MATH 228: Calculus III
SAMPLE MIDTERM I
Fall 2015
NAME : SOLUTIONS (Median: 77)
NOTE : There are 5 problems on this midterm (total of 6 pages). Use of
calculators will NOT be permitted. In order to receive full credit for any
problem, you must show work leading to your answer. You have 50 minutes
to complete this test.
Problem Possible
points
1
20
2
20
3
20
4
20
5
20
Total
100
Score
MATH 228 – MIDTERM I
Page 2
Problem 1. (20pts) Find the equation of the plane through (1, 1, −1),
(2, 0, 2) and (0, −2, 1).
Label the points A,B and C, respectively, so that
−→
AB = h1, −1, 3i
−−→
CB = h2, 2, −1i
so that the normal vector to the plane is
i j k
−→ −−→ AB × CB = 1 −1 3 2 2 1 = h−7, 5, 4i
and the equation of the plane is
−7(x − 1) + 5(y − 1) + 4(z + 1) = 0
or
7x − 5y − 4z = 6
MATH 228 – MIDTERM I
Page 3
Problem 2. (20pts)
(a) (7pts) Find parametric equations for the line ` containing (4, 3, −5)
and (3, 5, −2).
 
 
3
1
r(t) =  5  + t −2
−2
−3
(b) (7pts) Find the value(s) of the parameter you used in (a) for which the
vector from the origin to the point on ` determined by the parameter
is perpendicular to `.
r(t) · h1, −2, −3i = 0
(3 + t) − 2(5 − 2t) − 3(−2 − 3t) = 0
14t = 1
1
t=
14
(c) (6pts) Find the distance from the origin to `.

 

1
43/14
3 + 14
1
r( 14
) =  5 − 17  =  34/7 
3
−31/14
−2 − 14
√
1 1
distance = r( 14
) =
432 + 682 + 312
14
MATH 228 – MIDTERM I
Page 4
Problem 3. (20pts) Let F (x, y, z) = xyz and u = 2i + j − k.
(a) (10pts) Compute Du F at the point P0 (1, −1, 2).
∇F P0 = hyz, xz, xyiP0 = h−2, 2, −1i
Du F = h−2, 2, −1i · h2, 1, −1i
= −1
(b) (5pts) How fast is F increasing in the direction of u ?
Du F
−1
=√
|u|
6
(c) (5pts) Find the equation of the plane tangent to the level surface
F (x, y, z) = −2
at P0 .
−2(x − 1) + 2(y + 1) − (z − 2) = 0
−2x + 2y = z = −6
MATH 228 – MIDTERM I
Page 5
Problem 4. (20pts)
(a) (12pts) Find the length of the curve given by
√
r(t) = (cos t)i + ( 2 sin t)j + (1 − cos t)k,
Z
L=
π
−π ≤ t ≤ π.
q
√
(− sin t)2 + ( 2 cos t)2 + (sin t)2 dt
Z−π
π p
2 sin2 t + 2 cos2 t dt
=
Z−π
π √
=
2 dt
−π
√
= 2 2π
(b) (8pts) Find the velocity and acceleration at time t = 0.
√
dr
= h− sin t, 2 cos t, sin ti
dt
√
dv
a(t) =
= h− cos t, − 2 sin t, cos ti
dt
v(t) =
v(0) =
√
2j
a(0) = −i + k
2
(BONUS: 3pts) Describe the intersection of the infinite elliptic cylinder x2 + y2 = 1 and
the plane x + z = 1. √
It’s a circle of radius 2 parametrized by r(t).
MATH 228 – MIDTERM I
Page 6
Problem 5. (20pts) Find the absolute maxima and minima of the function
f (x, y) = 2x2 − 4x + y 2 − 4y + 1
on the closed triangular plate bounded by the lines x = 0, y = 2, y = 2x in
the first quadrant.
The vertices of the triangular region are A(0, 0), B(0, 2) and C(1, 2).
∇f = h4x − 4, 2y − 4i = 0
implies 4x − 4 = 0 and 2y − 4 = 0 so that x = 1 and y = 2.
There are no local extrema on the interior of the triangle because (1, 2) lies
on the boundary of the triangle (in fact, it is vertex C).
Along AB, x = 0, so that
f = y 2 − 4y + 1
with a minimum at y = 2 with value f (0, 2) = −3
and a maximum at y = 0 with value f (0, 0) = 1.
Along BC, y = 2, so that
f = 2x2 − 4x − 3
with a minimum at x = 1 with value f (1, 2) = −5
and a maximum at x = 0 with value f (0, 2) = −3.
Along AC, y = 2x, so that
f = 2x2 − 4x − 4x2 − 8x + 1
= 6x2 − 12x + 1
with a minimum at x = 1 with value f (1, 2) = −5
and a maximum at x = 0 with value f (0, 0) = 1.
Thus, the absolute maximum occurs at (0, 0) with value 1
and the absolute minimum occurs at (1, 2) with value −5.