Tagged Questions

2
votes
1answer
71 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 ...
4
votes
5answers
341 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, ...
2
votes
5answers
68 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.
3
votes
1answer
63 views

Summation of powers inequality

Can anyone provide a slick proof of the following? Let $0 < x \le 1$. Then $\displaystyle \sum_{k=0}^{n-1} x^k \ge \frac {1} {1 - (1 - 1/n)x}$.
3
votes
1answer
294 views

Comparing Powers with Different Bases Using Logarithms?

I looked all over to see if a question like this had already been answered, but I couldn't find it. So here goes: I need a general formula for comparing two (insanely huge) powers. I'm pretty sure ...
4
votes
3answers
273 views

Power function inequality

Let $x$ and $p$ be real numbers with $x \ge 1$ and $p \ge 2$ . Show that $(x - 1)(x + 1)^{p - 1} \ge x^p - 1$ . I recently discovered this result. I am sure it is known, but it is new to me. It is ...
16
votes
2answers
253 views

$2^x - a$ touches $\log_2(x)$

I was playing around with the functions $2^x$ and $\log_2(x)$. As they are the inversions of each other, I thought there was a simple number $a$ for which $2^x - a$ touches $\log_2(x)$. Using ...