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

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3answers
35 views

Show an exponential function has a valid density.

Given: Let $X$ be exponential with parameter $\lambda$, that is $$ f_X(x) = \begin{cases} \lambda e^{-\lambda x} & \text{if }x> 0, \\ 0 &\text{for }x\leq 0. \end{cases} $$ where ...
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1answer
50 views

Proving that the exponential function is its own derivative, using the limit definition of $e$

I saw the proofs on the derivative of $\frac{d e^x}{dx}=e^x$ from here and the one that was intriguing was this : $$e^x:=\lim_{n\to\infty}\left(1+\frac{x}{n}\right)^n \implies \frac{d(e^x)}{dx} = ...
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0answers
29 views

How does this exponential equation add up?

Here is the equation. How do we add up and get such a value? I can convert to a non-rad value, but I can't understand how 0.327 or -1.18 is gotten
3
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1answer
63 views

Details from a Proof that a Tournament has Property $S_k$

(Edit: While the context is not central to my question, I decided to include it anyway to make the question a little more searchable.) Some technical details are omitted from a theorem in Alon and ...
0
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1answer
39 views

the absolute value of $\frac{1}{e^{i\omega t}-1}$

I am told to get the absolute value of $$\frac{1}{e^{i\omega t}-1}$$ I sense that there's something ridiculously simple about this, but I tried working from the fact that if I square it, the absolute ...
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3answers
47 views

How to solve exponential inequality with $x$

I need to solve the following inequality. $$\ln(x) - x > 0.$$ I oddly remember that it can only be done by using the graph... Is it true? I have the same problem with $$e^x(x-1)>-2.$$ ...
2
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1answer
91 views

Summation of exponential series [duplicate]

Evaluate the limit: $$ \lim_{n \to \infty}e^{-n}\sum_{k = 0}^n \frac{n^k}{k!} $$ It is not as easy as it seems and the answer is definitely not 1. Please help in solving it.
2
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1answer
56 views

Why does a heating model work?

I am referring to: $T=T_0 e^{kt}$ where T=temperature,t=time and k=constant. It seems to work, I as just curios to why it works?
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2answers
138 views

Is there any proof for this formula $\lim_{n \to ∞} \prod_{k=1}^n \left (1+\dfrac {kx}{n^2} \right) =e^{x⁄2}$

Some times ago, In a mathematical problem book I sow that this formula. I don't no whether it is true or not. But now I'm try to prove it. I have no idea how to begin it. Any hint or reference would ...
2
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3answers
80 views

If $\ln x$ is defined via an integral and $e$ defined from $\ln x$, how would you prove that $\ln x$ is the inverse of $e^x$?

This is a somewhat technically specific question about the relationship between $\ln x$ and $e^x$ given one possible definition of $\ln x$. Suppose that you define $\ln x$ as $$\ln x = ...
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4answers
56 views

Proving the Derivative of $f'(x) = b^x$

Given $f(x) = b^x = e^{x\ln b}$ for $b > 0$, can someone show me how $f'(x) = \ln b e^{x\ln b}$ ?
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2answers
131 views

Constrained Newton-Raphson method

Peace be upon you, I want to solve a system of two equations in which the existence of $ln\left(\frac{\alpha}{\alpha+\beta}\right)$ function makes some limitations in iterations of the Newton-Raphson ...
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0answers
36 views

Overflow and underflow of a probability value

I am evaluating the probability that the minimum of a process is a above a a barrier $\log(H)$. The probability is given by $$P_i=1-\exp\left(-2\frac{(\log(H)-x)(\log(H)-x_b)}{\tau\sigma^2}\right).$$ ...
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0answers
64 views

Exponential of a polynomial of the differential operator

Given that $$\exp(aD)f(x)=f(x+a)$$ where $\exp(D)$ is the exponential of the differential operator $D$, is there a similar closed-form, general expression for $\exp(g(D))f(x)$, where $g(D)$ is a ...
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2answers
21 views

Special vs. General Case in Basic Algebraic Notation

From Rogawski, Jon (2011-04-01). Single Variable Calculus (Page 343). W. H. Freeman. : To compute the derivative of a function of the form eg(x), write eg(x) as a composite eg(x) = f(g(x) ), where ...
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1answer
123 views

Find exponential decay equation for tiger population model

I've forgotten how to do it first it starts.. In 1900, there were 100,000 wild tigers worldwide; in 2010 the number was 3200. (a) Assuming that the tiger population has decreased exponentially, find ...
2
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2answers
133 views

Solving integral that contain exponential function and lower incomplete gamma function

I have the following integral; $$y=\int_0^\infty\frac{e^{-xf}}{m+x}\gamma(a,hx)~dx$$ where $f,m,h\in\mathbb{R}^+$ , $a\in\mathbb{N}$ , $\gamma\left(a,h x\right)$ is the lower incomplete gamma ...
5
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1answer
91 views

