21-259: Calculus in Three Dimensions

Exam 1

Spring 2025

1. (8 pts) Determine whether the following two lines are intersecting, parallel, skew, or equal:

L1 : x = 2t,    y = t,    z = 2

              L2 : x = 1− s,    y = 8+ s,    z = 7+ s

2. (4 pts) A sled is pulled by exerting a force of 100 N on a rope that makes an angle of 30◦ with the horizontal. Find the work done in pulling the sled 20 m.

3. (8 pts) Find the limit, if it exists:

4. (8 pts) Find parametric equations for the tangent line to the graph of the vector function

at the point (5,9,−9).

5. (8 pts) Find an equation of the plane passing through the points P = (1,2,3), Q = (2,−1,4), and R = (4,3,−2). Write it in the form. Ax +B y +C z +D = 0.

6. (6 pts) Consider the surface given by the equation −x2 + y2 − 2y + z2 + 6z = −9. Categorize the surface and give a sketch.

7. (8 pts) Reparametrize the curve r (t) = ­<3sint, √7t,3cos t> in terms of arc length from a starting point of (0,0,3). Then find the point which is located exactly π units of arc length from the starting point.

8. (Math bonus, 4 pts) Find the ditsance between the parallel planes:

6z = 4y −2x,    9z = 1−3x +6y

9. (Pop culture bonus, 2 pts) Choose ONE of the following:

(a) “I could do this all day.” Who said it, and what movie is it from?

(b) What does “i.e.” stand for in Latin, and what is the English translation?

(c) What does “e.g.” stand for in Latin, and what is the English translation?