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The fall 2006 exam for math 2210-04, a university-level mathematics course focusing on vector calculus and parametric equations. The exam consists of 10 problems, each worth 10 points, and covers topics such as finding derivatives of parametric curves, unit vectors, vector-valued functions, cross products, and planes. Students are required to justify all their answers.
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Exam # 1 Fall 2006 MATH 2210-
Instructor: Oana Veliche
Time: 50 minutes
(1) Fill in your name and your student ID number.
(2) There are 10 problems, each worth 10 points.
(3) Justify all your answers. Correct answers with no justification will not be given any credit.
(4) No books, notes or calculators may be used.
Problem 1 2 3 4 5 6 7 8 9 10 Total
Points
1
Problem 1. Let
x = 2 + cos t
y = 1 − sin t
t 6 = nπ
be a curve given by parametric equations.
(a) Find
dy
dx
(b) Find
d
2 y
dx
2
at t =
π
Problem 3. Consider the vector-valued function ~r(t) = e
−t~ i + e
2 t~ j, with t ∈ R.
(a) Find
1
0
~r(t) dt.
(b) Show that D t
[|~v(t)|
2 ] = 2(~a(t) · ~v(t)), where ~v(t) is the velocity and ~a(t) is the acceleration.
Problem 4. Draw the circle of curvature of ~r(t) = (1 + t)
i − (1 + t)
j at t = 0.
(Hint: You may use that eliminating the parameter one obtains y = −x
2 .)
Problem 6. Consider the following vectors in three-space:
~a =< 1 , − 1 , 1 >,
b =< 0 , 2 , 1 >, and ~c =< 3 , − 2 , 1 >.
(a) Find the cross-product ~a ×
b.
(b) Find the volume of the parallelepiped with edges ~a,
b and ~c.
(c) Does the expression ~a × (
b · ~c) makes sense? Justify your answer.
Problem 7. Consider the plane given by the equation x − 2 y + 4z = 10.
(a) Find the parametric equation of a line parallel to this plane going through the point P (1, 1 , 1).
(Remark that there are infinitely many lines with these properties; choose one.)
(b) Find the distance from the line of part (a) to the plane. (Hint: Use that the distance formula from
a point (x 0 , y 0 , z 0 ) to the plane Ax + By + Cz = D is given by the formula:
|Ax 0
By 0
Cz 0
2
2
2
Problem 9. Name and sketch the graph in three-space of the surface given by
4 x
2
2
2 = 100
in the following cases:
(a) a = 4.
(b) a = −25.
Problem 10. Consider the equation: r = 3 sin θ.
(a) Write the equation in Cartesian coordinates.
(b) Write the equation in spherical coordinates.