If $e^{i\theta}=e^{i\varphi}$, then $\theta-\varphi=2k\pi$

This is pretty easy I think but I am having a tough time trying to prove this in a satisfying way to me. I am trying to show that $$e^{i\theta}=e^{i\varphi} \Rightarrow \theta-\varphi=2k\pi,\, \text{ ...
2
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4answers
188 views

The definition of e by limits of $(1+1/n)^n$ through series expansion

I think the problem I have is due to not being knowledgeable about limits. If I use binomial expansion to expand $(1+1/n)^n$ to $1 + \frac{n!}{(n-k)!k!}*(1/n)^k + ...$, I can imagine replacing $n$ ...
3
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0answers
49 views

Why is the base of an exponential function limited to the set of real numbers greater than zero?

From Rogawski, Jon (2011-04-01). Single Variable Calculus (Page 339). W. H. Freeman. An exponential function is a function of the form $f (x) = b^x$, where $b > 0$ and $b \neq 1$. Why is $b$ ...
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2answers
58 views

What is the method to correctly isolate $y$ as the dependent variable for $x = e^y$?

In this youtube video about 5:00 minutes in, the instructor makes the point that you can simply exchange the $x$ and $y$ values of the exponential form $x = e^y$ of the equation $y = ln x$ to make $y$ ...
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0answers
204 views

Complex exponential integral: Mathematica and MATLAB give unexpected results

I currently compare analytical vs. numerical evaluation of the complex exponential integral and find mismatches: The imaginary part differs by $\pm \pi$ and the real part has a large error when ...
0
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2answers
63 views

Matrix Exponent - equivalent of a rotation matrix

Every Rotation Matrixcan be represented as a power of e with exponent a skew symmetric matrix. In particular, if we have a rotation matrix ${R}\in\mathbb R^{3 \times 3,}$ then there will be a skew ...
2
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2answers
74 views

Evaluating exponential integral

I am struggling for some time to solve the following integral: $$ \int_{-n}^{N-n} \left( \frac{e^{-j\pi(\alpha-1)\tau}}{\tau} - \frac{e^{-j\pi(\alpha+1)\tau}}{\tau} \right) d\tau $$ $N$ is a ...
5
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3answers
121 views

Is $x^x$ a polynomial, an exponential or both?

If $c$ is a constant, and $x$ is a variable, we'd say that $f(x) = x^c$ is a polynomial function of order $c$. Conversely, the function $f(x) = c^x$ would be called an exponential function. Is there ...
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2answers
40 views

Solving equations having both log and exponential forms

How can one Solve equations having both log and exponential forms: For eg... $e^x$ $=$ $\log_{0.001}(x)$ gives $x=0.000993$ (according to wolfram-alpha ...
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0answers
104 views

Why is sequence $(1+\frac{1}{n})^{n+1}$ descending? [duplicate]

I was studying the proof of $e$ number when I noticed something: Why is the sequence $(1+\frac{1}{n})^{n+1}$ descending? It starts ascending with grater n but in one moment it starts descending? Why ...
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3answers
35 views

How do I rewrite a logarithm in exponential form, so as to plot it? $f(x) = 2\log x$

How do I write $f(x)=2\log x$ in exponential form? Is $2(10)^y=x$ correct?
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1answer
35 views

Exponential function, domain of definition

I have the function $\displaystyle f(x,y)=x^2e^{-x^2-y^2}$ with the domain of definition = $\{(x,y) \mid x^2+y^2=2\}$ The task is to decide $f$'s maximum and minimum value and the range. How do I get ...
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1answer
31 views

Integrals with an imaginary linear term in the argument of the exponent

in this entry on Wikipedia stays $$ ...
2
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0answers
32 views

Solving $n \int_{\mathbb{R}}{\left|\frac{1}{n}\sum_{j=1}^n{e^{(itY_j)}}-e^{-\frac{1}{2}t^2}\right|^2}\psi(t)dt$

I have to solve $$ n \int_{\mathbb{R}}{\left|\frac{1}{n}\sum_{j=1}^n{e^{(itY_j)}}-e^{-\frac{1}{2}t^2}\right|^2}\psi(t)dt $$ where $\psi(t)=(2\pi)^{-\frac{1}{2}}e^{-\frac{1}{2}t^2}$ is the density ...
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2answers
136 views

Exponential function, multivariable calculus

I got the function $f(x,y)=e^{-x^2-y^2}$ with the domain of definition $x^2+y^2\leq25$. The task is to decide the biggest and lowest value. How do I get there?
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1answer
74 views

Proving $E_{\theta}[T(X)] = \frac{\psi'(\theta)}{\eta'(\theta)}$

I'm trying to understand how to prove the following theorem: Let $\{P_{\theta}, \theta \in \Theta\}$ be a family of distributions in the one parameter exponential family with density (pmf) ...
2
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1answer
64 views

