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

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0
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1answer
21 views

Fitting a curve - trending line formula

In OSX Numbers I have a chart with these data points: 50 53 100 62 200 78 300 91 500 117 1000 192 2000 297 3000 412 5000 567 10000 990 Using the trending line ...
2
votes
1answer
68 views

Approximating the number $e$ through computer simulation - mathematical background

There is nothing original about this question. It was asked here. I am just curious about an answer that is beyond my mathematical level. In one of the simulations appearing in the comments to the ...
1
vote
4answers
86 views

Proving that the exponential inequality $e^x \ge x^e$ holds for all $x \ge 0$ [duplicate]

How does one prove that $$e^x \ge x^e$$ for all $x \ge 0$? I tried to do this by setting $f(x)=e^x-x^e$ Plotting this function shows this easily, as seen here. However, when I tried to prove ...
4
votes
1answer
96 views

How to solve $\ln(y)=\ln(x)e^{\ln(x+1)} $ for x?

I know that if I have had $y = x^{x+0} $ aka $y = x^x$ I could do $y = x^x$ // $x = e^{\ln(x)}$ $y=x^{e^{\ln(x)}}$ // $\ln$() $\ln(y) = \ln(x)e^{\ln(x)}$ then using Lambert's W function I ...
1
vote
0answers
27 views

Complex exponentials [duplicate]

How do I solve: $$ e^{4z}+e^{3z}+e^{2z}+e^z+1=0 $$ I'm getting lost on where to start. I tried using the definition $$ e^z=e^x(\cos(y) +i\sin(y)) $$ But that doesn't seem to do me any good. I also ...
1
vote
1answer
59 views

How do you differentiate the integral from $ \int_{e^{-x}}^{e^x} \sqrt{1+t^2}\,dt$ [duplicate]

How do you differentiate the integral from $e^{-x}$ to $e^x$ of $\sqrt(1+t^2)$ with respect to t? $$ \int_{e^{-x}}^{e^x} \sqrt{1+t^2}\,dt $$ I know the answer is $$ e^x\sqrt{1+e^{2x}} + ...
0
votes
1answer
32 views

Minimum of four exponential variables

Four accidents occur independently, with each accident following an exponential distribution with a mean of 22.5. What is the expected value of the minimum of the four accidents? Attempt: ...
-2
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2answers
59 views

find the value of $k$ in the term $2^{-k} = 1/n$

What is the value of $k$ if I have the following equation: $2^{-k} = \frac1n$? $$2^{-k} = \frac 1 n \implies n = 2^k \implies \log_{2} n = k$$ Is my solution correct?
1
vote
2answers
42 views

Limit of $\frac{e^{1/x}}{x^2}$ as x approaches 0 negatively

I know the following: $$\lim_{x\to 0^-} \frac{e^{1/x}}{x^2} =0$$ I cannot, however, see why. Is there a method that makes this result intuitively clear?
0
votes
2answers
28 views

Difficulty finding the sum of a hyperbolic function.

Can someone please point out where I am (If I am) going wrong during the solution process of the following question: I have been presented with the following : $$4sinh(2ln(2))-cosh(ln2)$$ and told ...
2
votes
3answers
57 views

Why Doesn't $2^{1/n}= 1/(2^n)$

Take $2^{1/n}$. Since $1/n$ can be simplified as $n^{-1}$, the original term can become $2^{n^{-1}}$. The exponents can then be multiplied to result in $2^{-n}$ which is $1/(2^n)$. However it is ...
0
votes
3answers
17 views

How to isolate X in ${A * B ^X = C * D ^ X}$

${A * B ^X = C * D ^ X}$ The idea is to find in how much time (X) a small (A) investment with a good tax (B) beats a big investment (C) with a bad tax (D). All values are nonzero and positive.
1
vote
2answers
49 views

taking the natural log of e^(2x) = (4/3)

