VICTOR VALLEY COLLEGE
Name
Department of Mathematics
Final Exam, Group A
Math 228 - Calculus 3
Spring 2026
Instructions: This is the take-home component for your final exam. You are to print and complete this by Wednesday, June 10th when many of you will begin the in-class component. It
can also be emailed if you’re not done by that date!
~ = h3, 0, 4i and v~ = h0, 12, −5i . Find the following.
1. Let u
a) 3~
u − 2~
v
b)
~
u
~.
c) A unit vector in the direction of u
~ and v~.
d) An expression for the angle θ between u
1
~ of u
~ onto v~.
e) The projection projv~ u
~ × v~
f) u
~ and v~.
g) A unit vector orthogonal to both u
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2. Consider the points P = (5, 7, 1), Q = (10, 3, 4), and R = (−2, 1, 1).
a) Find both the parametric and symmetric equations for the line through points P and Q.
b) Find the general form of the equation of the plane passing through P, Q, and R.
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3. Do the following.
a) Convert the point 8, − π4 , π4 from spherical to rectangular coordinates.
√
b) Convert the point 4 3, −4, 6 from rectangular to cylindrical coordinates.
c) Convert the equation 6x 2 + 6y2 − 8z = 0 into cylindrical coordinates.
d) Convert the equation ρ2 sin2 φ = 9 from spherical to rectangular coordinates. Note that it
has a nice expression!
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4. For the curve given by
D
E
r~ (t) = t 3 − 1, e2t−2 , cos(πt) ,
a) Find the general velocity vector r~ 0 (t).
b) Find the unit tangent vector when t = 1.
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5. For the curve given by
r~ (t) =
√ 2
1 3
t , 4t, 2t ,
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a) Set up and simplify the integral representing the length of the curve for t ∈ [0, 3].
b) Find the value of the integral above.
c) Calculate the curvature at the origin (assuming we consider t ∈ R).
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6. For the equation sin x 2 + z = 2xy + z, use implicit differentiation to find the first partial
derivatives of z.
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sd
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