I. Foundations/Comprehension (No technology allowed, show all work!) 1. Let r(t) = (V2 -t,, In(t + 1) a) Find the domain of this vector function (3 points). t - 0 b) Compute lim r(t) algebraically (4 points). c) Find r'(t), and evaluate when t = 1 (4 points).2. Suppose that a given line passes through (2, 3, 1) and (5, 5, 2): a) Find the paint where this line intersects the plane having equation x + 2}! + 42 = 6'? (4 points). b) Find the acute angle (in degrees, rounded to nearest whale number) between the given line and the normal vector of the given plane (4 points). 3. Find, and sketch, the domain of the following metion (6 points): Hm) =m+m II. ApplicationsiAnalysis (Technology allowed, show all work! i I) 1. Compute the volume of the parallelepiped that has corners located at the points (2, 3, 1), (1, 3, 2), (3, 2, 4), and the origin (4 points). 2. A particle starts at the origin with initial velocity {1, 1, 3). Its acceleration is given by a(t) = (6r, 12:2, 6t). Find its position function, T(t), showing all steps of the process (4 points). 3. Describe the spatial region that is enclosed by the sphere of radius 2 (centered at the origin) and lies bemeea the parallel planes 2: = 1 and z = 1, using inequalities with spherical coordinates (5 points). 4. Let r(t) = (at3, t2, 2t) be a given vector function: a) For the portion of the corresponding space curve that goes from the origin to the point (72, 36, 12), find the exact length (4 points). b) At the point (9, 9, 6), find T(t), the unit tangent vector (4 points). c) At the point (9, 9, 6), compute the curvature, using any appropriate curvature formula discussed in class (4 points)