I have been unable to answer the following question. I must solve for x: $$e^{2x} = (4/3)$$ I have been made aware that I must take the natural log of both sides, giving: $$ln(e^{2x}) = ln(4/3)$$ ...
3
votes
2answers
47 views

Proof of $a^x ≥ x+1 \; \forall x \in \Bbb R \implies a=e$

I'm trying to prove the following : Let $a>0$ a real number. Then : $\quad a^x ≥ x+1 \;\; \forall x \in \Bbb R \iff a=e$ I managed to prove the '$\Longleftarrow$' part : $x≥0$ then ...
2
votes
3answers
48 views

Solving inequlity with $e^x$

I'm studying differential calculus, but one of the questions involves solving an inequality: $$(x-2)e^x < 0$$ I intend to go deeper in solving inequalities later, but I just want to understand ...
1
vote
4answers
91 views

how to solve $x(e^{-{1\over x}}-1)=$ constant

As mentionned in the title, how to solve analytically the equation $x \cdot \left(e^{-\frac{c_1}{x}}-1\right)=c_2$ where $c_1$ and $c_2$ are known constants. I can easily find a solution ...
2
votes
1answer
60 views

Integral of $x^{-2}e^x$

This is the original problem. $$\int_2^1 \frac{x^2e^x - 2xe^x}{x^4}$$ My attempt at breaking it down $$\frac{x^2e^x}{x^4} - \frac{2xe^x}{x^4}$$ $$x^{-2}e^x - 2x^{-3}e^x$$ $$ ...
0
votes
2answers
67 views

Integral of exponential rational function

I'm asked to find $$\int_0^{\ln 2}{e^{2x}\over{e^{4x}+3}} \text{ d}x$$ I can't for the life of me figure out how to integrate this.
6
votes
6answers
157 views

why is the limit as n goes to infinity of $(1+\frac{1}{n}+\frac{200}{n^2})^n = e$?

I know that $$\lim_{n\to\infty}\left(1+\frac{1}n\right)^n = e .$$ But why does $$\lim_{n\to\infty}\left(1+\frac{1}n+\frac{a}{n^b}\right)^n = e ? \quad where\quad b\gt1$$ better yet, how can I ...
3
votes
2answers
89 views

Integral of $x^2 e^{-x^2}$

Like the title says, I'm trying to find $$\int_0^r x^2 e^{-x^2}\,dx$$ Where $r$ is some finite value. I've done one step using integration by parts with $u=x^2$ and $dv=e^{-x^2}dx$, which has left ...
4
votes
3answers
364 views

Eigenvector and eigenvalue for exponential matrix

$X$ is a matrix. Let $v$ be an eigenvector of $e^{X}$ with corresponding eigenvalue $a$. Show that $v$ is also an eigenvector of $e^{X}$ with eigenvalue $e^{a}$ If $X$ is diagonalizable, then we can ...
0
votes
0answers
26 views

Let $f(z) = e^{z^2}$. Find $\theta \in (-\pi,\pi]$ such that $\lim_{r \to \infty} f(re^{i\theta})=0$

Let $f(z)=\exp(z^2)$, with $z=re^{i\theta}$. Find $\theta \in (-\pi,\pi]$ such that $\lim_{r \to \infty} f(re^{i\theta})=0$. With the identity $e^z=e^x(\cos(y)+i \sin(y))$, I found that ...
7
votes
2answers
511 views

Why doesn't this infinite exponential growth go beyond 2.5?

