Knowledge Demonstration Opportunity 1: SM221, Calculus III

Name:

13 September 2018

Read all of the following information before starting:

• You are allowed pencils, pens, your TI-36X calculator, and your wits. That is all. In particular, no computers,notes, books, smartphones, iPads, or pocket-sized hobbits.

• To receive full credit, justify your work clearly and in order. I reserve the right to take off points if I cannotsee how you arrived at your answer (even if your “final” answer is correct).

• Use sentences to explain your reasoning. Please keep written answers brief; and simultaneously clear!

• Box or otherwise indicate your final numeric answers.

• Good luck!

Problem Possible Score

1 10

2 10

3 10

4 10

Total 40

Problem 1 (10 points). Let ` denote the line in 3 dimensions which passes through the points (1, 2,−1) and (0, 3, 1).

(a) Write parametric equations for `.

(b) Find an equation of the plane which contains ` and the point (1, 1, 1).

Problem 2 (10 points). Answer each of the following questions.

(a) Next to each boldface word/phrase, circle all of the subsequent italicized words/phrases/expressions which aresynonymous.

(i) perpendicular: colinear, obtuse, orthogonal, tangent, normal, dot product equals zero, dot product equalsone, dot product is negative

(ii) speed: velocity, magnitude of velocity, dot product with position, derivative of acceleration, derivative ofvelocity, derivative of position

(b) Which of the following sets of equations parameterize paths along a circle of radius 2021 in R2? Circle all thatapply.

I. x = cos(2021t), y = sin(2021t) III. x = 2021 cos(t), y = 2021 sin(t)

II. x = 2021 cos(2t), y = 3 + 2021 sin(2t) IV. x = 2021 sin(t), y = 2021 cos(t)

(c) Consider an ant whose position in the xy-plane at time t is given by the parametric equations

x = 2 sin(t) y = 2 cos(t).

Which of the following is a true statement regarding the ant’s movement?

(i) At time t = 0, the ant’s velocity vector points in the negative x direction.

(ii) At time t = 0, the ant’s acceleration vector points in the negative y direction.

(iii) At every time t, the ant’s velocity and acceleration vectors are perpendicular.

(iv) All of the above.

(v) More than one of (i), (ii), or (iii), but not all of them.

Problem 3 (10 points). In this problem, consider the quadric surface defined by the equation

x + y2 + z2 = 2

(a) Sketch traces for the values x = 0, x = 1 and x = −2 below:

x = 0 x = 1 x = −2

(b) Sketch traces for the values z = 0, z = ±1 and z = ±2 below:

z = 0 z = ±1 z = ±2

(c) Identify which picture corresponds to the quadric surface cut out by x + y2 + z2 = 2.

2 Bonus points. What is this quadric surface called? (Circle the correct name):

hyperboloid of 1 sheet, hyperboloid of 2 sheets, cone, ellipsoid, sphere, elliptic paraboloid, hyperbolic paraboloid,elliptic cylinder, circular cylinder, parabolic cylinder, hyperbolic cylinder

Problem 4 (10 points). Suppose that a magnetized particle travels through a magnetic field in 3D space withacceleration at time t ≥ 0 given by the vector function

a(t) = 〈6t, 20et, 21e−t〉.

(a) If its initial velocity is v(0) = 〈5, 20,−21〉 and its initial position is the origin, find the velocity vector v(t) andposition vector r(t).

(b) Write down, but do not evaluate, an integral which computes the arc-length of the particle’s path from timet = 0 to time t = 4.