Name:
APPM 2350 FINAL Exam Summer 2008
Be sure to include your name and a grading table on the front of your blue book. Show
ALL of your work and BOX IN YOUR FINAL ANSWERS. A correct answer with no
relevant work may receive no credit, a wrong answer with no work will receive no credit,
and an incorrect answer accompanied by some correct work may receive partial credit. Text
books, class notes, crib sheets, cell phones, calculators, or electronic devices of any kind are
NOT permitted. Please start of each new problem on a new page. Good luck! There are
three sections on this exam. Please read each section carefully or you will end
up doing way more work than you need to. Note that this exam is worth 150 points.
SECTION A: WORK QUESTIONS 1-3
1. (20 points) Find the flow along the curve given by r(t) =<2 cos(t),2t, −3 sin(t)>with
0≤t≤πin the vector field ~
F(x, y, z) =<20x3z+ 2y2,4xy , 5x4+ 3z2>.
2. (20 points) Find the flux of ~
F(x, y, z) =< x + cos(y), y + sin(z), z +ex>through the
closed surface bounded by z= 0, y= 0, y= 2 and z= 1 −x2.
3. (20 points) A liquid is swirling around in a cylindrical container of radius 2 oriented
along the z-axis and bounded by the planes z= 0 and z= 10. The fluid motion is
described by the velocity field ~
F(x, y, z) = −ypx2+y2ˆ
i+xpx2+y2ˆ
j. Find the
circulation around the curve defining the intersection of the top of the cylindrical
container and its rounded sides (i.e. upper surface of the cylindrical container).
SECTION B: WORK 4 OF THE NEXT 6 QUESTIONS
4. (15 points) Two unrelated questions.
(a) If ~v =<1,3,−1>and ~w =<−1,2,−2>, find the vector projection of ~v onto ~w.
(b) Find the distance between the parallel planes 3x−y−2z= 6 and 3x−y−2z=−2.
5. (15 points) Consider the surface given by z= 2x2+ 6y2.
(a) Find an equation for the tangent plane at the point (1,−1,8).
(b) Find an equation for the normal line at the point (1,−1,8).
