1
vote
0answers
43 views

How to prove that $b^{x+y} = b^x b^y$ using this approach?

Fix $b>1$. If $m$, $n$, $p$, $q$ are integers, $n > 0$, $q > 0$, $r = m/n = p/q$, then I can prove that $(b^m)^{1/n} = (b^p)^{1/q}$. Hence it makes sense to define $b^r = (b^m)^{1/n}$. I ...
0
votes
4answers
40 views

Why is a Constant added to front?

I made the differential equation : $$dQ = (-1/100)2Q dt$$ I separate it and get: $\int_a^b x (dQ/Q) = \int_a^b x (-2/100)dt$ this leads me to: $\log(|Q|) = (-t/50) + C$ I simplify that to $Q = ...
0
votes
1answer
58 views

How does one obtain the expansion of $e^{-x^2}$ in a power series?

So I know that the Power Series $y = \displaystyle\sum_{m=0}^\infty\displaystyle\frac{(-1)^m}{m!} x^{2m}$ is equivalent to $e^{-x^2}$. Could someone show me why this is?
1
vote
2answers
59 views

How does exponentiation relate to multiplication?

My book derives the logarithm function as a definite integral of $1/x$ and defines the exponential function as its inverse. It then extends this definition to other bases: $$b^x = e^{\ln (b) x}$$ ...
3
votes
1answer
71 views

Integral of an exponent of an exponent

For a homework problem, I have to integrate this: $$\int{4^{(4+x)^x}}dx$$ How would I go around to starting this question? I don't know how to evaluate this, and I've tried to use u-subs and ...
0
votes
2answers
48 views

Matrix Exponent - equivalent of a rotation matrix

For every Rotation Matrix,there is a Matrix Exponent representation where the power is a skew symmetric matrix. More clearly if I have a rotation matrix ${R}_{3 \times 3}$ then there will be a skew ...
6
votes
2answers
219 views

Evaluating a limit. What makes the equality right?

I'm reading a proof of a limit calculation. The limit is: $$\lim\limits_{x\to 0}\left(\frac{a^x+b^x}{2}\right)^\frac{1}{x}$$ where $a,b>0$. The aother claims that: $$\lim\limits_{x\to ...
0
votes
2answers
44 views

When $\ln(1+y) = y + o(y)$?

I was reading a proof which utilize the fact that: $\ln(1+y) = y + o(y)$ http://math.stackexchange.com/a/842557/160028 I'm not so sure what is the meaning of $\ln(1+y) = y + o(y)$. When is it ...
5
votes
3answers
121 views

Prove $\lim_{n\to\infty} \frac{(2^{2^n}+1)(2^{2^n}+3)(2^{2^n}+5)\cdots (2^{2^n+1}+1)}{(2^{2^n})(2^{2^n}+2)(2^{2^n}+4)\cdots (2^{2^n+1})}=\sqrt{2}$

Question: Prove or disprove $$I=\lim_{n\to\infty} \frac{(2^{2^n}+1)(2^{2^n}+3)(2^{2^n}+5)\cdots (2^{2^n+1}+1)}{(2^{2^n})(2^{2^n}+2)(2^{2^n}+4)\cdots (2^{2^n+1})}=\sqrt{2}$$ I know ...
3
votes
1answer
39 views

Calculating limit in parts. Why possible?

Let $f$, continuous function, differentiable at $x=1$ and $f(1)>0$. Consider the following equation: $$\lim \limits_{x\to 1} ...
2
votes
4answers
125 views

Proving exponential is growing faster than polynomial

Let $P(x)$, a polynomial which isn't the zero-polynomial. I want to prove the following limits $$\lim \limits_{x\to\infty} \left|P(x)\right|e^{-x} = 0$$ $$\lim \limits_{x\to-\infty} ...
2
votes
2answers
93 views

Is $\lim_{x\to 0+} \frac{\ln(x)}{\ln(x)} = \frac{-\infty}{-\infty} = 1$

$$\lim_{x\to 0^+} \sin(x)^\frac{1}{\ln(x)} = ... = \exp \left(\lim_{x\to 0^+} \frac{\ln(\frac{\sin x}{x}) + \ln(x)}{\ln(x)}\right)$$ Now, from continuity we can evaluate each term separately. ...
0
votes
1answer
52 views

How do they integrate this exponential?