Solving exponential equation $e^{x^2+4x-7}(6x^2+12x+3)=0$

How would you find $x$ in: $e^{x^2+4x-7}(6x^2+12x+3)=0$ I don't know where to begin. Can you do the following? $e^{x^2+4x-7}=1/(6x^2+12x+3)$ and then find $ln$ for both sides?
1
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1answer
45 views

Find real-valued sequences $x(n)$ for which $c^{x(n)} = o(1/n )$

For which $x=x(n)$ does it hold that $$c^x = o\left(\frac{1}{n}\right)$$ where $c\in(0,1)$ is a constant. So clearly, for $x=n$, this is true. But for which $x =o(n)$ does this hold? I thought ...
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1answer
376 views

Approximating the exponential function

I have found experimentally something that seems graphically like an approximation of the exponential function. However, it is totally experimental and I have no idea whether it really converges ...
0
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1answer
29 views

How to scale a equation e.g. by log

I'm currently trying to scale an equation since the numbers I have to calculate with are pretty large and Matlab outputs Infinity (Inf). However, the question here is more about the mathematics behind ...
0
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2answers
70 views

Construction of complex numbers and exponent rules for them

I have some questions about the construction of the complex numbers in this Wikipedia article, especially of the exponents of complex numbers. $1$. Is it enough to define it as $a+bi$, where a,b are ...
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4answers
47 views

Exponential function (t)

I got the function $8.513 \times 1.00531^{\Large t} = 10$. The task is to solve $t$. The correct answer is $t = 31$. How do I get there ?.
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1answer
57 views

How can we know that x^x is an exponential function or not without drawing the graphic?

In general, exponential function is defined as a.b^x, where a=coefficient, and b= base. I only knew that the function is exponential function or not, just by drawing the graphic. But, how can we ...
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1answer
58 views

Where do I make mistake on this derivative containing e^x^2

My brother is preparing for the university and asked me the following multiple choice question. $$\frac{d}{dx}(x^3 * e^{x^2})$$ a) $e^{x^2}*x^2*(1+2x)$ b) $e^{x^2}*x^2*(3+2x)$ c) ...
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1answer
39 views

Explicit and Recursive Exponential Growth

The population of a certain organism triples every hour. Write a function that models this growth. By what factor does the population grow in one-half hour? I'm unsure of how I should approach the ...
2
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4answers
67 views

How to solve this kind of equation $(x^y=y^x)$

I'm little bit stuck with this system of equations : $x^y=y^x$ and $x^3=y^2$ An obvious solution is $(x,y) = (1,1)$ but what about the solution $(9/4,27/8)$ ? I know the relation $a^r=e^{r ...
1
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1answer
49 views

Solution of $d^2u/dx^2 + u/A = 0 \ (\text{or } \ C),$ with conditions

Does the following ODE: $$d^2u/dx^2 + u/A = 0 \quad (\text{or } \ C),$$ have a solution with the conditions: $$ \left.\frac{d^2u}{dx^2}\right|_{x=0} = 0, $$ $$u(x=0) = B$$ and $$ ...
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0answers
33 views

Log(x) or Exp(x) with limits 0,0 and 1,1

trying to find an equation or function that displays an accelerated progression. It may be an Exp or Exp-like function. Input is X, ranging from 0 to 1. Output is Y ranging from 0 to 1. Note Y must ...
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2answers
38 views

What kind of mathematical operation is used to repeatedly increase a number by a certain percentage?

I am sure that this is an easy question to answer for most of you. I need to take a number, let's say $10$, and then increase it by a percentage, let's do $25\%$. $10 \times 1.25 = 12.5$ Easy ...
2
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1answer
82 views

Exponential Diophantine: $2^{3x}+17=y^2$

Is there a way of solving the following equation, in integers $(x,y)$, by hand? : $2^{3x}+17=y^2$. You can also try: $2^{2x}+17=y^2$ or more generally $2^x+17=y^2$; each of these has at least 1 ...
2
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2answers
62 views

Solve the inequality $(1/2)^x-(1/2)^{-1-x}\ge1$ for real $x$

I have to solve in $\Bbb{R}$ the following inequality : $$ \left(\frac{1}{2}\right)^{x} - \left(\frac{1}{2}\right)^{-1 - x} \ge 1 \qquad(E) $$ So far I have : For $x=0$ this inequality if not ...
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0answers
16 views

Explicitly relating two functions containing exponential terms

I have two functions related to the distribution of administered drugs in the body: $$\begin{align}c_1(t) &= a_1\exp(-k_{11}t) - b_1\exp(-k_{21}t)\\ c_2(t) &= a_2\exp(-k_{12}t) - ...
1
vote
5answers
96 views

What is the reason to introduce and study logarithmic functions?

I don't understand why logarithms exist when we have exponential functions. Exponential functions seem to be an easier and less convoluted way to write something. Why invent logarithms to do something ...