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

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0
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2answers
38 views

How to prove the following? $\frac{d}{dx}a^x=(\ln a)a^x$

How to prove that the following holds? $$\frac{d}{dx}a^x=(\ln a)a^x.$$ Just a hint will do it.
3
votes
2answers
97 views

Show that $f(x)=e^x$

In this case $f(x)=1+x+x^2/2!+x^3/3!+x^4/4! + ... = \sum_{n=0}^\infty \frac{x^n}{n!}$. I understand it conceptually in terms of the Taylor series, but I have no idea how to prove it rigorously.
26
votes
3answers
855 views

Limit with a big exponentiation tower

Find the value of the following limit: $$\huge\lim_{x\to\infty}e^{e^{e^{\biggl(x\,+\,e^{-\left(a+x+e^{\Large x}+e^{\Large e^x}\right)}\biggr)}}}-e^{e^{e^{x}}}$$ (original image) I don't ...
3
votes
2answers
74 views

Non-integral power of a singular matrix

I know, that if $A$ is nonsingular matrix, so $\det{A} \ne 0$, then $A^p=\exp\left(p\ln A\right)$ is true for any real exponent, but what about if $A$ is singular? Then $A$ has a zero eigenvalue, so ...
0
votes
2answers
45 views

Compound interest coumpounded n time per year formula. $A=P\left(1+\frac{r}{n}\right)^{nt}$ intuition behind it.

I know that the compound interest formula for the interest compounded annually is given by $$A=P(1+r)^t$$ I know the intuition behind it. But why the compound interest formula for the interest ...
0
votes
1answer
29 views

Exponential percentage decrease based on time

I have a bar that shows the time left for a task to finish and I want it to decrease faster as it gets closer to the end time. Example: Let's assume that the total time required for Task A to ...
1
vote
2answers
38 views

How do I transform an r.v. using the floor function? (exponential distribution)

Just had a bash at this question for my Intro to Maths Stats module...I got to the end with a probability density function rather than a probability mass function, namely $f_Y(y) = \lambda a ...
1
vote
1answer
17 views

Mean and STD of a max/min of an exponentially distrubuted iid random variable

Let $S_1, S_2, S_3, ...$ be a sequence of independent, identically distributed (iid) random veriables, each exponentially distributed with a mean of $\mu_S$ (hence $\sigma_S = \mu_S$). Let $M_n = ...
3
votes
1answer
55 views

Definite integration of a exponential function mixed with rational functions

Suppose $a>0$ , I am interested in a solution of the following definite integral: $$\int_{1}^{\infty}\frac{\exp({-az})}{z \sqrt{z^2-1}}{\,dz}$$ Thank you.
3
votes
2answers
67 views

Calculation of integral $\int\exp \left(-\alpha \sin^2 \left(\frac{x}{2} \right) \right) dx$

Given $\alpha$ is a constant. How to calculate the following integral? \begin{equation} \int \exp \bigg(-\alpha \sin^2 \bigg(\frac{x}{2} \bigg) \bigg) dx \end{equation} Thanks for your answer.
1
vote
3answers
98 views

Prove by induction that (5^(n))-1 is divisible by 4 for all natural numbers n.

Prove by induction that $5^n-1$ is divisible by $4$ for all natural numbers $n$. I got $P(k+1)=5^{k+1}-1$ but I don't where to go now.
1
vote
1answer
36 views

Solving $Ae^x=Bx$ analytically, where $A$ and $B$ are constants?

This equation mixes both exponential terms and linear terms, something which I do not know how to deal with. Any pointers?
0
votes
0answers
33 views

Problem with commutator relations

part a) is fine. part b) is not. A commutator is defined as, for operators $A$ and $B$, $[A,B]=AB-BA$. [SOLVED]I get that $H(\lambda)=e^{-\lambda D}Ce^{\lambda D}$, $H'(\lambda)=-De^{-\lambda ...
1
vote
1answer
100 views

Double integrals of exponential functions

I need to find the double integral of $$e^{\frac{x}{y^2}}$$ bound by the $y\mbox{-axis}$, $x=y^2$, $y=1$, and $y=2$. The limits of integration were easy to find, but I am pretty confused about how to ...
2
votes
1answer
57 views

Exponentiation of imaginary operator

It is very easy to prove that if $D=\dfrac{d}{dx}$, then $(e^{nD}f)(x)=f(x+n)$ about $x=m$ in the real numbers. Proof: $$(e^{mD}f)=\sum^\infty_{n=0}\dfrac{D^nf}{n!}m^n\\ \implies ...
0
votes
4answers
71 views

What is the name of the answer to exponentiation?

