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MATH 251- Fall 2006 Midterm 2 Exam, Exams of Calculus

A midterm exam for math 251 at simon fraser university, fall 2006. It includes five questions covering topics such as vector calculus, calculus of functions of several variables, and implicit differentiation. Students are required to find velocities, positions, accelerations, curvature, unit tangents, principal normals, derivatives, partial derivatives, directional derivatives, and tangent planes.

Typology: Exams

2012/2013

Uploaded on 02/18/2013

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MATH 251-3, Fall 2006 Simon Fraser University
Midterm 2 1 November 2006, 8:30–9:20am
Instructor: Ralf Wittenberg
Last Name:
First Name:
SFU ID:
Signature:
Instructions
1. Please do not open this booklet un-
til invited to do so.
2. Write your last name, first name(s) and
SFU ID 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 14
2 6
3 7
4 11
5 12
Total 50
pf3
pf4
pf5

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Download MATH 251- Fall 2006 Midterm 2 Exam and more Exams Calculus in PDF only on Docsity!

MATH 251-3, Fall 2006 Simon Fraser University

Midterm 2 1 November 2006, 8:30–9:20am

Instructor: Ralf Wittenberg

Last Name:

First Name:

SFU ID:

Signature:

Instructions

  1. Please do not open this booklet un- til invited to do so.
  2. Write your last name, first name(s) and SFU ID 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. [14 points] The acceleration of a particle moving along a helical path is

a(t) = 〈4 cos 2t, 0 , −4 sin 2t〉.

At time t = 0, the particle passes through the origin (0, 0 , 0) with initial velocity v(0) = r′(0) = 〈 0 , 3 , 2 〉.

(a) Find the velocity v(t) and position r(t) of the particle, and also the speed of the particle, at time t.

(b) Reparametrize the curve with respect to arc length measured from the origin in the direction of increasing t.

(c) Find the tangential and normal components of the acceleration vector.

Question 1 continued on next page...

  1. [7 points] The period T (m, k) of a simple harmonic oscillator of mass m and spring constant k is given by

T = 2π

m k

(a) Find the differential of T.

(b) Estimate the percentage change in the period of oscillation if the mass m in- creases by 3% and the spring constant k decreases by 2%.

  1. [11 points] Let z = f (x, y), where f has continuous second-order partial derivatives, and x = r cos θ, y = r sin θ.

(a) Find

∂z ∂r

and

∂z ∂θ

(b) Show that ( ∂z ∂r

r^2

∂z ∂θ

∂f ∂x

∂f ∂y

(notation) = f (^) x^2 + f (^) y^2.

(c) Find

∂^2 z ∂r∂θ