MATH261 EXAM I FALL 2013
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
20
2
25
3
25
4
10
5
12
6
8
Total
100
Score
1. Consider the three points P (3, −2, 3), Q(1, 3, 4), and R(3, −1, 2).
(a) (8pts) Find an equation of the plane containing the points and write it in the
form ax + by + cz = d.
(b) (6pts) Find the area of the triangle with vertices P, Q and R.
(c) (6pts) Find a vector equation of the line perpindicular to the plane and containing
the point S(3, −2, 1).
2. (25pts, 5pts each part) given the equations of the planes
P lane 1 :
P lane 2 :
P lane 3 :
2x − 3y + z = 1
5x + 3y − z = 2
4x − 6y + 2z = 2
and the equation of the lines
Line 1 :
Line 2 :
r1 (t) = ht, −t, 1 − 5ti
r2 (s) = h1 + s, −5 + 3s, 1 − 10si
SHOW WORK to conclude:
(a) True or False: The planes P1 and P3 never intersect.
(b) True or False: The planes P1 and P2 are perpindicular along the line of intersecton.
(c) True or False: The line r1 (t) lies on plane P1
(d) True or False: The line r1 (t) intersects P1 exactly once.
(e) True or False: The lines r1 (t) and r2 (s) intersect.
3. (25pts) Given r(t) = sin(t)i + 21 tan(t)j + cos(t)k find (a)-(e). Note: To get full credit
you must know the values of certain trigonometric functions at π/4.
(a) (6pts) Find v(π/4).
(b) (6pts) Find a(π/4).
(c) (6pts) Find T(π/4).
(d) (4pts) Find aT (π/4).
(e) (3pts) Find aN (π/4).
4. (10pts) Find v(t) given a(t) = h
−1 3t−3
1
i with initial condition
,e
,√
2
t
2t − 1
v(1) = h1, 1, 2i.
5. (12pts; 6pts each part) Find the limit if it exists (showing your calculations). If the
indicated limit does not exist, then written arguments, with supporting calculations,
must be given as reasons for your answer.
cos(x2 − y)
(a) lim
(x,y)→(0,0)
x2 − y
x2 − y
(b) lim
(x,y)→(0,0) 2x + y
6. (8pts, 2pts each problem) Next to each equation write the letter corresponding to its
graph.
(I) x2 +
y2 z2
+
=1
16
9
(III) y 2 − 1 = x2 + z 2
(II) y = x2 − z 2
(IV) 9x2 + y 2 = 4
(A)
(B)
(C)
(D)
(E)
(F)
(G)
(H)