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

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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})} + ...
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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 ...
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1answer
26 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 ...
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1answer
31 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 ...
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3answers
36 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.
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1answer
83 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?
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1answer
31 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 ...
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votes
5answers
142 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 ...
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2answers
44 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!
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0answers
16 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) = ...
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0answers
78 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
51 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 $0\leq b,r \leq 1$. ...
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1answer
30 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
180 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} ...
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2answers
62 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.
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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, ...
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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?
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2answers
62 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 ...
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4answers
46 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 ...
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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 ...
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2answers
31 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
99 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 ...
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2answers
59 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!
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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
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2answers
189 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
139 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 ...
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2answers
24 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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1answer
71 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 ...
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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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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!
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1answer
31 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 $2$ ????
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1answer
28 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 ...
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1answer
44 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.
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1answer
39 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?
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1answer
32 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 ...
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1answer
88 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 ...
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2answers
43 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 ...
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2answers
103 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 ...
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0answers
63 views

Exponential Function and simultaneous equations

I have this maths problem for school that I cannot solve. $a(x) = Ne^{kx}$ This exponential function can be calculated by looking at the maximum height of each bounce. $$\begin{array}{|c|c|} ...
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3answers
55 views

Find solution of differential equation $y'(t)=-2y(t)+1$

Could you help me explain how to find the solution of the differential equation $$ y'(t)=-2y(t)+1, $$ with $$y(0)=1.$$ I know that the solution is $$y(t)=\frac12 (1+e^{-2t}).$$ How about the IVP ...
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1answer
48 views

Partial derivative of the trace of matrix entry-wise exponential?

Just checking my math here and getting some help for the exponential part. $\renewcommand{\v}[1]{\mathrm{vec}\left(#1\right)} \renewcommand{\m}[1]{\mathbf{#1}} ...
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0answers
225 views

Hidden Markov Model, transition probabilities which are modeled with an exponential distribution

I'm looking at implementing an algorithm described in a paper, but I'm having trouble understanding how the transition probabilities for a Hidden Markov Model are defined. In the first sections, I ...
0
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1answer
31 views

Exponential relationship between data points; trying to find matching equation

I have a list of data points that I'm trying to find an equation to To be honest, I can't remember where to begin. It's been years since I did this kind of stuff. Is there an online site that can do a ...
1
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1answer
23 views

Definite Integrations involing volumes

suppose the area of the region is bound by the curve y= (e^x + e^-x)/2 , the lines x=-1 and x=1 and the x-axis. Find the volume of the solid of revolution obtained when the region is rotated about the ...
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3answers
59 views

Evaluating $ \int {e^x \sin (k \pi x) } dx $

I'm trying to integrate $$ I = \int {e^x \sin (k \pi x)} dx. $$ I've used Matlab and Wolfram Alpha, which have both given me the result $$ I = \frac{e^x(\sin (k \pi x) - \cos (k \pi x))}{k^2 \pi^2 +1 ...
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1answer
18 views

exponential equation (solve for x)

We started review in my calculus class and I have mostly forgotten everything about exponential equations. $$ e^{-x} * (3e^{2x}-(5/4))^{1/2} = e^x $$ Would x just equal 0, when -x+x=0 (two sides of ...
0
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1answer
36 views

Bessel Function to Exponential Function

Solving this equation : $ y'' + (1/x)y' - (k^2 - (m^2/x^2) )y=0$ gives $y(x) = J_m (-ikx) + Y_m (-ikx)$ which is further written as $y(x) = \frac{1}{\sqrt{x}}\left(\exp(-kx) + \exp(kx) \right)$ ...
0
votes
1answer
21 views

amplify/exaggerate by zero at the start increasing to double at end

My values are as follows -0.030880035897248945, -0.030881151955902714, -0.030882268014556485, -0.03088355292653248, -0.030884837838508476, -0.030885793201895988, -0.0308867485652835, ...
1
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2answers
30 views

Inverse of a function containing the ceiling function over the natural numbers

I am wondering if there exists an inverse function for $\lceil{e^{x}}\rceil$ over the natural numbers. I don't think it is a trivial task to derive an inverse function for a function containing a ...
2
votes
1answer
31 views

Exponent laws with Negative bases

$-49 =7^x$ is the question. Here I m supposed to solve for what power of $7$ will give me $-49$. Or in other words, I have to solve for $x$. This looks fairly simply when thinking about the ...