MIDTERM 1
MATH2210-004
This is a 1-hour exam. No calculators are allowed.
Note: Your final answers may involve quantities such as:
not make approximations (eg. π ≈ 3.14).
√
p
√
5 − π, ln(2), tan−1 ( 53 ), etc. You should
Name:
Q1 (worth 15 marks):
π π
Consider the curve defined parametrically by x = 3 sin(2t), y = 3 cos(2t), with t ∈ [ 12
, 6 ].
(a) Graph the curve. Is it simple? Is it closed?
(b) Find the arc length of the curve.
1
2
MIDTERM 1
Q2 (worth 20 marks):
(a) Tom is initially at the origin O, and is headed in the direction h1, 0, −1i with speed 2m/s. A pole is
situated at the point P , (3, 2, 0). At which point in time is Tom closest to the pole?
−→
Hint: Find the projection of OP onto h1, 0, −1i.
(b) Let Q be the point (−2, −2, −1). The line P Q intersects the line of Tom’s motion at ( 21 , 0, − 12 ). Find
the (smaller) angle between the two lines.
MIDTERM 1
Q3 (worth 15 marks):
Use the following method to determine whether or not the three planes P1 , P2 , P3 intersect in a line.
P1 : x − y − z = 3, P2 : 2x + 2y − z = 4, P3 : 4x − 3z = 10
(a) Find parametric equations for the line l of intersection between P1 and P2 .
(b) Determine whether l lies on P3 or not.
3
4
MIDTERM 1
Q4 (worth 15 marks):
Suppose the position vector at time t is given by
→
−
→
−
→
−
r(t) = ln(et + 1) i + tet j + (2t − 1) k
Find the acceleration vector, and the parametric equation of the tangent line, when t = ln(3).