Midterm 1: Practice Exam
Math 2210-003
Fall 2015
Warnings:
(1) Please bring your student ID with you during midterm exams.
(2) No references or calculators can be used for midterm exams.
Practice Exam
Z
3
xydx along the curve x = t − 1, y = t + 1.
(1) Calculate
1
(2) Find the arc length of this curve:
x = t3/2 , y = t3/2 , z = t; 2 ≤ t ≤ 4
(3) Two vectors are given: u =< 1, −3, 4 > and v =< −2, 1, −1 >
(a) Calculate the angle between the two vectors;
(b) The projection of u onto v and that of v onto u;
(c) Suppose u is orthogonal to v +cw, where w =< 0, 0, 1 >. Determine
the scalar c.
(4) Find the plane through (2, 4, 3) and is parallel to:
(a) xy plane;
(b) the plane x − y + 6z = 213
(5) Find the equation of the plane through (1, 3, −2) that is perpendicular
to both x − 4y + 2z = 0 and 2x − 3y + z = −1.
(6) Find the distance:
(a) from the point (1, 0, −3) to the plane 3x − y + z = 4
(b) between two parallel planes x − 2y + z = 3 and x − 2y + z = 5
(7) The position vector:
µµZ
r(t) =
t
¶ µZ t
¶
¶
2/3
e dx ,
sin πθdθ , t
x
1
π
is given. Calculate the following quantities at t = t1 = 1:
(a) The velocity v(t);
(b) The acceleration a(t);
(c) The speed s.
(8) Find the symmetric equation of the line of intersection of the given pair
of planes: 4x + 3y − 7z = 1 and 10x + 6y − 5z = 10.
(9) Find the equation of the line through (5, −7, 2) and perpendicular to
both < 2, 1, −3 > and < 5, 4, −1 >.
(10) Find the equations of the line through (2, 4, −5) that is parallel to the
plane 3x + y − 2z = 5 and perpendicular to the line
x+8
y−5
z−1
=
=
2
3
−1
(11) Find the equation of the plane containing the line x = 3t, y = 1 + t,
z = 2t and parallel to the intersection of the planes 2x − y + z = 0 and
y+z+1=0
(12) (a) Turn ρ = 2 cos φ into Cylindrical coordinates;
(b) Turn r2 cos 2θ = z into Cartesian coordinates.
(13) Prove that this limit doesn’t exist.
x2 − 6y
(x,y)→(0,0) 3x2 − y
lim
(14) A function w = f (x, y, z) = x2 + sin (xyz) +
z+1
is given.
x2 + 1
(a) In what direction does the function w increases most rapidly at any
point (x, y, z).
(b) Calculate the directional derivative in the direction of (0, −1, 0).