PRACTICE QUIZ #1 for math127  Warren D. Smith    12 Feb 2003
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Real quiz is Tues Feb 18.
Should be similar coverage and difficulty level to this.

Several remarks about my teaching, students, etc.:
A. The students seem now to be roughly in 2 groups, getting 45-65
out of 70 on the HW and 10-30 out of 70, and the latter is a big 
danger sign for me.  I am worried the ones in the low group
probably in trouble and should come to me for extra help, send me
emailed questions, whatever. Or maybe they have inadequate prerequisites
for this course. I don't know.

B. There are worries I am going too fast thru the syllabus. That
criticism may be correct. This is the first time I am teaching this
course so I feel I need to start out somewhat fast to provide a 
safety margin at the end.  (If end without covering stuff, that
worse than: end having got done early, then do review.  I think.
Maybe I'm wrong.) Also, the syllabus can be a bit deceptive since the 
"2nd half" of the course (according to syllabus) contains
(what probably will be) harder and newer material. 
Hence it has to be covered slower than the 1st half.
Well, we'll see.  If things go downhill I will have to readjust my
approach...

C. Also, note, I think HWs and related reading should consume time
comparable to in-class time, per week.

D. Finally, I do want to cover material before I give HW on it, which 
I have mostly but not totally succeeded at.  But the more advanced
the math course, the less things can or will be of the form
"here is a formula, here is an example of me putting numbers in;
now you put other numbers in, your job is over."  Instead,
the goal is that you need to view formulas as tools and recipes
to use to derive your own formulae, and you need more and more
original thinking and less and less plug and chug.  Nevertheless,
in this course there still is a substantial amount that is plug and chug.

OK, on with the practice quiz. NOTE I have made a REVIEW OF MATERIAL
web subpage of the math127 page, describing what quiz covers and a
quick review of it! Helpful!

-------------------------------------THE PRACTICE QUIZ----

Find  (1,2,5) X (8,-2,1).
What is the area of the parallelogram these two vectors determine?

Find  (1,2,5) . (8,-2,1).
What is the angle between these two vectors (write as a simplified formula
involving arcsin or arccos, but do not need to then use calculator to deduce
approximate numerical value of arcsin(3/5) or whatever).

The plane  1x+2y+5z=1  and the line  (x,y,z) = (8t, -2t, t+7)
make what angle?  (write as a simplified formula
involving arcsin or arccos, but do not need to then use calculator to deduce
approximate numerical value of arcsin(3/5) or whatever).

If A,B,C are vectors, simplify the formula
   (A X B) . (5A + 7B + 2C) 
as much as possible.

Is there something wrong with this formula-fragment? What?  
5 (A.B) X C - D     (where A,B,C,D vectors and X means cross product,
and . means dot product)

Consider the curve (x,y,z)=F(t) where F(t)=(cos(t), sin(t), 7t+2)
and t is time.
What is the velocity vector at time t?
What is the arc length from t=0 to t=7?

Consider the function
                     2           2
F(x,y,z) = sin(x) + y  + yz + 3 z  .

What is  grad F?
What is the unit-normalized version of the vector (2,1,1)?
What is  F's directional derivative in that (unit) direction?
Find conditions obeyed by a local minimum of F(x,y,z).
Solve those conditions to actually find the location x,y,z of
some local minimum of F(x,y,z).

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