Below, I tired to integrate te^(-j2pi*t) from 0 to 1. But am not getting what my professor got for n not equal to zero, which is also shown. I tried LIATE but am always getting something with an ...
1
vote
1answer
29 views

Calculus / exponential

Find values of a and b so that $y = a·b^x$ and the line $y = x + 2$ are tangent at $x = 0$. I tried to substitute with the zero and it seem that the $B=1$ at all time but what about the $a$?
0
votes
2answers
47 views

No real solution to logarithmic equation?

$$e^x + 1 = 2e^{-x}$$ Wolfram Alpha claims no real solution and my text book claims the solution $x=0$. Why can't I simply multiply each side by $e^x$: $$e^{2x} + e^x = 2$$ $$\ln(e^{2x}) + ...
1
vote
4answers
80 views

How to find the integral $\int4^{-x}dx$?

What approach would be ideal in finding the integral $\int4^{-x}dx$?
1
vote
2answers
22 views

Difference between growth formulas

What is the difference between $$N = N_0 \cdot e^{kt}$$ and $$N= N_0(1+r)^n$$ I'm trying to find the best formula to calculate population growth and sources seem to vary between these two?
2
votes
2answers
124 views

First derivative of multiplied powers

Wolfram Alfa shows $\frac{d}{dx}e^{4y} = 4e^{4y}$ but I do not understand how to get to that answer I have $e^{4y} = (e^4)^y$ So by the chain rule is it not the case that \begin{align} ...
8
votes
1answer
249 views

Exponential of a function times derivative

Exponential of a derivative $e^{a\partial}$ is simply a shift operator, i.e. \begin{equation} e^{a\partial}f(x)=f(a+x) \end{equation} This can be easily verified from a Taylor series \begin{equation} ...
3
votes
2answers
72 views

Exponents with the same power

I've wanted to practice solving simple operations on exponents, so I've made a couple of equations to which I know the answers. $$5^x -4^x = 9$$ I feel really stupid, because I can't solve this one ...
0
votes
3answers
65 views

$(5^{2x}-1)(5^x)=1/5^x$ solve

I have the problem $(5^{2x}-1)(5^x) = 1/5^x$. I have already simplified it to $5^{3x}-1=1/5^x$ My question is when I do $\log$ base $5$ to the left side of the equation to get $3x-1$ by itself so ...
0
votes
2answers
114 views

Show that $\displaystyle \lim_{n \rightarrow \infty} \sqrt[n]{a} =1$

Hi guys this was a practice problem I was given, can anyone help me out on it? This is the problem: Show that $\displaystyle \lim_{n \rightarrow \infty} \sqrt[n]{a} =1$ and the following is what I ...
3
votes
2answers
98 views

Limit of $\frac {n^n}{n!}$ [duplicate]

I have to prove that $$\lim_{n\to \infty} \frac {n^n} {n!}=\infty$$ I've tried to look for a lower bound that also converges to $\infty$ (I don't know if I'm explainig myself correctly), but I ...
-1
votes
1answer
40 views

How to solve the integral $\int x\cdot 9\cdot x^{2x^2} dx$

How to solve the integral $$\int x\cdot 9\cdot x^{2x^2} dx$$ I tried $u=2x^2$ and $du= 4x\;dx\Longrightarrow$ $$\int x\cdot9\cdot x^{2x^2}\;dx=\frac94\int x^u du$$But it was unable to pass.
1
vote
3answers
71 views

Explanation for limits equality.