My calculus book says that with: $$a=x^{x^{x^{.^{.^{.}}}}}$$ (exponent tower goes on forever), then: $$x=a^\frac{1}{a}$$ I tried it out with $a=3$ so $x=3^\frac{1}{3}$ and then ran a python program ...
1
vote
3answers
76 views

Solving equation with infinite exponent tower

How to solve this equation for $x$ where $a>0$? The exponent tower goes on forever: $$a=x^{x^{x^{.^{.^{.}}}}}$$ My Calculus book gives the following reasoning: ...
0
votes
0answers
116 views

Mapping in the complex plane

I have the following two circles in the complex plane, $z = x + iy$, which bound a region, $R$. The equations for the circles and a sketch of the region is given as follows: $$ x^2 + (y-1)^2 = 1\\ x^2 ...
1
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2answers
21 views

Manipulating Complex Exponentials

I am trying to show that $$ sin(5\pi t) = \frac 12 e^{-j \frac {\pi}2}e^{j5\pi t} - \frac 12 e^{-j \frac {\pi}2}e^{-j5\pi t} $$ I am aware that $$ sin(\theta) = \frac {e^{j\theta} - ...
3
votes
1answer
89 views

Which is larger, $e^\pi$ or $\pi^e$? [duplicate]

I don't know how to approach this. I tried expanding $e^{\pi}$ using the power series but that was a dead end since I didn't know what to do with it. I tried estimating if $e \log({\pi})$ was ...
4
votes
3answers
94 views

Trying to understand the function $y = x^x$

I am trying to understand the function $y=x^x$: 1) Why is $0^0$ not defined? Why isn't it defined as $0^0=1$? The limit of the function for $x\to0$ also goes to $1$ 2) Why is it only defined for ...
5
votes
6answers
96 views

Finding the limit $\lim_{n\to \infty} \left({\frac{n+1}{n-2}}\right)^\sqrt n$

I have to find: $$\lim_{n\to \infty} \left({\frac{n+1}{n-2}}\right)^\sqrt n$$ But, to be honest, I haven't got a faintest idea how to even begin. Is there a way to evaluate this radical exponent?
1
vote
2answers
101 views

Prove that the exponential $\exp z$ is not zero for any $z \in \Bbb C$

How can the following been proved? $$ \exp(z)\neq0, z\in\mathbb{C} $$ I tried it a few times, but I failed. Probably it is extremly simple. If a draw the unit circle and then a complex number ...
3
votes
4answers
55 views

asymptotic behavior of the two sequences defining exponential function

There are two definitions of exponential function: $$e^x=\lim_{n\to\infty} S_n=\lim_{n\to \infty} a_n \text{ ,}$$ where $$S_n=1+x+\frac{x^2}{2!}+\dots+\frac{x^n}{n!}$$ and ...
2
votes
2answers
46 views

Common factor out from a sum of exponential functions

From the below equation, which is a sum of two exponential functions I would like to compute the common factor $n$ $$ d = \exp\left(\frac{-x}{n}\right)+\exp\left(\frac{-y}{n}\right)$$ Unfortunately, ...
0
votes
2answers
37 views

exponential regression for bacteria growth

I'm studying regression lines and curves, and I've learn the methods for working with curves of the types $ax^2+bx+c$ and $ax+b$ as well as $a\sin(x)+b\cos(x)$. Now I'm asked this: $$(0,32), ...
0
votes
2answers
65 views

Can you easily simplify these terms?

I need to simplify these terms step by step to prove they are equivalent $$(100^{(2n+1)}-1+99×10^{(4n+2)}-99×10^{(2(n+1)-1)})/(11×10^{(2n+1)}-11) $$ and ...
1
vote
1answer
100 views

prove that the following function is decreasing?

I am trying to prove that the following function is decreasing. \begin{align}&f(t)=\frac{1-g(t)}{\sqrt{1+e^t}}\cdot\exp\left(-\frac{te^t}{2(1-e^t)}\right)&t<0\end{align}where $ ...
1
vote
1answer
84 views

Which function to kill: Sine or Cos?

