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

learn more… | top users | synonyms (1)

2
votes
2answers
43 views

Calculate $\int_{-\pi}^{\pi} \frac{xe^{ix}} {1+\cos^2 {x}} dx$

So I'm trying to calculate $$ \int_{-\pi}^{\pi} \frac{xe^{ix}} {1+\cos^2 {x}} dx $$ knowing that if $f(a+b-x)=f(x)$ then $$ \int_{a}^{b} xf(x)dx=\frac{a+b}{2} \int_{a}^{b} f(x)dx, $$ but it doesn't ...
0
votes
0answers
22 views

How to evaluate or approximate this kind of recursion: $a(n+1) = m \cdot \exp(-K \cdot a(n) / m)$?

I came up on a recursive definition of a function, given by $$a(n+1) = m \cdot \exp(-K \cdot a(n) / m),\ n \geq 2$$ with $m$ and $K$ being fixed integers ($m$ large). The first terms of the recursion ...
1
vote
1answer
24 views

How to calculate this integral $\frac{1}{2} \int_0^1\ 1.5 e^{-ik\pi \ t} \ \ dt, \, k \in \mathbb{Z} $

$$\frac{1}{2} \int_0^1\ 1.5 e^{-ik\pi\ t} \ \ dt, \, k \in \mathbb{Z} $$
0
votes
0answers
18 views

Exponential-ish function from 0,0 to 1,1: how to push the turning point of the curve

I am trying to find a weighting function to map $x$ values $0 < x < 1$ to a $y$ values $0 < y < 1$, following something similar to an exponential curve. So far, I have been using the ...
2
votes
1answer
23 views

Problems Calculating Fractional Derivative

I have been trying to calculate the fractional derivative of $e^{ax}$ using the Liouville Left-Sided derivative, which states that, for $x>0$ and $0<n<1$, $D^n f(x) = \frac{1}{1-n} ...
2
votes
2answers
35 views

Infinite differentiability of a function with a removable discontinuity

How would I prove that $\frac x{e^x-1}$ is infinitely differentiable? (This question came up since the No 1 answer in Maclaurin series for $\frac{x}{e^x-1}$ states that the function is infinitely ...
0
votes
1answer
17 views

Limit of trig functions

We have to evaluate $$\lim_{x\to 2} \frac{\cos^x a +\sin^x a -1}{x-2}.$$ I am working on it for hours I tried using series , replacing $\cos a$ by $t$ and $\sin a$ by $\sqrt{1-t^2}$ but not got any ...
0
votes
2answers
24 views

Graphing log with number in front of “log”

When I have something like $y = log_2(x)$ I know that I have to turn it into exponential form and get: $2^y = x$. Next, I make a table for $X,Y$ and choose about 5 values for $y$, typically $-1, 0, 1, ...
0
votes
2answers
39 views

What will the value of an account be after $12$ years if the account earns $4.91\%$ a year and if someone invests $\$20,000$?

Second National Bank offers an account that earns $4.91\%$ per year, compounded continuously. If a person invests $\$20,000$ in this account, what will be the value of the account at the end of $12$ ...
5
votes
1answer
67 views

Entire function $f$ such that $\lim\limits_{z\rightarrow \infty}f(z)=0$ and $f(0)=1$?

The question is this: Does there exist an entire function $f$ such that $\lim_{z\rightarrow \infty}f(z)=0$ and $f(0)=1$. I immediately would point to $f(z)=e^{-z}$. It is entire and satisfies the ...
0
votes
3answers
18 views

isolating x with two variables and negative exponents

I have: $$ 4^y = x^{-2} $$ Can someone hint to me what I need to do to isolate $x$? I'm not sure what to do.
0
votes
1answer
36 views

Convergence of $\sum_{k \geq 1} e^{-tk} \cos kz$

I would like to find the convergence of the series $\sum_{k \geq 1} e^{-tk} \cos kz$. Clearly, this series converge in using the comparison test or the integral. How could I get an explicit function ...
2
votes
2answers
36 views

Limit of indeteminate form $1^{(∞)}$

If we consider the function $f(x)=[(ax+1)/(bx+2)]^{x}$ where $a$,$b$ >$0$ and a I tried as follows]1 But at end i got stuck .
0
votes
0answers
22 views

Integral computation with Mathematica and Sympy differ

To compute the integral: $I = \int_{0}^{+oo} ue^{Au^{2}+Bu}du$ where $A<0$ and $B>0$ I have tried both Mathematica and Sympy but they yield different results: Mathematica yields: $ I = ...
1
vote
1answer
46 views

Express $y = KC^x$ as a linear function

Consider an exponential relationship of the form $y = KC^x$ where $K$ and $C$ are constants. Express the exponential function $y = KC^x$ as a linear function and describe how you would obtain the ...
0
votes
2answers
18 views

How to show that: $\log_a (x^{a}-x)-\log_a \Big(\dfrac{x^{a}-x}{a}\Big)=1$, where $a$ and $x$ are positive integers.

