MATH 2414 CALC 2 FINAL EXAM | QUESTIONS AND ANSWERS | 2026 UPDATE | WITH COMPLETE SOLUTIONS.
Question 1
Your answer is CORRECT.
Which of the following is a description for the plane which includes the point P (3, −2, 3) , while also being
perpendicular to the line below?
x−4 y z−2
= =
3 2 2
a) 3 x + 2 y + 2 z − 11 = 0
b) 3x+2y+2z−7=0
c) −3 x + 2 y − 2 z + 19 = 0
d) 3 x − 2 y + 2 z + 19 = 0
e) 3x+2y+2z+7=0
f) None of the above.
Question 2
Your answer is INCORRECT.
Given the points P (−1, −1, −4) and Q(−3, −2, −2) , find the vector of length 6 in the direction opposite
→
QP .
2 1 2
a) − i− j+ k
3 3 3
b) −4i − 2j + 4k
2 1 2
c) i+ j− k
3 3 3
d) 4i + 2j − 4k
2 2
e) − i + j − 2k
3 3
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, f) None of the above.
Question 3
Your answer is CORRECT.
Calculate by changing to polar coordinates.
4 √ 16−y2
∫ ∫ √ x2 + y2 dx dy
−4 0
64π
a) 3
32π
b) 3
128π
c) 9
64π
d) 9
128π
e) 3
f) None of these.
Question 4
Your answer is CORRECT.
Find an equation for the tangent plane at the point P .
x3 + y3 = 3 xyz
3
P (2, 1, )
2
a) 5x−4y−4z=0
b) −4 x + 5 y + 5 z − 7 = 0
c) 5 x − 4 y − 4 z + 20 = 0
d) 5 x − 4 y + 5 z + 16 = 0
e) 5 x − 4 y − 4 z − 12 = 0
f) None of these.
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, Question 5
Your answer is CORRECT.
Which of the following is a representation for line which is perpendicular to the the plane
P : −4 x − 4 y = −6 and contains the point (1, 2, 2) ?
a) x(t) = 1 + 2 t, y(t) = 2 + 2 t, z(t) = 2
b) x(t) = 1, y(t) = 2 + 2 t, z(t) = 2 + 6 t
c) x(t) = 1 + 2 t, y(t) = 2 + 4 t, z(t) = 2 − t
d) x(t) = 2 + t, y(t) = 2 + 2 t, z(t) = 2 t
e) x(t) = 2 + t, y(t) = 4 + 2 t, z(t) = −1 + 2 t
f) None of the above.
Question 6
Your answer is CORRECT.
Find the measure of the angle of intersection between the lines:
r 1(t) = i + 2j + 3k + t(−2i + 3j − k) and r 2(u) = i + 2j + 3k + u(−i − 2j + 3k)
5π
a) θ=
6
2π
b) θ=
3
π
c) θ=
6
π
d) θ=
3
π
e) θ=
4
f) None of the above.
Question 7
Your answer is INCORRECT.
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Question 1
Your answer is CORRECT.
Which of the following is a description for the plane which includes the point P (3, −2, 3) , while also being
perpendicular to the line below?
x−4 y z−2
= =
3 2 2
a) 3 x + 2 y + 2 z − 11 = 0
b) 3x+2y+2z−7=0
c) −3 x + 2 y − 2 z + 19 = 0
d) 3 x − 2 y + 2 z + 19 = 0
e) 3x+2y+2z+7=0
f) None of the above.
Question 2
Your answer is INCORRECT.
Given the points P (−1, −1, −4) and Q(−3, −2, −2) , find the vector of length 6 in the direction opposite
→
QP .
2 1 2
a) − i− j+ k
3 3 3
b) −4i − 2j + 4k
2 1 2
c) i+ j− k
3 3 3
d) 4i + 2j − 4k
2 2
e) − i + j − 2k
3 3
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, f) None of the above.
Question 3
Your answer is CORRECT.
Calculate by changing to polar coordinates.
4 √ 16−y2
∫ ∫ √ x2 + y2 dx dy
−4 0
64π
a) 3
32π
b) 3
128π
c) 9
64π
d) 9
128π
e) 3
f) None of these.
Question 4
Your answer is CORRECT.
Find an equation for the tangent plane at the point P .
x3 + y3 = 3 xyz
3
P (2, 1, )
2
a) 5x−4y−4z=0
b) −4 x + 5 y + 5 z − 7 = 0
c) 5 x − 4 y − 4 z + 20 = 0
d) 5 x − 4 y + 5 z + 16 = 0
e) 5 x − 4 y − 4 z − 12 = 0
f) None of these.
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, Question 5
Your answer is CORRECT.
Which of the following is a representation for line which is perpendicular to the the plane
P : −4 x − 4 y = −6 and contains the point (1, 2, 2) ?
a) x(t) = 1 + 2 t, y(t) = 2 + 2 t, z(t) = 2
b) x(t) = 1, y(t) = 2 + 2 t, z(t) = 2 + 6 t
c) x(t) = 1 + 2 t, y(t) = 2 + 4 t, z(t) = 2 − t
d) x(t) = 2 + t, y(t) = 2 + 2 t, z(t) = 2 t
e) x(t) = 2 + t, y(t) = 4 + 2 t, z(t) = −1 + 2 t
f) None of the above.
Question 6
Your answer is CORRECT.
Find the measure of the angle of intersection between the lines:
r 1(t) = i + 2j + 3k + t(−2i + 3j − k) and r 2(u) = i + 2j + 3k + u(−i − 2j + 3k)
5π
a) θ=
6
2π
b) θ=
3
π
c) θ=
6
π
d) θ=
3
π
e) θ=
4
f) None of the above.
Question 7
Your answer is INCORRECT.
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