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Date

Vector Calculus Independent Study

Unit 2 Sample Test

1. [20 points] Show that the arclength of the line segment connecting the points P and Q is the same as the distance between P and Q.

2. [20 points] Say a particle starts at time 0 at the point (1, 2, 3) and then moves such that its velocity vector at time t is (sin t, t^{2}, e^{t}). Give the position function of the particle.

3. [20 points] Find the tangent line to the curve σ⃗(t) = (1 − 5t, te^{t}, log(t + 10)) at t = 1.

4. [20 points] Find the arc length of the spiral

σ⃗(t) = (e^{t} cos 2πt, e^{t} sin 2πt)

for t = 0 … 2π.

5. [20 points] Find the work done by a particle moving on the path σ⃗(t) = (cos t, sin t, t), 0 ≤ t ≤ 2π through the force field F⃗ (x, y, z) = (−y, x, z).