Math 210 E1 S92 — Calculus III — Exam #1 — Hanson — page 1 of 2 Show work on this sheet in 50 minutes ((5 × 10 + 50 = 100)pts).

Name . SocSecNo. . 1. Reduce to standard form, Classify, and Sketch the Surface, 4x^{2} + y^{2} − z^{2} − 2y + 1 = 0.

2. Show that v⃗ = a⃗ − (a⃗ · b⃗) b⃗/ |b⃗|^{2} is Orthogonal to b⃗.

3. Find the Normal to and Vector Equation of the Plane passing through the three points (1, 0, −1), (2, 4, 5) and (3, 1, 7).

4. Find the Vector Equation of the Line passing through the two points (1, 3, −5) and (−5, 6, −8).

5. Identify and Sketch the Surface given in Spherical Coordinates by ρ sin(ϕ) = 3.

Math 210 E1 S92 — Calculus III — Exam #1 — Hanson — page 2 of 2

6. Given the position vector r⃗(t) = cos(3t)i⃗ + sin(3t)j⃗ + 4tk⃗, find the (a) Velocity

(b) Acceleration

(c) Speed

(d) Unit Tangent Vector

(e) Arclength when π ≤ t ≤ 2π

(f) Curvature (g) Unit Normal Vector