$$\mathop {\lim }\limits_{x \to 0} {\left( {\frac{{{a^x} + {b^x}}}{2}} \right)^{\frac{1}{x}}} = \exp \left( {\mathop {\lim }\limits_{x \to 0} \frac{{\frac{{{a^x} + {b^x}}}{2} - 1}}{x}} \right)$$ I am ...
0
votes
1answer
90 views

Exponential equation+derivative

I saw here on math.stackexchange.com an equation which has very nice solutions (by solutions I mean a proof): $3^x+28^x=8^x+27^x$, where $x$ is a real number. However, I think there must be an ...
2
votes
3answers
101 views

How can we differentiate $(x^{-1})^{({x^{-1})^{x^{-1}}}}$ wrt $x$?

How can we differentiate $(x^{-1})^{({x^{-1})^{x^{-1}}}}$ with respect to $x$?
12
votes
7answers
353 views

integral of $x^2e^{-x^2}~dx$ from $-\infty$ to $+\infty$

I know that the $$\int^{+\infty}_{-\infty}e^{-x^2}~dx$$ is equal to $\sqrt\pi$ It's also very clear that $$\int^{+\infty}_{-\infty}xe^{-x^2}~dx$$ is equal to 0; However, I cannot manage to ...
0
votes
0answers
21 views

I need to find $n$ that $\frac{1}{(n+1) \cdot \ln(n+1)} <10^{-4}$

$\frac{1}{(n+1) \cdot \ln(n+1)} <10^{-4}$ So what I did is this: $(n+1)\ln(n+1) > 1000 \Rightarrow n>190$ When I put it back I see that $\frac{1}{192 \cdot ln(192)} \not < 10^{-4}$. ...
0
votes
2answers
395 views

Help with limit of radical function

$$\lim_{x \to \infty} \frac{\sqrt{4x^{4}+3}}{5x^2+3}$$ $$= \lim_{x \to \infty} \frac{(4x^{4}+3)^{1/2}}{5x^2+3}$$ $$= \lim_{x \to \infty} \frac{(\frac{4x^{4}}{x^{1/2}} +\frac{3}{x^{1/2}})^{1/2} ...
0
votes
0answers
114 views

Integral involving Modified Bessel function, exponential and power

I am trying to evaluate the following integral: $$ \int_{b y}^\infty \frac{e^{-x}(-1+I_0[2\sqrt{bx}]-\sqrt{bx}I_1[2\sqrt{bx}])}{x^2}dx $$ Even though there are a lot of integrals involving the ...
1
vote
0answers
65 views

Double summation of elementary functions

I am finding some trouble on calculating the following double summation: $ \sum_{k=1}^\infty \frac{b^k}{k!k}\sum_{n=0}^{k-1}\frac{(b*x)^n}{(n-1)!} $ Note that the inside sum gives: ...
8
votes
1answer
157 views

Need a general formula for $\frac{d^n}{dx^n}\left(f(x)^m\right)$

Let $m,n\in\mathbb{N}$. I need to express the derivative $\displaystyle\frac{d^n}{dx^n}\left(f(x)^m\right)$ in terms of sums/products of the derivatives of the function $f$ itself. Here are results ...
10
votes
1answer
579 views

What is wrong with this funny proof that 2 = 4 using infinite exponentiation?

Out of boredom, I decided to recall the following equation: $$x^{x^{x\cdots}} = 2.$$ Which, I simply rewrote like this: $x^2 = 2$, and therefore $x = \sqrt{2}$. Then I took a look at the more ...
2
votes
1answer
178 views

Raise a power series to a fractional exponent?

