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In the interest of enhanced buffering capacity, what is the best way to prepare an acidic buffer? [HINT: Read the each answer in its entirety]

Consider the function on R2 (R is the set of Real numbers) u(x,y)=min{x2y,xy}. What is the maximizer of f in the set {(x,y): x+5y=10}? (Hint: consider a vector in the hyperplane (1 5).(x,y)=10; when can this vector be a maximizer of u if x2y>xy? What if x2y

Given a function g:X->R (R is the set of Real numbers), the binary relation associated with g, is defined as follows: {(x,y):g(x) ≥ g(y)}. Let f:R->R be the function f(x)=x2. Which of the following functions has the same associated binary relation as f?

Let f:Rn->R (here R is the set of Real numbers) be a continuous functions. Let x in Rn and consider the set A={y in Rn: f(y) ≥ f(x)}. Then,

Consider the function u(x1,x2)=min{x12,x2}. Let p=(1 2) and w=1. Among the vectors {x≥0:p.x≤1}, which one maximizes u?

For each n in the Natural numbers, let An=[1/n,2/n]. Then the intersection of this collection is?

Let f:R2->R2x2 defined by (x,y)|->f(x,y)=[x x+y2; y xy] (here x x+y2; is the first row of f(x,y), and y xy the second). Let v=(2,1) be a vector in R2. Then:

Which of the following sets is compact?

For each Natural number, n, let Bn=(-2n,4n). Then the union of the family?

Your instructor has hired you to work as an assistant in the lab.  For the upcoming lab experiment you are asked to prepare a 0.05000 M carbonic acid solution from a 12.00 molar “stock solution”.  The lab will contain 15 students and each will need 100 mL of the “diluted” carbonic acid solution.  Therefore, you should prepare a little extra, namely 2.000 liters.  Briefly (2 to 3 sentences) explain how you would prepare the “diluted” solution.  Of course, I will be looking for a particular value in your explanation.  [Note, you can find an example problem in the PowerPoint slides]