I got an equation which was a solution to some familiar Differential Equation I am solving, the solution takes the form of: $$V=Ce^{-ix}$$ but $$Ce^{-ix}=A\cos(x)+B\sin(x)$$ so ...
-1
votes
2answers
66 views

Integral of $\int\frac{1}{1+2e^x}dx$

It seems there are two ways to find the integral of this function $f(x) = \frac{1}{1+2e^x}$. In both paths I only do operations that I know are true, but for some reason one of them gives me the right ...
0
votes
1answer
63 views

solving for a variable that exist inside as well as outside of natural log or exponent

can the following equation be solved for K analytically? If not, then what other approaches I could try out? K*ln[(C2-K)/(C1-K)] = -(F/V)*t The original equation ...
0
votes
1answer
42 views

Complex numbers converge if their absolute values and arguments converge

Let the sequence $\{z_n\}_{n>0}$ and $w \not=0$ be such that $|z_n| \to |w|$ and $\operatorname{Arg}(z_n) \to \operatorname{Arg}(w)$. Show that $z_n \to w$. My proof: $z_n= |z_n|e^{i \arg(z_n)} ...
0
votes
2answers
58 views

Closed form of $\int_{\delta_1}^{\delta_2}(1+Ax)^{-L}x^{L}\exp\left(-Bx\right)dx$

Is there a closed-form expression for the following definite integral? \begin{equation} \mathcal{I} = \int_{\delta_1}^{\delta_2}(1+Ax)^{-L}x^{L}\exp\left(-Bx\right)dx, \end{equation} where $A$, $B$, ...
-4
votes
2answers
57 views

$e^{\mathrm{Re}\,z}$ not analytic in complex plane

In my textbook I found a text where it says that $e^z$ (z is a complex number) is analytic everywhere. But $e^x=e^{\mathrm{Re}\,z}$ is not. How can I prove that about $e^x$ and what is the ...
7
votes
9answers
344 views

Why $e^x$ is always greater than $x^e$?

I find it very strange that $$ e^x \geq x^e \, \quad \forall x \in \mathbb{R}^+.$$ I have scratched my head for a long time, but could not find any logical reason. Can anybody explain what is the ...
0
votes
0answers
28 views

Evaluate the area of the intersection of two amoebas

Here is defined what is a amoeba and what is a tentacle, see page 3. I believe that this could be a nice exercise in multivariable calculus. Question. Can you give an example to show how compute ...
-1
votes
1answer
38 views

Solving an integral with exponential function

I try to solve the following integral $$\int_a^b \exp\left\{-\lambda \left(\frac{y}{2x^2}-\frac{1}{x}\right)\right\} dx$$ for $\lambda>0$ and $y \in \mathbb{R}$. Do you see any relation to any ...
1
vote
3answers
66 views

Exponential of a square matrix [closed]

I need to find the matrix exponential $\exp(At)$ where $$A= \begin{pmatrix} -a & 0 \\ 1 & 0 \\ \end{pmatrix}. $$
0
votes
2answers
117 views

Simplify $(2^{2015})(5^{2019})$

Question : $(2^{2015})$$(5^{2019})$ How do I simplify/solve that without a calculator? I have no idea how to continue, I know it's important to get the Base number the same so I can add the ...
1
vote
0answers
56 views

Three almost-integers of the form $ce^{H_a+H_b}\approx 2^k\pm1$

The approximation $$H_n\approx log(2n+1)$$ http://math.stackexchange.com/a/1602945/134791 suggests that the harmonic number for composite odd numbers might be close to the sum of the harmonic ...
0
votes
1answer
21 views

Evaluating an integral that contains a positive definite matrix

Let $A \in \mathbb{R}^{d \times d}$ be a positive definite matrix (meaning that $z^t A z > 0 \space \forall z \in \mathbb{R}^d, z ≠ 0$, although I'm also allowed to use any other of the "usual" ...
2
votes
1answer
36 views

Unable to process Large numbers [closed]

A small spherical cell of diameter $1.616E^{-35}$ is exponentially multiplying as $2^n$ where n is the generation number. The duration of 1 generation is $5.39E^{-44}$ second. And the cells cluster ...
0
votes
0answers
15 views

Which one is the right answer if this was on the SAT Math IIC?

If $300 is invested at 3%, compounded continuously, how long (to the nearest year) will it take for the money to double? (If P is the amount invested, the formula for the amount, A, that is available ...