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The instructions and problems for the final examination of mathematics 217 at the university of british columbia, held on december 16, 2011. The examination covers various topics in multivariable calculus, including gradient vectors, directional derivatives, unit tangent and normal vectors, curvature, triple iterated integrals, and line integrals. Students are required to solve problems related to these topics and demonstrate their work.
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The University of British Columbia
Final Examination - December 16, 2011
Mathematics 217
Time: 2.5 hours
LAST Name
First Name Signature
Student Number
Special Instructions:
One formula sheet allowed. No communication devices allowed. One calculator allowed.
Show all your work; little or no credit will be given for a numerical answer without the
correct accompanying work. If you need more space than the space provided, use the back
of the previous page.
Rules governing examinations
UBCcard for identification.
except in cases of supposed errors or ambiguities in examination
questions.
after the expiration of one-half hour from the scheduled starting
time, or to leave during the first half hour of the examination.
est practices shall be immediately dismissed from the examination
and shall be liable to disciplinary action.
(a) Having at the place of writing any books, papers
or memoranda, calculators, computers, sound or image play-
ers/recorders/transmitters (including telephones), or other mem-
ory aid devices, other than those authorized by the examiners.
(b) Speaking or communicating with other candidates.
(c) Purposely exposing written papers to the view of other can-
didates or imaging devices. The plea of accident or forgetfulness
shall not be received.
rial; must hand in all examination papers; and must not take any
examination material from the examination room without permis-
sion of the invigilator.
rections communicated by the instructor or invigilator.
Total 60
Page 1 of 10 pages
[10] 1. Recall that a direction can be given by a unit vector of the form ~u θ
= 〈cos θ, sin θ〉
for some angle θ measured counterclockwise from the positive x-axis. Let
f (x, y) = e
−x
2 − 2 y
2
and consider the point P (− 1 , 0).
(a) Find ∇f at P.
(b) Find the angles θ that determine the directions of maximum increase, maximum de-
crease, and zero change of the function f.
[10] 2. Consider the parabola y = x
2
, which we parametrize as ~r(t) = 〈t, t
2
〉 for −∞ < t < ∞.
(a) Find the unit tangent and unit normal vectors
T(t) and
N(t), respectively, for this
parabola.
(b) Show that the curvature of the parabola y = x
2 at its vertex is κ = 2.
(c) Find the equation of the osculating circle at the vertex of the parabola. Sketch both
the parabola and this osculating circle on the same set of axes.
[10] 4. Let C 1 be the circle (x − 2)
2
2
= 1 and let C 2 be the circle (x − 2)
2
2
= 9.
Suppose
F(x, y) =
−y
x
2
2
x
x
2
2
Find the integrals
C 1
F · d~r and
C 2
F · d~r, where both circles are oriented counterclockwise
in computing the line integrals.
[10] 6. Find
S
F · ~n dS, the flux of the vector field
F(x, y, z) = 〈x sin y, − cos y, z sin y〉
across the surface S, where S is the boundary of the region in R
3 bounded by the planes
x = 1, y = 0, y = π/ 2 , z = 0, and z = x.
The End