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]

A colleague in your lab has a bottle of sulfuric acid but the molar concentration (M) is smudged (not legible).  Using a 2.000 M potassium hydroxide solution and you noticed that 58.56 mL was required to neutralize a 50.00 mL sample of your colleague’s sulfuric acid of unknown concentration.  Help your colleague determine the concentration of the sulfuric acid solution.  Your response should briefly (2 to 3 sentences) describe the steps you took and, of course, include the calculated concentration of the sulfuric acid solution. 

Your supervising partner proposes that any commission received on arranging the life policies should be offset against the client’s bill. Can this be done?