COALITION MATH 152, SPRING 1998

              (S. A. Fulling, assisted by Vera Rice)

                             Day 16.T


    1.  Restructuring:  Merger and downsizing
THIS WORKED FAIRLY WELL -- CONSIDERING THAT MOST OF THE STUDENTS 
FROM ONE OF THE OLD GROUPS DIDN'T SHOW UP TILL THE END OF THE 
PERIOD.  THE PICTURES TURNED OUT TO BE ESSENTIAL TO ENABLE THE 
TWO HALVES OF THE TEAMS TO FIND EACH OTHER!

        A.  Form temporary teams by merging parallel teams from 
        the two groups and making minor adjustments.  (Review the 
        jigsaw concept.)

        B.  Return final exams, projects, etc.  Exam will be 
        discussed in recitation tomorrow.  THEN YOU ARE EXPECTED 
        TO RETURN (making copies for yourself if you want):

            1.  Final exam

            2.  Project report packet, unless you were in Group 
            II and the report is labeled "OK".  (Your critique 
            of another team's project doesn't need to be 
            returned.) 

            3.  Team pictures

        C.  Announce E-mail addresses, Web page, tentative office 
        hours, recitation and help session plans.  Homework will 
        be due on Tuesdays:  Assignments 16.3, 17.1, and 17.2 on 
        day 18.T, etc. 

    2.  Three-dimensional vectors, determinants, and the vector 
    cross product:  Read Sec. 11.4, expect RAT.

    3.  Optimization (applied max-min word problems)

        A.  Example  

            i.  Your dream house will be built on a rectangular 
            lot.  Along the side facing the street you will have 
            a stone wall which will cost $10 per foot 
            (horizontal).  The other 3 sides will be enclosed by 
            a steel fence costing $5 per foot.  You have $2500 to 
            spend on wall and fence together.  Find the 
            dimensions of the lot with maximum area consistent 
            with your plans.

           ii.  Now suppose your budget is flexible and you want 
            exactly 10,000 square feet.  Find the dimensions of 
            the lot with minimal wall-fence cost.

        B.  Strategy

            i.  Read the problem carefully.  Understand

                a.  What QUANTITY is TO BE EXTREMIZED?  (Let's 
                call it Q.)

                b.  What other quantities can vary?  What 
                quantities are fixed?

           ii.  Introduce NOTATION.  Draw a DIAGRAM if 
            appropriate.

          iii.  Write down the RELATIONS among the variables.  
            (They may be given in the problem, or deducible from 
            general knowledge.)  You need

                a.  an OBJECTIVE FUNCTION expressing Q in terms 
                of other variables.

                b.  CONSTRAINT EQUATIONS relating the other 
                variables so that you can write Q as a function 
                of JUST ONE independent variable.  (Let's call 
                that one x.)

           iv.  SOLVE THE CONSTRAINTS and substitute the results 
            into the objective function.

                a.  DON'T differentiate the constraints.

                b.  DON'T differentiate Q UNTIL you have 
                eliminated all variables but x.

            v.  Diffferentiate Q(x) and find its CRITICAL POINTS.

           vi.  Verify that your favorite critical point is the 
           correct extremum.

                a.  Is it a max or a min?  Or neither?  How?

                    (1) second-derivative test

                    (2) comparison of values (cf. b, c)

                    (3) first-derivative test

                b.  Is it in the physically allowed interval?

                c.  Remember to check the endpoints.

(TIME DID NOT PERMIT.)
        C.  Second example if time permits:  A service station 
        owner knows that if he sets the price of gasoline at x 
        dollars per gallon, he will sell  1000(3 - 2x)  gallons 
        of gas tomorrow.  The maximum price permitted by law is 
        one dollar per gallon. 

            1.  Write down the function, y = f(x), that gives the 
            TOTAL INCOME (cash taken in by the station) if the 
            price is x.

            2.  Find the legal price that will maximize the 
            income.

            3.  Resolve the problem when the "inverse demand 
            function" is changed to  1000(3 - x).

        D.  Prepare for jigsaw:  Multiply your temporary team 
        number by 2.  Solve and be prepared to explain on 
        Thursday the corresponding problem from Sec. 3.8. 
PROMISED MORE TIME FOR TEMPORARY TEAMS TO PREPARE ON THURSDAY.