Math 75  HOMEWORK #1     DUE  Monday 9 Sept    Prof. WARREN D. SMITH
   Show your work.     (No COW problems this time, I'm taking pity on you.)
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0. FIND OUT how to (and do it!) browse the web and reach the course
   web page from
       http://math.temple.edu/~wds
   All kinds of important info on this page!  Also take a look at the 
   Prerequisites/Background Knowledge page (1 mouseclick away) and try
   to make sure you know that stuff.  If not, get help from me or the
   T.A. on whatever you don't know.
   (This one is not a homework problem. It is a demand. :)

1. The temperature in Fahrenheit is a linear function of
    the temperature in Celsius.
       Water boils:    100 Celsius     212 Fahrenheit
       Ice melts:        0 Celsius      32 Fahrenheit

  (a) Derive a formula saying what C is in terms of F.
  (b) Derive a formula saying what F is in terms of C.
  (c) At what temperature do both a Fahrenheit and a Celsius
  thermometer read the same?
  (d) plot a graph of F as a function of C.
  (e) What is the SLOPE of this graph?

2. Old myth: the king asks the wise man how much to pay him for his services.
WM: "put 1 grain of rice on the first square of a chessboard, then 2
on the next, then 4 on the next, then 8 on the next, and so on (keep
doubling every time)."  (A chessboard has 64 squares.)
K: "sure, I'll pay you that, no problem."
  (a) How many grains of rice total? Write as an exact formula.
(Hint. Can you find a formula for the answer when there are N squares,
where here N=64?  Your formula will be very short. Hint. Geometric
series or induction.)
  (b) Express that number (to 1% accuracy) in scientific notation.
  (c) Would this much rice fit inside a sphere the size of our planet?

3. What is the area of a triangle with legs 5 & 8, and angle 20
   degrees?  (Write as a formula, then compute to 1% error.)

                 2
4. (a) Plot y = x  + 2  .

                  1
   (b) Plot y = -------  
                 2
                x  + 2

                   1
   (c) Plot y = --------   (can be on same plot; 
                 2            if so label your curves a,b,c.
                x  - 2        Do: the x's with  -5 < x < 5, at least. )

What are the domains and the ranges of these functions?  Which of
these appears to be concave-up (if any)?

5.  You have 1000 dollars. Each year you lose 30% of whatever you had
at the start of that year (bad investments). What function of N says your
wealth after N of these losing years?  What value is it when N=5?  
Now after 5 of these losing years you start earning, each year, 30% 
of whatever you had at the start of that year. What is your wealth 
after 5 winning years? (As formula, and as number).  Hint: the 
correct answer is NOT 1000 dollars back again.  (Important to
understand this before you invest your money!)

6. The total area of a square with sidelength S, plus a rectangle
of sides S and 1, is 7.  What is S (as a formula, then to 1%
accuracy)?  (Hint. quadratic equation.)