Math 127(3) - Warren D. Smith - Homework #3 - due Tues 11 Feb 2003
------------------------------------------------------------------
Finish reading chapters 12 and 13. Don't need to read the 13 stuff on
planetary orbits, at least not yet. Start 14.

1. Do exercises 8, 12, 14, 40
of section 13.1 ("vector functions")


2. Do exercises 4, 8  of section 13.2 ("differentiation formulas")


3. Consider the plane curve whose curvature (=1/RadiusOfCurvature)
c is, as a function of arc length s along the curve, c(s)=s.
Assume this curve starts out at the origin x=y=0 moving in the
x-axis direction.
Write this curve in parametric form (that is, x,y are functions,
which you give in closed form, of some other variable t; also
you can, if desired, just write y as a function of x).  

HINT/WARNING: The parameterized curve will end up as a formula involving
certain integrals.  You should at least reach that point.
Actually doing these integrals is not so easy.
But if you consult good tables of integrals, or Abramowitz and
Stegun "handbook of mathematical functions" (also available online 
hyperlinked to my math127 web page!) you will be able to do it. Hint:
look under "Fresnel integrals".  If you do this you will get extra
credit.  This is anyway a good thing for you to do to get used to the
mechanics of using tables of integrals.

4. A thrown object describes a parabolic curve of the form
                                 2
     x = A t        y = B t - C t
where t is time since throw it, and A,B,C are positive constants.
Find the arc length of this curve, i.e. the total distance the object
travels, as a function of t.  (Should be 0 when t=0 and should be
positive when t>0.)  If you cannot do the integral leave it
unevaluated and get less credit, but you should be able to do the
integral with some work to get a closed form in terms of A,B,C, and
t...

HINT/WARNING: Again the easiest way is to look in a table of
integrals... again I will give extra credit to those who accomplish that.

---------

FOR THOSE WITHOUT BOOK... THESE ARE AGAIN:

13.1 8:
find t-derivative of this vector-valued fn of t: 
  ( (t+1)/(t-1),  t exp(2t),  sec(t) )


12 find second t-derivative of
    1/2     3/2
 ( t     , t    , ln(t) )

14 integrate from t=0 to pi, dt: (sin t,   cos t,  t)

40: find a parameterized curve (vector-valued fn of t) describing the
curve
        2     2
   (x-1)   + y   = 1
and going (a) clockwise (b) anticlockwise.


13.2 4: find the 1st and 2nd derivative with respect to t of this
vector-valued fn  (the . means dot product)
    2                 2
[ (t , -1, 0) . (1, -t , 0) ] (1,0,0)

8: and this one  (the X means vector cross product):
      2             3
(t, -t ,  1) X (1, t , 5t)