What is the name of the answer to exponentiation? Adding two numbers produces a sum. Multiplying two numbers produces a product, but I cannot think of or find the name for the solution to ...
1
vote
1answer
20 views

Mean and STD of a sequence based on Exponential Random Variable

Say I have a sequence $S$ that is a exponentially random variable with mean $\mu$. Now say I create another other sequences from this: $T$ which is $T(n) = 2(S(n+1)-S(n))$. I know that theoretically, ...
2
votes
2answers
83 views

Integral of exponential using error function

I'm trying to solve some integrals below $$\int_{-\infty}^{\infty} {x^n e^\frac{-(x - \mu)^2}{\sigma^2}}dx$$ I am interested in the solutions where n = 0, 1, 2, 3, 4. I have learned that ...
1
vote
1answer
46 views

Exponential problems

A ship embarked on a long voyage. At the start of the voyage, there were 300 ants in the cargo hold of the ship. One week into the voyage, there were 600 ants. Suppose the population of ants is an ...
0
votes
1answer
82 views

Series proof for $e^x$.

Problem: Prove $$\sum_{n=0}^\infty \frac{1}{n!}x^n=e^x$$ I am a bit confused on how I should start this proof. Any pointers on how I should start would help.
2
votes
1answer
80 views

Gradient of matrix exponential function

Grateful if somebody could help me with the following. I am trying to find the gradient of the next expression: $$f(a_1, a_2, a_3, a_4)=\Vert R*y-x \Vert $$ where $y$ and $x$ are known 4x1 column ...
1
vote
1answer
68 views

Solving Exponential Function for termites vs spiders

The populations of termites and spiders in a certain house are growing exponentially. The house contains 120 termites the day you move in. After four days, the house contains 210 termites. Three days ...
1
vote
1answer
50 views

Rewriting in $y=A_0\cdot e^{at}$

How do you rewrite $y = −8(1.589)^{t − 3}$ in $y=A_0e^{at}$ form for appropriate constants $A_0$ and $a?$ For other problems I took the $\ln$ of the number inside the parenthesis. So for example I ...
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vote
2answers
25 views

Exponential Growth Rates

So if you are given two different numbers to determine a growth rate, do you use to largest number compared to the value when x=0. For example the problem I am working on is: Your grandfather ...
0
votes
2answers
36 views

Logs of a complex number

Write a solution in Cartesian for of What should come next?
0
votes
1answer
15 views

prove doubling time in an exponential function

I am currently working my way through Hughes-Hallet et al., Calculus- Single and Multivariable. I am having trouble with the following problem. Show algebraically that if P=P0a^t doubles between ...
0
votes
1answer
66 views

how to solve equation $x^x=5$ [duplicate]

How can I calculate the equation $x^x=5$ Is it an exponential function? Thank you.
2
votes
8answers
272 views

How come $1^{\infty}$ = undefined, while $2^{\infty} = \infty$ and $0^{\infty} = 0$? [duplicate]

$1^\infty$ = undefined $2^\infty = \infty$ $0^\infty = 0$ Why is $1^\infty$ undefined? People were trying to explain to me that infinity isnt part of the Real numbers, yet, $2^\infty$ and ...
2
votes
1answer
31 views

Finding Complex Zeros

I have to find how many zeros $3e^z - z$ has in $abs(z) < 1$. I know a function has a zero of order m if $f(z) = (z-z_0)^mg(z)$, where $g(z)$ does not equal 0. I was thinking of maybe applying ...
-14
votes
4answers
179 views

How is the limit definition of e, actually equal to e? [duplicate]

$$\lim_{n\to\infty} (1+\dfrac{1}{n})^n = e.$$ To me this seems like $\dfrac{1}{n}$ goes to $0$, then you get $1^\infty$, which is equal to $1$. So why is it $e$?
0
votes
1answer
30 views

Rescaling, or finding logarithmic equivalent of exponential functions

I'm working on an algorithm that gets the weighted least squares of some data and outputs really big, or really small numbers. I'm talking about numbers like $3.55114473577E+256$. Now, part of the ...
0
votes
1answer
16 views

Basic Integration + Root + Exponential issue

I have the following question : $\int ( 27e^{9x} + e^{12x} )^{1/3} dx $ However when I solved it I simplified it first to: $\int ( 27e^{9x} + e^{12x} )^{1/3} dx = \int \sqrt[3]{27e^{9x} + ...
1
vote
1answer
24 views

%reduction in a decaying exponential function

I am working my way through a calculus book I purchased- Calculus- Single and Multivariable (3rd edition) by Hughes-Hallet et al. I am having issues with the following question "When the olympic ...
0
votes
2answers
68 views

