QUIZ 2 SOLUTIONS
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1. The given series [sum a_n (-2)^n] and [sum a_n 4^n] are values
at x=-2 and x=4 of the power series [sum a_n x^n], which is centered
at 0. Recall that the domain of a power series is a symmetric
interval about its center, hence the radius of convergence is between
2 and 4. Then the series certainly converges for x=1 when its value
is [sum a_n]; hence the answer is (B).
Note that (C) is wrong because the radius could be exactly 2 and
convergence at one endpoint does not imply convergence at the other
endpoint.


2. Two lines are perpendicular when their direction vectors are,
and two vectors are perpendicular when their dot product is zero;
hence (C) is the answer because (1,1,2)*(1,1,-1) = 1+1-2=0.


3. I like this problem a lot. It tests your understanding of what
it means to parametrize a curve. There are two main ways you can
solve this problem.
  (i) notice that the z-value is oscillating with 5 peaks; this
matches the answer in (B) (notice that the x,y part describes
a circle in the x-y plane and for each t-interval of length 2*pi,
the x-y part goes in one complete circle and the z part goes up
and down 5 times).
  (ii) notice that in the picture, all three coordinates are bounded; 
this means that each of the functions x(t), y(t), z(t) must have bounded
range, and neither y(t)=t (in (A)) nor z(t)=ln(t) (in (C)) satisfies
this requirement.

comments: what i am doing is using the projection trick we have
discussed in class. for another example, look at (A): if this were 
the right equation, then the projection of the curve onto the x-z plane 
would have to be a circle since (x,z) = (cos(4t), sin(4t)) (the 4 just 
increases the "speed" by a factor of 4). We can easily see from the picture
that this is not the case. So there are many ways to rule out
wrong answers using projections.