MATH261 EXAM I FALL 2014
NAME:
CSU ID:
SECTION NUMBER:
You may NOT use calculators or
any references. Show work to receive full credit. Circle the answer
for each problem.
GOOD LUCK !!!
Problem Points
1
21
2
18
3
24
4
10
5
15
6
12
Total
100
Score
1. (a) (15 pts; 5 pts each part) Given the plane P: 3x−4y +2z = 4, classify the following
planes as either perpendicular to P, parallel to P, or neither.
(i) 6x − 8y + 4z = 9
(ii) 4x + 2y − 2z = 1
(iii) 3x − 4y + z = 3
(b) (6 pts) With respect to the plane P and the plane Q: 2x + y + 5z = 0, using a
parametric representation, define the line of intersection.
2. (a) (6 pts) In the form Ax + By + Cz = D, give an equation for a plane passing
through P(1, 1, 6), Q(0, 1, 4), and R(6, 0, 5).
(b) (6 pts) Compute the distance from the point (1, 4, 1) to the plane in part (a).
(c) (6 pts) Find the point of intersection of the line r(t) = h2 − 3t, −1 + 2t, −2ti and
the plane found in (a).
3. (24 pts) Consider r(t) = t2 i+cos(t)j+sin(t)k. Using vector notation, find the following:
(a) (10 pts) v(t), |v(t)|(best to simplify), and a(t).
(b) (9 pts) T(t) and aT when t =
(c) (3 pts) aN when t =
π
4
π
4
(d) (2 pts) N(t) when t =
π
4
4. (10 pts; 5 pts each part) Given a(t) = (6t3 + t)i + (π cos(2πt) + 1)j + π cos( π2 t) + et k,
complete the following:
(a) If v(0) = h−1, 2, 7i, find v(t).
(b) Find the equation of the tangent line at t = 0 assuming r(0) = 2i + 4j − 7k
5. (15 pts; 5 pts each part) Find the limit if it exists. If it does not exist then show why
it does not.
x2 + 4y
sin t √
x2 − 4y 2
, 1 − cos ti (b) lim
(c) lim
(a) lim ht + 1,
t→0
(x,y)→(0,0) x + 2y
(x,y)→(0,0) x + 2y
2t
6. (12 pts; 3 pts each part. No partial credit given.) Next to each equation write the
letter corresponding to its graph.
(I) z 2 = x2 + y 2 + 1
(III) 2x + 2y + z = 4
(II) y = z 2 − x2
(IV) x = y 2 + z 2
(A)
(B)
(C)
(D)
(E)
(F)
(G)
(H)