Muddiest Point 11/16/17
Upper and Lower Sums
We illustrated lower sums in class to show that, given a function 
![[a,b]](https://s0.wp.com/latex.php?latex=%5Ba%2Cb%5D&bg=ffffff&fg=000&s=0&c=20201002)
![L(f,[a,b],Q)\geq L(f,[a,b],P)](https://s0.wp.com/latex.php?latex=L%28f%2C%5Ba%2Cb%5D%2CQ%29%5Cgeq+L%28f%2C%5Ba%2Cb%5D%2CP%29&bg=ffffff&fg=000&s=0&c=20201002)
These are our observations about the areas in the upper sum and lower sum:
We illustrated lower sums in class to show that, given a function
These are our observations about the areas in the upper sum and lower sum:
![U(f,[a,b],Q)\leq U(f,[a,b],P)](https://s0.wp.com/latex.php?latex=U%28f%2C%5Ba%2Cb%5D%2CQ%29%5Cleq+U%28f%2C%5Ba%2Cb%5D%2CP%29&bg=ffffff&fg=000&s=0&c=20201002)
![L(f[a,b],P)\leq U(f,[a,b],P)](https://s0.wp.com/latex.php?latex=L%28f%5Ba%2Cb%5D%2CP%29%5Cleq+U%28f%2C%5Ba%2Cb%5D%2CP%29&bg=ffffff&fg=000&s=0&c=20201002)
Thank you for the post, though I will admit I am a little confused about what feature you are identifying as the “muddiest point”…