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

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1answer
16 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
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5answers
53 views

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

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
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3answers
28 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
49 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
27 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
116 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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0answers
10 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 ...
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2answers
13 views

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

$$V = 2e^{5t-9}$$ I'm not sure how to solve this.
-1
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0answers
18 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
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2answers
35 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
6 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
72 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
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2answers
41 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
41 views

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

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

What do you do to simplify this radical expression? [closed]

This is the expression that needs to be simplfied Look at the image. Does the parenthesis have effect on the simplification of this problem?
0
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1answer
25 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
169 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
58 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
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2answers
74 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
56 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
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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
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2answers
43 views

Represent $3^x$ as a sum of even and odd function [closed]

Represent $3^x$ as a sum of even and odd functions. And also represent $a^x$ as a sum of even and odd functions, where $a$ and $x$ are real numbers. Use simple mathematics!
1
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2answers
51 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
38 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
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
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2answers
22 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
93 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
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2answers
18 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
31 views

Natural logarithm problem

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

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

How to solve this equation $x^{2}=2^{x}, x \in R$ ? por tentativa erro consegui descobri que $2$ é uma solução, mas não encontrei um método pra isso. Alguma sugestão?
2
votes
5answers
130 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
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2answers
19 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
66 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
54 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
38 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
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1answer
28 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
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1answer
20 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
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1answer
43 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
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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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2answers
37 views

Pre-Algebra: How to solve this in simple way? [closed]

I wanted to know what's the easiest way to solve this math problem. I've got my tests tomorrow and I would be grateful if you could help me out. $$\frac{5^{a-2b} \cdot 125^{2a-b}}{25^{-a-b}}$$ ...
1
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1answer
28 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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2answers
23 views

Solve this problem using logarithms… [closed]

The doubling period of a bacterial population is $10$ minutes. At time $t = 90 \text{ minutes}$, the baterial population was $70000$. With t representing minutes, the formula for the population is ...
0
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1answer
25 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
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2answers
37 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
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1answer
44 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
55 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|} ...
1
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3answers
45 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 ...
0
votes
1answer
34 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}} ...
1
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0answers
83 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
votes
1answer
21 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 ...