In showing that $\log^\alpha{(1+x)}$ is $O((x)^\alpha)$ at $1$, for $\alpha>0$, one can note that $$\left ( \frac{\log{(1+x)}}{x} \right )^\alpha \overset{x\to 0}{\longrightarrow} \left ( 1\right ...
5
votes
2answers
297 views

Geometric Identities involving $π^2$

Are there any known geometric identities that have $π^2$ in the formula?
1
vote
3answers
137 views

Limit of ${x^{x^x}}$ as $x\to 0^+$

Can you please explain why \begin{align*} \lim_{x\to 0^+}{x^{x^x}} &= 0 \end{align*}
1
vote
2answers
117 views

I just found out that $0^0$ equals $1$, why is this? [duplicate]

I have done a lot of math so far, but I never stumbled on something this simple and yet mind boggling. Can someone tell me why $0^0$ equals $1$? I always knew that everything raised to a power of $0$ ...
1
vote
0answers
107 views

Double summation including power and factorial [duplicate]

I am finding some trouble in computing the following sum: $$\sum_{k=0}^\infty \frac{x^k}{k!}\;\sum_{m=0}^k\frac {y^m}{m!}$$ Could you please provide a result? Thanks in advance
6
votes
5answers
264 views

What does $x^\pi$ mean? [duplicate]

I was just wondering, what does $x^\pi$ or for that matter, $x$ raised to any irrational number mean? For example, I want to represent $x^2$ then that would mean $x * x$ or if I want to do ...
4
votes
1answer
62 views

Simplifying $y=2^{2/3} + 2^{-1/3}$

I am working on a calculus problem where I have to find the local minimum. The value I got was $$y=2^{2/3} + 2^{-1/3}.$$ I simplified it and got this: $$ y=2^{2/3} + \frac{1}{2^{1/3}}$$ ...
2
votes
1answer
123 views

Definite integral including the ratio and power functions of a single variable

I find trouble in calculating the following integral: $$ \int_0^R \frac{m\cdot x}{m+s\cdot x^a} \,dx $$ Mathematica does not provide an output for this function, however, there seems to be an output ...
0
votes
1answer
137 views

For what $ \alpha \in \mathbb R$ is $ |x|^\alpha $ differentiable in $x=0$?

I came across the following question: For what $ \alpha \in \mathbb R$ is $ |x|^\alpha $ differentiable in $x=0$? What I have tried: Since for $ \alpha = 1 $ is clearly non-differentiable in ...
3
votes
2answers
147 views

infinite derivative of $e^x$

i have started thinking about one topic a few days ago and i am confused if i am wrong or what happens,generally we know that function $e^x$ is somehow 'magic',which means that ...
5
votes
5answers
1k views

Why does the power rule work?

If $$f(x)=x^u$$ then the derivative function will always be $$f'(x)=u*x^{u-1}$$ I've been trying to figure out why that makes sense and I can't quite get there. I know it can be proven with limits, ...
0
votes
2answers
148 views

upper bound for $e^{ax^2}$

I want to find a upper bound for $$e^{ax^2}\leqslant \: ?$$ "a" is a constant and $a\geqslant 0$ . x is a variable. I prefer to have a polynomial function or power function (like $ x^{k}$) is there ...
2
votes
5answers
76 views

Proof for power functions

Which is greater? $\sqrt{n}^{\sqrt{n+1}}$ or $\sqrt{n+1}^\sqrt{n}$ I know that $\sqrt{n}^{\sqrt{n+1}}$ is greater but I tried using induction and I couldn't figure it out. Thanks for the help.
1
vote
1answer
105 views

Solving for x with radicals and negative exponents

How do I go about solving for $x$ in this equation? $$\displaystyle -x^{-\large\frac{3}{4}} + \frac{15^{\large\frac{1}{4}}}{15} = 0$$
5
votes
3answers
122 views

Differentiate $y=x^x$

How do you differentiate $$\large{f(x) = x^x}$$ The working I got was $$\ln f(x) = x \ln x$$ which I am pretty fine...but I do not know how it advances to $$\frac{f'(x)}{f(x)} = x\begin{pmatrix} ...
3
votes
1answer
199 views

Summation of powers inequality

Can anyone provide a slick proof of the following? Let $0 < x \le 1$. Then $\displaystyle \sum_{k=0}^{n-1} x^k \ge \frac {1} {1 - (1 - 1/n)x}$.