Download MATH261 EXAM I SPRING 2014 NAME: CSU ID: SECTION NUMBER:

MATH261 EXAM I SPRING 2014
NAME:
CSU ID:
Problem Points
1
18
2
24
3
20
4
18
5
12
6
8
Total
100
SECTION NUMBER:
You may NOT use calculators or
any references. Show work to receive full credit.
GOOD LUCK !!!
Score
1. (a) (6 pts) Find an equation of the plane through the three points P = (−1, 1, 0), Q =
(2, 0, 1), R = (4, 1, 0). Write your final answer in the form ax + by + cz = d.
(b) (6 pts) Find the distance from the point P (−1, 4, 0) to the line r(t) = h1, 2, 0i +
th4, 3, −1i.
(c) (6 pts) Find a vector equation for the line of intersection of the planes x−2y+z = 1
and 4x + y − 2z = 0.
2. (24 pts, 6 pts each problem) Given the equations of the planes
P lane 1 :
x − 2y + z = 1
P lane 2 :
2x − 4y + 2z = 2
P lane 3 :
x + 2y + z = 4
and the equations of the lines
Line 1 :
r1 (t) = h1 + 2t, 2t, 2ti
Line 2 :
r2 (s) = h1 + 2s, 1, si
SHOW WORK to conclude:
(a) True or False: Planes 1 and 2 never intersect.
(b) True or False: Planes 1 and 3 intersect in a line.
(c) True or False: The point (1,0,0) is common to both Plane 1 and Line 1.
(d) True or False: The point (1,1,0) is common to both Plane 3 and Line 2
3. Given
r(t0 ) = πi + 3ej − 100k
−4
3
v(t0 ) =
i+ j+k
5√
5√
√
a(t0 ) = 2 2i − 2 2j + 2 2k
complete the following
(a) (12 pts) Find T(t0 ) and aT (t0 ).
(b) (5 pts) Find |a(t0 )| and aN (t0 ).
(c) (3 pts) Find N(t0 ). Do Not Simplify.
4. Given that a(t) = h
1
, e1−t , cos ti, v(0) = h2, e, 1i, r(0) = h3, 2e, 0i, complete the
(t + 1)2
following.
(a) (12 pts) Find v(t).
(b) (6 pts) Find r(t).
5. Find the limit if it exists. If it!does not exist then show why it√does not.
1 + x + 2y
x− y+1
(a)
lim
ln
(b)
lim
(x,y)→(2,−1)
(x,y)→(2,3) x2 − y − 1
3y 2 − x
3
x +y
(c) lim
(x,y)→(0,0)
y
6. (8 pts; 2 pts each part. No partial credit given.) Next to each equation write the letter
corresponding to its graph.
(I) z 2 − x2 = y 2
(II) z 2 = x2 + y 2 − 4
(III) y 2 + z 2 = 4
(IV) x2 +
(A)
y2
16
+
z2
16
(B)
=1
(C)
(D)
(E)
(F)
(G)
(H)