I was studying Fermat's Little Theorem and Logarithm to see if there is any interesting result or correlation exist between the two. So I came up with this equation. I know few basic logarithmic ...
4
votes
2answers
68 views

Check that $\lim\limits_{n\to\infty}\sum\limits_{i=1}^{n}\left(\frac{i+x}{n}\right)^n=\frac{e^{x+1}}{e-1}$

Show that $$\lim_{n\to\infty}\sum_{i=1}^{n}\left(\frac{i+x}{n}\right)^n=\frac{e^{x+1}}{e-1}$$ Any hints how I can tackle this problem? Although I checked on a sum calculator that it converges ...
0
votes
1answer
27 views

Prove, for every $l \geq 3$ , the $\Big( 1- \dfrac{1}{2 \cdot l}\Big)^{2 \cdot l} < \dfrac{1}{e}$ holds

I need to prove that for every $l \geq 3$, the $\Big( 1- \dfrac{1}{2 \cdot l}\Big)^{2 \cdot l} < \dfrac{1}{e}$ holds. ($l$ is integer) This is what I tried so far. $$ \begin{align} x &= ...
2
votes
1answer
42 views

$(-1)^0$ , calculating zeroth power of a negative number

I wish to calculate the zeroth power of a negative number $(-1)^0 = (-1)^{2-2}$ =$\frac{(-1)^2}{(-1)^2} = 1$ But when I put it in a calculator, it comes out to be $-1$.
0
votes
2answers
33 views

Equations with variable Exponents

I am struggling to find a solution to $x^{x-5}=5$, although clearly from plotting the graph of $f(x)=x^{x-5}-5$ I can see that there are two real solutions, but I have no idea how to evaluate them, or ...
0
votes
3answers
56 views

exponential function and mathematical induction

May I ask how to solve the problem? Use mathematical induction to prove that for $x\geq0$ and positive integer $n$, $$e^x \geq 1 + x + \frac{x^2}{2!} + \cdots + \frac{x^n}{n!}$$
0
votes
1answer
28 views

A basic question about the decay rate of $te^{-t}$ as $t$ tends to infinity

It is well-known that $te^{-t}$ tends to $0$ as $t$ tends to infinity. But I want to know the decay rate of $te^{-t}$ as $t$ tends to infinity. Using Taylor expansion of $e^{t}$ we have: $${t ...
0
votes
3answers
31 views

Subtracting powers with variable in exponent

I am having some troubles with a question that subtracts powers. Solve for unknown: $$3^{x+4} - 5(3^x) = 684$$ I have a hunch that I should apply factorization somehow. Do I multiply 5 and 3 to ...
2
votes
1answer
25 views

What is the solution for $y(t)=e^{-\frac{t}{\tau y(t)}}$?

A simple quadratic flow model leads to the following apparently simple equation $$y(t)=e^{-\frac{t}{\tau y(t)}}$$ where the flow, $y$ is a function of time, $t$ and $\tau $ is a constant. But is ...
0
votes
0answers
16 views

Can I bound this $N((1+\delta)^{2N}+1)\leq N^j$? for any $j\in \mathbb{N}$, where $\delta<1$

For $N\in\mathbb{N}$ large enough. Can I bound this $N((1+\delta)^{2N}+1)\leq N^j$ for any $j\in \mathbb{N}$, where $\delta<1/2$? I tried using Matlab for j=5, but I'd like some ideas for a proof. ...
0
votes
0answers
21 views

Is $(1 + \tau / (3p))^k \ge e^{k\tau/(4p)}$ really true?

In the paper "Preserving Statistical Validity in Adaptive Data Analysis", it says that if $p \in (0,1], \tau \in [0, 1/3]$ then $$(1 + \tau / (3p))^k \ge e^{k\tau/(4p)}$$ I understand that $(1 + \tau ...
0
votes
2answers
29 views

Calculating the mass xkg of radio-active substance pertaining to days after starting timing

Just testing myself with some tricky questions in my further maths textbook. This one states that the mass xkg of a radio-active substance remaining in a sample t days after starting timing is given ...
0
votes
2answers
24 views

multiplying powers with variable in exponent and different bases

I am having trouble sorting out where to begin with solving for unknown value in this equation: $16^{5a−1} \times 256^{3a} = 128$. I imagine I would need to change to logarithmic form, but am ...
0
votes
1answer
58 views

If $x>0$ and $x^a=x^b$ can one assume that $a=b$?

If $x>0$ and $x^a=x^b$ can one assume that $a=b$? The answer says it's not right. I've tried coming up with a counter-example but keep failing. Thanks in advance!
1
vote
2answers
38 views

$(5 + (24)^{\frac{1}{2}})^x + (5 - (24)^{\frac{1}{2}})^x = 10$ , solve for $x$

I have been stuck to this question lately $(5 + \sqrt{24})^x + (5 - \sqrt{24})^x = 10$ , solve for $x$
-1
votes
1answer
19 views

fill in a table with a 7% yearly growth [closed]

There is $1000$ put in a account and it grows $7\%$ each year starting at $1000$. I'm confused on how to get the calculation and I have several problems just like this.
0
votes
0answers
18 views

