Alexander

You have a linear function [math]f(x)=-\frac{2}{3}x+4[/math] where [math]0\leq x\leq 6[/math] and [math]0\leq y\leq 4[/math]. What is the maximum area of a rectangle that has one side on the line of the function? I know how to optimize this, I am just having trouble finding the equation for area of such a rectangle in terms of the function. I have included an image of what I am talking about. Thanks a lot.

I'm not sure, that is all the information that was provided. Someone else plugged it into Mathematica and got [math]\frac{3}{2}[/math]. They used the inequality [math]\int_{1}^{2^n}\sqrt{x} dx < \sum_{k=1}^{2^n}\sqrt{k} < 1+\int_{1}^{2^n}\sqrt{x} dx[/math]. I came across these problems on the internet. I just wanted to know if I'm missing something (being in Calculus) and not knowing these.

Dual-Enrollment. My high school only offers through AP Calculus so I have to take my math classes at a local community college. Thanks for your help everyone.

[math]\displaystyle \lim_{n\rightarrow 0}\frac{\log_{2}\sum_{k=1}^{2^n}\sqrt{k}}{n}[/math] [math]\displaystyle \lim_{n\rightarrow 0}\frac{\log_{2}\sum_{k=1}^{2^n}\sqrt{2k-1}}{n}[/math] I have had people give me the results after plugging it into Mathematica/Maple, but I was wondering if anyone knew how to solve these without a computer.

You know, I'm in Calculus II also. This is the first time I have seen anyone use the concept of modulo. I am a junior in high school now, and I would expect that questions like these would appear on AIME/USAMO tests. Where/when in the sequence of math courses does someone usually encounter the uses of modulo (and these types of problems)? Would this be an analytic number theory course?