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A midterm exam for math 251 at simon fraser university, fall 2006. It includes five questions covering topics such as vector calculus, calculus of functions of several variables, and implicit differentiation. Students are required to find velocities, positions, accelerations, curvature, unit tangents, principal normals, derivatives, partial derivatives, directional derivatives, and tangent planes.
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Midterm 2 1 November 2006, 8:30–9:20am
Instructor: Ralf Wittenberg
Instructions
a(t) = 〈4 cos 2t, 0 , −4 sin 2t〉.
At time t = 0, the particle passes through the origin (0, 0 , 0) with initial velocity v(0) = r′(0) = 〈 0 , 3 , 2 〉.
(a) Find the velocity v(t) and position r(t) of the particle, and also the speed of the particle, at time t.
(b) Reparametrize the curve with respect to arc length measured from the origin in the direction of increasing t.
(c) Find the tangential and normal components of the acceleration vector.
Question 1 continued on next page...
T = 2π
m k
(a) Find the differential of T.
(b) Estimate the percentage change in the period of oscillation if the mass m in- creases by 3% and the spring constant k decreases by 2%.
(a) Find
∂z ∂r
and
∂z ∂θ
(b) Show that ( ∂z ∂r
r^2
∂z ∂θ
∂f ∂x
∂f ∂y
(notation) = f (^) x^2 + f (^) y^2.
(c) Find
∂^2 z ∂r∂θ