Math notation that I am not familiar with

I am working on reverse engineering a game and have come across the following formula as a string in a config file: A*(B^xt)+C; xt=A2*x*(T>x)*(B2^x+C2) It ...
0
votes
1answer
26 views

Definition of Derivative And Exponential Functions

Given $f(x) = 5^{3x}$. Find $f'(x)$ using definition of a derivative. The definition of the derivative of $f(x)$ is $f'(x) = \lim_{h \to 0} \dfrac{f(x + h) - f(x)}{h}$ The derivative of $f(x) = ...
2
votes
1answer
46 views

Solving an equation with curly brackets and entering into excel

Hi I am sorry if this is a trivial question but I am trying to follow a book to create a mathematical model but I can not get the same result. The equation has curly brackets and I am not sure what ...
4
votes
1answer
70 views

Applications of the Exponential Integral?

this is my first time asking a question on here so please forgive me if I have made any formatting mistakes. I have the integral $f(x) = \int_0^\infty \frac{e^{-t}}{x + t} \; dt$ and I have shown the ...
0
votes
2answers
48 views

Is this exponential equation solvable? natural logarithms, exponential

$$\displaystyle{a=\frac{e^{-cos(\frac{b}{x})}-e^{-\frac{1}{x}}}{(1-e^{-\frac{1}{x}})}}$$ I'm trying to solve for $x$. $a$ and $b$ are constants. Any help is really appreciated. Thanks Ghassan
2
votes
2answers
68 views

How can I define $e^x$ as the value of infinite series?

I understand the definition of $e^x$ by limit. But I do not know how to come up with: $$e^x = \sum_{n=0}^{\infty}\frac{x^n}{n!}$$ without using Taylor series. more explicitly without using calculus. ...
1
vote
1answer
43 views

Biasing sigmoid curve

I wish to use the sigmoid function $1-{1\over1+e^{-x+c}}$ to obtain a value from 0 to 1 (to be used for a probability value), where $c$ is a constant. The higher this constant, the lower the ...
0
votes
0answers
20 views

Trying to reverse engineer a formula called “exponential_flat”

Here are the values I know: A = 200 B = 1.75 C = 0 A2 = 1.8 B2 = 0.93 C2 = -0.64 T = 14 It is possible that some of these values are not used. The other formula ...
0
votes
1answer
33 views

Simulation - Find the maximum of a function with exponential decay

I need to run a program to calculate the integral of the following function with exponential decay $$t(x) = \exp(-Lx)(a\sin(bx) + d\cos(ex))$$ and for a simulation purpose, I need to find maximum of ...
4
votes
3answers
63 views

Game With 21 Squares, How Many Possible Answers? Function Building

We played this game in our math class, okay, I'll explain how it's played. There are 21 squares in a straight line across, the first person shades in 2 adjacent squares. The next player shades in 2 ...
0
votes
1answer
23 views

Exponential Random Variables and Confidence

Assume that the amount of evidence against a defendant in a criminal trial is an exponential random variable X. If the defendant is innocent, then X has mean 1, and if the defendant is guilty, then X ...
0
votes
1answer
59 views

Finding time constants of a circuit?

So this is a homework question and I am having trouble figuring out what they are asking. 'The potential difference (voltage) across the capacitor at time t > 0 is given by $V_C(t) = q(t)/C$. The ...
1
vote
2answers
34 views

Integrating two exponentials produces a cosine integral? Can somebody explain?

I discovered the following conversation that I do not understand. It reads: $$\int_{-U_1}^0 {(\frac {u_1} {U_1}+1)e^{-j\omega_1u_1}}~du_1+\int_0^{U_1} {(-\frac {u_1} {U_1}+1)e^{-j\omega_1u_1}}~du_1 = ...
2
votes
2answers
18 views

Exponential growth precalc population

The population of City A increases by 8% every 10 years. The population of City B triples every 120 years. The two cities had equal populations of 10,000 residents each in the year 2000. In what year ...
1
vote
1answer
39 views

What is the integral containing decaying exponential function?

I am trying to figure out properties of the following integral: $$p(t)=\int_{0}^{t} e^{\alpha(t-t')} f(t')dt', \hspace{1 cm} t>t'$$ I would google and read more info about this integral but I do ...
2
votes
0answers
26 views

Yacov Perelman Nepero game

This is my first question, so sorry if I'll make any mistake in using the site formatting. I found this game on a book by Yacov Perelman and I thought it could be nice to introduce Nepero number to ...
-1
votes
2answers
68 views

Integration of $g(x) = e^{f(x)}$ [closed]

Is there any way of simplifying this integral? $$ f(x) = \int e^{2x^3}\,dx $$