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Exam Practice Problems
Math 352, Fall 2011
- Let γ : R → R^3 be the curve γ(t) = (t^2 + t, sin t, et). Compute the Frenet vectors t, n, and b at the point (0, 0 , 1).
- Let γ : R → R^2 be the curve defined by
γ(t) = (2etcos t, 2 etsin t, et).
(a) Find the arc-length of γ from the point (2, 0 , 1) to the point (− 2 eπ, 0 , eπ). (b) Find a unit-speed reparametrization of γ.
- Suppose that a unit-speed curve γ : R → R^2 has signed curvature κs(s) = s^2. Given that ˙γ(0) = (0, 1), find a formula for ˙γ(s).
- Let γ : [0, 2 π] → R be the following curve:
γ(t) = (cos t + cos 2t, sin t − sin 2t)
Let κs denote the signed curvature of γ.
(a) Compute κs at the point (2, 0). (b) Find the equation of the osculating circle for γ at the point (2, 0). (c) What is the value of
γ κs^ ds?
- Let γ be a simple closed curve in the plane. Given that the total length of γ is 10, use
Green’s Theorem to find the maximum possible value of
γ
2 y dx + 5x dy.
- Let γ : R → R^3 be a unit-speed curve, and suppose that ‖γ(t)‖ = 1 for all t ∈ R. Prove that γ(t) · γ¨(t) = −1 for all t ∈ R.
- A circle of radius 6 is rolling inside a fixed circle of radius 16:
H0,0L
H0,- 10 L
H10,0L
P
The fixed circle is centered at the origin, and the smaller circle is initially centered at the point (0, −10). A point P lies on the circumference of the smaller circle, and initially has coordinates (− 6 , −10). Find the coordinates of P when the center of the small circle reaches the point (10, 0).
- Let f : R^2 → R be a smooth function, and let γ(t) =
γ 1 (t), γ 2 (t), γ 3 (t)
be a curve that lies entirely on the graph z = f (x, y). Given that γ(0) = (1, 2 , 3), fx(1, 2) = 9, fy(1, 2) = 4, γ˙ 1 (t) = 2, and ˙γ 2 (t) = −3, find the unit tangent vector to γ at the point (1, 2 , 3).
- A unit-speed curve γ : R → R^3 has constant curvature 1 and constant torsion 0. Given that γ(0) = (2, 0 , 0), ˙γ(0) = (0, 0 , 1), and ¨γ(0) = (1, 0 , 0), compute γ(π/2).
- The following figure shows a circle in R^3 , as well as two perpendicular diameters:
H3,3,4L H5,7,8L
H2,7,5L
Find a constant-speed parametrization γ for this circle satisfying γ(0) = (5, 7 , 8) and γ(π/2) = (2, 7 , 5).
