MIDTERM 1
MATH2210-001
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):
Consider the curve defined parametrically by x = t2 , y = t3 + 1, with t ∈ [1, 2].
(a) Graph the curve. Is it simple? Is it closed?
(b) Find the area between the curve and the x-axis on this interval.
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MIDTERM 1
Q2 (worth 15 marks):
Paul and Jack are both at the origin O = (0, 0). Paul steers his boat√in the direction h−1,
√ 1i, with
velocity 1 m/s; Jack steers his boat in the direction h1, 1i with velocity 3 m/s. After t = 2 seconds,
Paul is at point A, and Jack is at point B. Find the bearing of A from B, and the angle ∠AOB.
MIDTERM 1
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Q3 (worth 20 marks):
(a) Let l be the line with symmetric equations:
x+1
y−1
z−2
=
=
3
4
−1
Show that l is parallel to the plane P with equation 2x − y + 2 = 7. Find the distance between the line
and the plane.
(b) Let m be the line with symmetric equations:
y−5
z−8
x+3
=
=
2
1
−2
Find the equation of the plane that contains both of these lines.
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MIDTERM 1
Q4 (worth 15 marks):
Suppose the position vector at time t is given by
r(t) = sin(t2 ) + cos(t2 ) + (2t − 1)
√
Find the speed, and the parametric equation of the tangent line, when t =
π
.
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