MATH 251-Spring 2006 Midterm 1 Exam, Exams of Calculus

A midterm exam for math 251 at simon fraser university, held on 7 february 2006. The exam contains 5 questions on vector calculations, angular momentum, helical paths, and parametric equations. Students are required to solve problems related to finding vector products, areas, angles, equations of planes, distances, derivatives, and curvature.

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2012/2013

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MATH 251-3, Spring 2006 Simon Fraser University
Midterm 1 7 February 2006, 5:30–6:20pm
Instructor: Ralf Wittenberg
Last Name:
First Name:
Student Number:
Signature:
Instructions
1. Please do not open this booklet un-
til invited to do so.
2. Write your last name, first name(s) and
student number in the box above in block
letters, and sign your name in the space
provided.
3. This exam contains 5 questions on 5 pages
(after this title page). Once the exam be-
gins please check to make sure your exam
is complete.
4. The total time available is 50 minutes, and
there are 50 points, so allow about a minute
per point; for example, you should aim to
spend about 10 minutes on a 10-point ques-
tion. Attempt all problems!
5. This is a closed book exam. Only non-
programmable scientific calculators are al-
lowed.
6. Use the reverse side of the previous page if
you need more room for your answer, and
clearly indicate where the solution contin-
ues.
7. Show all your work, and explain your an-
swers clearly.
8. Good luck!
Question Maximum Score
1 18
2 4
3 10
4 10
5 8
Total 50
pf3
pf4
pf5

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MATH 251-3, Spring 2006 Simon Fraser University

Midterm 1 7 February 2006, 5:30–6:20pm

Instructor: Ralf Wittenberg

Last Name:

First Name:

Student Number:

Signature:

Instructions

  1. Please do not open this booklet un- til invited to do so.
  2. Write your last name, first name(s) and student number in the box above in block letters, and sign your name in the space provided.
  3. This exam contains 5 questions on 5 pages (after this title page). Once the exam be- gins please check to make sure your exam is complete.
  4. The total time available is 50 minutes, and there are 50 points, so allow about a minute per point; for example, you should aim to spend about 10 minutes on a 10-point ques- tion. Attempt all problems!
  5. This is a closed book exam. Only non- programmable scientific calculators are al- lowed.
  6. Use the reverse side of the previous page if you need more room for your answer, and clearly indicate where the solution contin- ues.
  7. Show all your work, and explain your an- swers clearly.
  8. Good luck!

Question Maximum Score

Total 50

  1. Given the four points

A(1, 2 , 1), B(2, 0 , 1), C(− 1 , 2 , 0), and D(3, 3 , −1) :

(a) [2 points] Compute

−→ AB ×

−→ AC.

(b) [2 points] Find the area of the triangle ABC.

(c) [4 points] Find the angle θ between the vectors

−→ AB and

−→ AC.

(d) [6 points] Find an equation of the plane that passes through A and B and is parallel to the line through C and D.

  1. The velocity of a particle moving along a helical path is given by the vector

v(t) = 3 i + 4 sin t j + 4 cos t k , t ≥ 0.

(a) [3 points] At time t = 0, the particle passes through the point P (1, 0 , 0). Find the position vector r(t) for the particle.

(b) [3 points] Find the distance travelled (arc length) along the helical curve from t = 0 to t = 2π.

(c) [4 points] Find the tangent vector T and principal normal vector N to the helical curve.

  1. A particle moves in space with parametric equations

x = t, y = t^2 , z = 43 t^3 /^2 , t > 0.

(a) [4 points] Find the velocity and acceleration vectors, and the speed of the particle as a function of t.

(b) [6 points] Determine the curvature κ and radius of curvature of the curve traced out by the particle at time t = 1.