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

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1
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1answer
52 views

Natural Logarithm - solve the equation

I am having problems understanding how to solve $e^{4x}+4e^{2x}-21 = 0$.
0
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1answer
32 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
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1answer
55 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
39 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
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1answer
85 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
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2answers
126 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
37 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
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0answers
69 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
49 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
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1answer
118 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 ...
1
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0answers
47 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
86 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
57 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
30 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
27 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
38 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 ...
1
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3answers
37 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
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1answer
84 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
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1answer
32 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 ...
5
votes
5answers
143 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
2answers
50 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
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0answers
29 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
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0answers
80 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: ...
2
votes
2answers
56 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 $0\leq b,r \leq 1$. ...
0
votes
1answer
31 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
191 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
63 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.
2
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3answers
59 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
60 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
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2answers
63 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
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4answers
50 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
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2answers
60 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
42 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
101 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
128 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
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2answers
33 views

Natural logarithm problem

I'm kind of confused on how to solve this problem! Any guidance/advice would be appreciated. Thanks! $e^{−9}e^{−2}e^{9}$
4
votes
2answers
213 views

How to solve this equation $x^{2}=2^{x}$?

How to solve this equation $$x^{2}=2^{x}$$ where $x \in \mathbb{R}$. Por tentativa erro consegui descobri que $2$ é uma solução, mas não encontrei um método pra isso. Alguma sugestão?(*) ...
2
votes
5answers
144 views

How to solve this exponential equation? $2^{2x}3^x=4^{3x+1}$.

I haven't been able to find the correct answer to this exponential equation: $$\eqalign{ 2^{2x}3^x&=4^{3x+1}\\ 2^{2x} 3^x &= 2^2 \times 2^x \times 3^x\\ 4^{3x+1} &= 4^3 \times 4^x \times ...
0
votes
2answers
25 views

Integral $\int_{-\infty}^0 e^{(-3i+\omega)t} $

Let's say I am integrating this function: $e^{(-3i+\omega)t}$ from $t=-\infty $ to $t=0$ [Note: $\omega$ is just a constant] The same function could be rewritten in this form(i believe?) : ...
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votes
1answer
74 views

How to Solve $-3^x+617x+1625=0$

can anyone please help me solve this : $$-3^x+617x+1625=0$$ I can't do it analytically. originally the problem was to find intersection point of $$y=1625+617x$$ and $$y=3^x$$ i did the regular ...
3
votes
2answers
59 views

How to find the number of solutions of equation $x^n - a^x = 0$?

I have to find the number of solutions of the equation $x^4 - 5^x = 0$ Since it is only asked to find the number of solutions and not the exact solution, what is the best way to approach such ...
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votes
3answers
40 views

Why is $3^{(x-5)} + 3^{(x-7)} + 3^{(x-9)} = 91$?

So far I think that this is somehow related to that $(x-7) - (x-5) = (x-9) - (x-7) = 2$, but is it ? What steps do you take to add $3^{x-5} + 3^{x-7} + 3^{x-9}$ up ? Thank you!
0
votes
1answer
32 views

How to solve an exponential equation

$8^X + 7 (2^{X+1}) = 7 (4^X) + 8$ $2^{3X} + 7 (2^{X+1}) = 7 (2^{2X}) + 2^3$ $3X = 3$ $X =1$ OR $X+1 = 2X$ $X=1$ BUT answer $X = 0$, or $1$ or $2$ ????
0
votes
1answer
29 views

Power function and involution $f(x) = x^a$

For power functions we have a variable $x$ and a constant $a$; we get that $f(x) = x^a$. Find all involutions for $f(x)$. I started out with basic functions such as $f_1(x) = f_1^{-1}(x) = x^1$ and ...
0
votes
1answer
45 views

Simplification of a formula with several exponential functions

I'm trying to work this problem my teacher gave as practice, I have the answers but I'm not sure what I'm doing wrong. I'd be grateful if anyone could help me out. Thank you so much.
0
votes
1answer
41 views

Evaluate the limit of $e^{\pi-\ln \frac{x+4}{-x}}/x$ as $x\to 0$

Evaluate:$$\lim_{x \to \ 0^-}\frac{e^{\pi-\ln \frac{x+4}{-x}}}{x} $$ I tried Hopital's rule, even the Taylor series of the function $e^x$ without success. So how can one solve it?
1
vote
1answer
39 views

Find the time required for an investment to grow to a given amount with compound interest

Find the time required for an investment of 5000 dollars to grow to 7400 dollars at an interest rate of 7.5 percent per year, compounded quarterly. Your answer is t= years. I got to the point ...
0
votes
1answer
163 views

Stuck on derivative of logarithm of sum of exponentials

let's say that I need to calculate the following expression: $$ \frac{\partial\mathrm{log}(\mathrm{exp}(w_1 * x_1 + b_1) + \mathrm{exp}(w_2 * x_2 + b_2))}{\partial w_1} $$ How do I start? The rules ...
1
vote
2answers
50 views

stuck on logarithm of derivative of sum $\frac{\partial\mathrm{log}(a+b)}{\partial a}$

I need to evaluate an expression similar to the following: $\frac{\partial\mathrm{log}(a+b)}{\partial a}$ At this point I don't know how to proceed. $b$ is a constant so there should be some way to ...
1
vote
2answers
200 views

Proof that sum of power series equals exponential function?

I have found that the Sum series equal an exponential function as below, however I have not found a proof for it: $$ ze^z = \sum_{k=0}^{\infty} k \frac{z^k}{k!} $$ I have though managed to prove ...