$? Suppose you have defined the Euler number $\mathrm{e}$ as limit of the sequence $(a_n)$ where $a_n = \left (1 + \frac{1}{n} ...
0
votes
2answers
59 views

Let$\ x$ be a real number between$\ 0$ and$\ 1$. Is it possible to write$\ e^{x}$ as a function of$\ \Gamma \left(x+1\right)$?

In particular, I'm looking for a relation between$\ e^x$ and$\ \frac{1}{ \Gamma \left(x+1\right) }$, which would be of help for a proof.
0
votes
2answers
75 views

What is the general solution for $y''e^{-y} =1$? [closed]

how can I find the general solution for an ODE $$y''e^{-y} =1?$$ Thanks.
2
votes
3answers
58 views

l'Hôpital and it's use in derivation

In for example $$\lim_{x\rightarrow 0} \frac{e^{ax} - 1 - ax}{1 - \cos x}$$ We would use l'Hôpital rule and derive it twice to get $a^2$ How do you see this when just looking at the given function, ...
0
votes
2answers
58 views

How to understand $2^{e^{x}}$?

$2^{e^{x}}$ is an exponent over a exponent. It is confusing. How to understand it? Can I simplify it?
1
vote
2answers
52 views

When all solutions of $y''+ay'+by=0$ are bounded in R?

Could you please help me solve this problem. Suppose $y''+ay'+by=0$ is differential equation with $a,b$ are real numbers. I need to find conditions when all solutions of this equation are bounded. I ...
1
vote
4answers
39 views

Criterion to satisfy Rolle's Theorem.

$f(x) = \begin{cases} x^a\log x, & \text{if $x \neq 0$,} \\[2ex] 0, & \text{if $x=0$. } \end{cases} $ What should be the value of $a$ so that f satisfies Rolle's theorem in [0,1] ?? What I ...
1
vote
2answers
57 views

find $\displaystyle \int \dfrac{e^{-2x-x^2}}{\left( x+1\right)^2}\hspace{1mm}dx$

find $\displaystyle\int \dfrac{e^{-2x-x^2}}{\left( x+1\right)^2}\hspace{1mm}dx$ If I do Integration by parts, I end up with $\displaystyle\int e^{-2x-x^2}\hspace{1mm}dx$ Which I believe cannot be ...
0
votes
2answers
23 views

Help me solve this exponential function problem…

The temperature of a cooling liquid over time can be modelled by the exponential function $$T(x)=60\left(\frac12\right)^\frac x{30}+20$$ where T(x) is the temperature, in degrees Celsius, and x is ...
2
votes
3answers
94 views

Find a particular solution of $\,\,y''+3y'+2y=\exp(\mathrm{e}^x)$

I already solved for the homogeneous one, but I'm still looking for the particular solution of the differential equation: $$y''+3y'+2y=\exp(\mathrm{e}^x)$$ The homogeneous solutions of this system ...
1
vote
2answers
19 views

Sketch the graph and Determine the domain and range of $h(x)=3+e^{-2x}$.

How do I even start on this? How do I sketch the graph and find the domain and range? I am really lost on how to do this problem! Please walk me through this question!
0
votes
2answers
32 views

Natural logarithm problem

I'm kind of confused on how to solve this problem! Any guidance/advice would be appreciated. Thanks! $e^{−9}e^{−2}e^{9}$