MATH 261 EXAM I FALL 2015
NAME:
CSU ID:
Problem Points
1
12
2
18
3
18
4
16
5
18
6
18
Total
100
SECTION NUMBER:
You may NOT use calculators or
any references. Show work to receive full credit.
GOOD LUCK !!!
Score
1. (12 pts; 3 pts per part) Let
u = h2, 1, 3i, v = h1, −1, −1i, w = h1, 0, −1i, z = h3, 4i.
Compute each of the following or write “invalid operation.”
(a) u + v
(b) w · z
(c) 7 + u
z
(d)
|z|
2. (18 pts; 6 pts per part) Consider the line L given by
L : r(t) = h3t + 1, t + 2, 2t − 1i
and the two planes P1 and P2 given by
P1 : x + y − z = 6
P2 : 2x − 2y + 2z = 2.
(a) Does the line L intersect the plane P1 ? If so, find the point of intersection. If not,
explain why.
(b) Find the angle between P1 and P2 . You should leave your answer as the arccosine
of some number.
(c) Find the distance between the origin and the line L.
3. (18 pts; 6 pts per part) For parts (a) and (b), consider the two planes P3 and P4 given
by
P3 : x + y + z = 1
P4 : 3x + y − z = 5.
(a) Find a vector parallel to the line of intersection of planes P3 and P4 .
(b) Find a point on the line of intersection of planes P3 and P4 .
(c) Given point (2, 4, 6) on the line L and vector h1, 3, 5i parallel to L, write parametric equations for the line.
4. (16 pts; 8 pts per part) Given a(t) = h12t2 + 2, 6t − cos(t), et i, v(0) = h1, 0, −5i, and
r(0) = h3, −3, 0i, find the following:
(a) Find v(t).
(b) Find r(t).
5. (18 pts; 6 pts per part) The three parts of this problem are independent. In other
words, these are three distinct problems that do not rely on each other.
(a) Suppose r(t) = (3 cos t)i + 4tj + (3 sin t)k. Compute v(t) and T(t).
3
4
, 0,
(b) Suppose that a = h2, 1, 1i and T =
at some value of t. Compute
5
5
aT and aN at that value of t.
π
π
π
π
(c) Find N
, given that a
= h2, 4, −2i, T
= h1, 0, 0i, aT
= 2, and
2
2
2
2
π
= −2.
aN
2
6. (18 pts; 9 pts per part) Evaluate the limit or show that the limit does not exist. Proper
work must be shown.
(a)
(b)
lim
(x,y)→(0, π2 )
sin(y) cos(y)
yex
x2 y 2
(x,y)→(0,0) x4 + 3y 4
lim