** Math 200**
** Section 106**
** Fall 2018**

** Multivariable Calculus**

** Section Specific Information**

**Time/Place: **MWF 15:00-16:00, LSK 200

**Instructor: ** K. Behrend

** Office hours during final exam period:**

Wed, Dec 5, 10:00-12:00

Fri, Dec 7, 10:00-12:00

Math Annex 1213.

**TA: ** N. Lawrence

contact info

No office hours during final exam period.

**Math Learning Centre: **

Open during exam period on Dec 3, 4, 10, 11.

LSK 301 and 302

**Central
Course Page** contains the syllabus, policies,

practice problems, textbook information, webwork link, etc.
** **

**Tests: ** There will be four in-class tests, each
approximately

20 minutes in duration.

The following applies to all tests in Math 220:

No aids of any kind: no calculators, no notes, no books.

No cell phones, no ipods, no electronic devices of any kind.

If you miss one of the tests for medical reasons, you need to

promptly inform the instructor, and provide a physician's note

specifically stating that you were medically unfit to write the missed

exam on that day. No make-up exams will be given. Your grade

will be based on the other course components.

** Test 1** Friday, Sep. 21: 3-D geometry

** Material:** All Material covered in class up to and inluding
Friday, September 14. In the textbook CLP3 this corresponds to
sections 1.1, 1.2 (skip the optional ones), 1.7, 1.8, 1.9. In
particular, know the distance formula between two points, understand
the technique of sketching a surface by identifying its 'traces' with
constant coordinate planes. Know the basics of vector arithmetic.
Know the dot product, the angle formula, and projections. Know the
cross product, its geometric meaning and the corresponding formulas
for the area of parallelograms and volume of parallelepipeds.

Review in-class examples, webwork problems from the first two
assignments, as well as suggested problems listed on the central
course page.

**Note. ** You must at least know how to sketch the following in
the xy-plane: parabolas, ellipses, hyperbolas, and graphs of sine,
cosine,
exponential and logarithm. Do not memorize names of various 'quadtric
surfaces' or other families of surfaces in 3D.

**
Solutions.**
Problem 1
Problem 2
Problem 3
Problem 4
The class average was 12/16=75%.

** Test 2** Friday, Oct. 12: Differentiation I

** Material:** All Materical covered in class up to and including
Friday, October 5. In CLP3, this corresponds to Sections 2.1, 2.2,
2.3, 2.4 (skip 2.4.3 and 2.4.4), 2.5, 2.6 (skip 2.6.1 and 2.6.2). In
particular, be able to compute limits, or show that they don't exist,
compute partial derivatives, and know Clairaut's theorem. Know that
continuity of partial derivatives implies * differentiability*,
that differentiability means that the graph looks more and more like its tangent plane
if you zoom in, an also that the error in *f* relative to the
errors in *x* and *y* goes to zero as the errors in *x*
and *y* go to zero, justifying the linear approximation procedure
for differentiable functions. Be able to compute equations for tangent
planes, and parametric equations for normal lines. Be able to linearly
approximate values of several variable functions, and estimate the error
terms. Read off simple properties of partial derivatives from contour
plots. Apply the chain rule in several variables.

** Practice Problems:** The following problems from previous
final
exams are especially relevant:

2016WT1 #3 a

2015WT1 #2 ii

2011WT2 #2 a

2011WT2 #2 b

2011WT1 #1 b, c

2013WT2 #2 a

2013WT1 #1 c

2013WT1 #1 d

**
Solutions.**
Problem 1
Problem 2
Problem 3
Problem 3(c)
The class average was 10.4/16=65%.

** Test 3** Friday, Oct. 26: Differentiation II

** Material:** All Material covered in class up to and including
Friday, October 19. In CLP3, this corresponds to 2.7, 2.9. In
particular, know the gradient vector of a function of 2 or 3 variables
and its geometric interpretation, its use in computing tangent
lines/planes and normal lines, the directional derivative and how
to compute it, the definitions of local maxima/minima and the fact
that they only occur at critical points (in class we used the term critical point
for both stationary points and singular points), or on the boundary of
the domain, Second Derivative Test for functions of 2
variables.

** Practice Problems:** The following problems from previous
final
exams are especially relevant:

2016WT1 #2

2016WT1 #3

2015WT1 #1 c

2015WT1 #2

2015WT1 #4

2014WT1 #1

2014WT1 #4

2014WT1 #5

2013WT2 #2 b

2013WT2 #3 b

**
Solutions.**
Problem 1
Problem 2
Problem 3
Problem 4
The class average was 12/18=67%.

** Test 4** Friday, Nov. 16: Integration

** Material:** All Material about multiple integrals and iterated
integrals covered in class up to and including
Friday, November 9. In CLP3, this corresponds to 3.1 (including the
optional section 3.1.5), 3.2 (excluding the optional section 3.2.4),
and 3.3 (skip moment of inertia). In
particular, know the definition of multiple integrals in terms of
Riemann sums, and how to use Riemann sums to approximate the
integral. Be familiar with Fubini's theorem, which says that we can
evaluate multiple integrals by doing iterated integrals (and that we
can choose in which order to perfrom the iterated integrals). Be able
to evaluate multiple integrals and iterated integrals, with particular
care to finding correct boundaries for the integrals. Be able to use
multiple integrals to find volumes, areas, averages, total mass,
centre of mass (this includes memorizing the necessary formulas).

** Practice Problems:** The following problems from previous
final
exams are especially relevant:

2015WT1 #6 c

2014WT1 #6

2013WT2 #5

2013WT2 #6 a

2013WT1 #5

2013WT1 #6

2012WT1 #7

2012WT1 #8

**
Solutions.**
Problem 1
Problem 2
Problem 3
The class average was 12/15=80%.

** Final Exam** Wednesday, Dec. 12, 8:30-11:00.

** See the
common course page for important information about seating
arrangements at the final exam.**

** Material:** The final exam will cover all material treated this
semester. In addition to the subjects listed above, this includes
Lagrange multipliers (Section 2.10, without the optional 2.10.1, in
CLP3) and triple integrals (Sections 3.5, 3.6, 3.7 in CLP3).

No electronic devices of any kind are allowed at the final exam. No
books or notes are allowed. One formula sheet will be provided. This
is the same formula sheet which was provided in previous years, see
2016WT1 here.

You are expected to be familiar with basic integration techniques:
simple substitutions, integration by parts, antiderivatives of
standard functions (polynomials, sin, cos, exp, ln, etc).

The UBC Math Club may be selling exam packages. Check them
out.

Another resource for solutions to past exams is
here.