Coefficient Correlation r of Exponential Functions Regression

I'm writing an exponent regression calculator $Ae^{Bx}$ Sample Data Set (X,Y) is (9, 1) (7, 10) (6,11) (20, 10) (15, 1) A = 5.287 and B = -0.0232. So $F(x) = ...
4
votes
2answers
64 views

Prove $\exp(\mathrm{Tr}(X))=\det(\exp(X))$

Show that $\exp(\mathrm{Tr}(X))=\det(\exp(X))$ where $X$ is a matrix using the concept of the Jordan normal form I realised this formula by considering that: $\det(\exp(X))=\exp(\lambda_1) ...
0
votes
0answers
26 views

Shift logistic function without moving inflection point from x=0

As a biologist that did not do much maths lately, formulation of my problem may be a bit strange. Sorry in advance and thanks for helping me improving my vocabulary. I am using logistic function from ...
0
votes
1answer
52 views

How do you solve a system of equations with e^x

How do you solve a system of equations with e^x. For example
-3
votes
2answers
87 views

Why exp(x) is so special that- [on hold]

$\exp(x)=\int \exp(x) \; \text{d}x$ = derivative of $\exp (x)$ with respect to $x$. I'm curious to know this? Do other such functions exist?
3
votes
0answers
105 views

Solving equation involving factorials

I have this particular equation $\frac{(\alpha-1)!(\beta-1)!}{(\alpha+\beta-1)!} = \frac{\Gamma(p)(1+q)^{n+2p} 2^n}{q^{p}(2+q)^{n+p}}$. Now, given the values of $\alpha$ and $\beta$, I need to find ...
1
vote
2answers
32 views

Combining linear and exponential functions

I am struggling with this problem: At $x=0$ I own $b$ units. Every year I deposit $a$ units. The bank pays an interest rate of $c$ every year. After how many years I will own $d$ units? ...
0
votes
4answers
53 views

Exponential decay 'proof' that $.\overline{9}\neq 1$?

I have doubts about $.\overline{9}$ being equal to 1 due to the following proof: To get a decimal containing $c$ 9's after the decimal point, the equation f(c) = $1-10^{-c}$ can be used. For ...
0
votes
3answers
46 views

Any smoother version of the exponential function?

Often one needs to express some quantity of interest in a scale other than its original one. One can use the exponential function to map $(-\infty,0)\to(0,1)$ and $(0,+\infty)\to(1,+\infty)$, but ...
0
votes
2answers
26 views

Simplifying a term

I want to know how we got the end term. It may be trivial, but I just can´t figure it out. Thanks for the help. Term
1
vote
1answer
42 views

Symmetric Matrix in SO(3) : Exponential Formula

Let $R\in $ SO(3), that is $R$ is real $3\times 3$ orthogonal matrix with determinant $+1$. I am trying prove that if $R= R^\top$, and $R\in $ SO(3) then $R \in \{exp(k\pi \hat{a}) | k\in \mathbb{Z}, ...
0
votes
1answer
38 views

How to compute the exponential of this matrix?

I am trying to prove all the results regarding linear algebra in my ODE class. I have already convinced myself that if I have a matrix $T$ which has an eigenvalue $\lambda = a + ib$ and an associated ...
1
vote
4answers
25 views

Half-life of Am-$241$, $3$ micrograms decays over $9$ years, how much if left?

$3$ micrograms of Americium-$241$, which has a half life of $432$ years. After $9$ years how much will remain? I'm not sure of the formula to use or how to calculate it. I'm assuming it's exponential ...
0
votes
1answer
28 views

How to solve $ax+be^{cx}=d$

Hi I have been trying to derive the formula for the range of a projectile with air resistance, and I've been trying to solve for time when $y=0$, and I have been left with an equation of the form ...
0
votes
1answer
32 views

Newton's Law of Cooling (and Heating)

The Formula for the equation is as follows: $$T(t)=\frac {\int^t(−T_s)ke^{-kt'}dt'+C}{e^{-kt}}$$ This formula is needed to determine the temperature at time $t$, $T(t)$, of an object as it begins to ...
2
votes
2answers
52 views

Prove that $e$ is the root of the equation $\int_0^{\infty} \frac{dt}{(t+x)\sqrt{4t+(x+1)^2}}=\frac{1}{x-1}$

It seems numerically that $e$ is the only real root of the equation: $$\int_0^{\infty} \frac{dt}{(t+x)\sqrt{4t+(x+1)^2}}=\frac{1}{x-1}$$ Mathematica confirms it at least to the large number of ...
1
vote
1answer
25 views

Newtons Law of Cooling in Forensic Science

Question goes: Law enforcement would like to know the time at which a person died. The investigator arrived on the scene at 8:15pm, which we will call $t$ hours after death. At 8:15 (i.e $t$ hours ...
2
votes
2answers
32 views

Questions about Exponentiation and roots and logarithms.

in this page a few questions I want to ask you about the Exponentiation and roots and logarithms: What and how the Exponentiation definition can be defined by real numbers.? What is the overall ...
3
votes
4answers
254 views

Finding limit without using limit

If we have to find the value of $$ \lim_{x \to 0} \frac{e^x-1}{x}$$ I tried to solve this by using series i.e by expanding $e^x$ and got the result. But if there is another method to solve this