University:
Seattle Pacific University
Course:
MAT 1236 | Calculus III
Academic year:

2024
Views:

433

Pages:

2
Author:

Yasmin Shaffer

Workshop #18 1. We can describe the surface 2x2 + 4y 2 = 45 − z 2 using by writing it as F (x, y, z) = 0, where F (x, y, z) = 2x2 + 4y 2 + z 2 − 45. We have a theorem that tells us that ∇F (x, y, z) is normal (perpendicular) to the surface. Use this fact to find the tangent plane to this surface at the point (2, −3, 1). 2. The following set of problems is about graphing regions in polar coordinates. Recall that r represents distance from the origin (and maybe be negative, which denotes going in the opposite direction), and θ represents the angle with the positive x-axis in the counterclockwise direction (a negative angle represents the clockwise direction). (a) Give three different ways to represent the point (x, y) = (2, 2) in polar coordinates (r, θ). (b) Each of the circular regions have rings 1 unit apart, and the lines are drawn at the “standard” angles used in trigonometry. Graph the indicated regions. i. 2 < r ≤ 5, 3π 5π ≤θ≤ 4 4 ii. −1 ≤ r ≤ 1, π 3π ≤θ≤ 4 4 3. Graph the set of points in R2 . (a) 1 ≤ r ≤ 2 (b) r ≥ 1 (c) 0 ≤ θ ≤ π, r = 1 11π (d) θ = , −1 ≤ r 4 3π π ≤θ≤ , 1≤r<2 (e) 4 4 −π π (f) <θ< , 2≤r≤4 4 6 −π π (g) ≤θ≤ , 0