MATH 251.504
Examination 1
February 23, 2011
NAME
SIGNATURE
This exam consists of 7 problems, numbered 1–7. For partial credit you must present your
work clearly and understandably and justify your answers.
The use of calculators is not permitted on this exam.
The point value for each question is shown next to each question.
CHECK THIS EXAMINATION BOOKLET BEFORE
YOU START. THERE SHOULD BE 7 PROBLEMS ON
5 PAGES (INCLUDING THIS ONE).
Do not mark in the box below.
1–4
5
6
7
Total
Points
Possible Credit
24
15
12
24
75
NAME
MATH 251
Examination 1
Page 2
Multiple Choice: [6 points each] In each of Problems 1–4, circle the best answer.
1.
Let L be the line given by parametric equations x = 3 + 2t, y = −2 + 5t, and
z = −1 − 4t. What is the equation of the plane that contains the point (3, −1, 5) and
is perpendicular to L?
(A) 2(x − 3) + 5(y + 1) − 4(z − 5) = 0
(B) 2(x − 3) + 5(y + 2) − 4(z + 4) = 0
(C) 3(x − 3) − 2(y + 1) − (z − 5) = 0
(D) 3(x − 3) − 2(y + 2) − (z + 4) = 0
(E) 3(x − 3) − (y + 2) + 5(z + 4) = 0
2.
Let f (x, y) = 2 cos(xy 2 − 3y). What is fxy ?
(A) −2y 4 cos(xy 2 − 3y)
(B) −4y(2xy − 3) cos(xy 2 − 3y)
(C) −4x sin(xy 2 − 3y) − 2(2xy − 3)2 cos(xy 2 − 3y)
(D) −4y sin(xy 2 − 3y) − 2y 2 (2xy − 3) cos(xy 2 − 3y)
(E) −4x(2xy − 3) sin(xy 2 − 3y) − 2y 2 cos(xy 2 − 3y)
February 23, 2011
NAME
3.
MATH 251
Examination 1
Page 3
Consider the curve in 3-space defined by the vector function r(t) = h3 sin(t), 4 cos(t), 4ti.
What is the unit tangent vector T to this curve when t = 0?
(A) h0, −4, 4i
(B) h3, −4, 4i
3
4
(C)
, 0,
5
5
3
4
4
(D) √ , − √ , √
41
41 41
4
4
(E) 0, − √ , √
32 32
4.
A curve in 3-space is defined by a differentiable vector function r(t). Let T, N, and
B be the unit tangent vector, the unit normal vector, and the binormal vector at
some point P0 on the curve, with P0 corresponding to t = t0 . Consider the following
statements:
I. N is perpendicular to B.
II. T is parallel to r0 (t0 ).
III. B is perpendicular to the osculating plane at P0 .
Which of these statements is true?
(A) I only
(B) II only
(C) I and II only
(D) II and III only
(E) I, II and III
February 23, 2011
NAME
5.
MATH 251
Examination 1
Page 4
[(a) 10 points; (b) 5 points] Let P = (3, 1, 1), Q = (1, 1, −1), and R = (1, 5, −1). Let
−→
−→
a = QP and b = QR.
(a) Find a × b.
(b) Is the triangle 4P QR a right triangle? Why or why not?
6.
[12 points] Suppose f (x, y) is a differentiable function such that
fx = 3yexy ,
Let
fy = 3xexy .
3u
g(t, u) = f tu ,
.
t
Use the multivariable Chain Rule to find
2
∂g
∂g
and
in terms of t and u.
∂t
∂u
February 23, 2011
NAME
7.
MATH 251
Examination 1
Page 5
[(a) 3 points; (b) & (c) 8 points each; (d) 5 points] Let
2
f (x, y) = 5 − x − 3y
2
and
E
8
r(t) = 8t − 6, 6t − 5, −
+2 .
t+1
D
2
(a) The graph of z = f (x, y) is what kind of quadric surface?
(b) Let P0 = (2, 1, −2). Find the equation of the tangent plane to the graph of
z = f (x, y) at P0 .
(c) The curve defined by r(t) also contains P0 . Find a tangent vector to this curve
at P0 .
(d) Is the tangent vector from (c) perpendicular to the tangent plane from (b)? Why
or why not?
February 23, 2011