In Mathematics 1000 and 1001, you were introduced to differential and
integral calculus, but solely in the context of functions of a single
variable. In Mathematics 2000, you'll revisit these topics, but in the
more general context of functions of several variables. All the techniques
and ideas you've already learned will still be important, but we'll see
how to extend them to a much wider array of functions. First, though,
we'll study brand-new subjects -- sequences and series -- and establish
the connection between these concepts and the calculus of functions with
which you're familiar. Mathematics 2000 is a key step in the development
of your understanding of calculus, and opens the door to such varied
topics as vector calculus, real analysis, and ordinary and partial
differential equations.
If you have any questions about the course, remember that I can always be
contacted at
[Your browser cannot view this email address; please turn on JavaScript] .
- This course has now concluded. Best of luck with your future
studies.
I do not make lecture notes available outside of class; if you miss a
lecture for any reason, I strongly encourage you to borrow the
corresponding notes from a classmate. However, as a guide, here is a brief
summary of what was covered in each lecture.
- September 5th: Finite and infinite sequences, notation for sequences,
describing a sequence with a formula (Section 1.1)
- September 7th: Common sequences (constant sequences, p-sequences,
geometric sequences), operations on sequences, finding a formula for a
sequence, starting indices other than i=1 (Section 1.1)
- September 10th: Definition of a function of two variables, identifying
and sketching the domain of a function of two variables, surfaces and
level curves (Section 2.1)
- September 12th: Alternating sequences, sequences involving factorial
and factorial-like expressions (Section 1.1)
- September 14th: Functions of three or more variables (Section
2.1); limits of functions of two variables, showing that a limit does
not exist by finding two paths along which it differs (Section
2.2)
- September 17th: Recursively-defined sequences (Section
1.1); limits of sequences (including constant, p- and
geometric sequences) (Section 1.2)
- September 19th: Continuity for functions of two variables (Section
2.2); Partial derivatives (Section 2.3)
- September 21st: Using basic properties to evaluate limits of
sequences, rewriting sequences to find the limit, the Evaluation Theorem
and L'Hôpital's Rule, the Squeeze Theorem (Section 1.2)
- September 24th: Higher-order partial derivatives, Clairault's Theorem,
partial differential equations (Section 2.3)
- September 26th: The General Chain Rule, application to implicit
differentiation (Section 2.4)
- September 28th: The Absolute Sequence Theorem (and its proof),
monotonic (increasing/decreasing) sequences (Section 1.2)
- October 1st: Monotonic tails, bounded sequences, the Bounded Monotonic
Sequence Theorem (Section 1.2); infinite series (Section
1.3)
- October 3rd: Implicitly-defined functions of two variables
(Section 2.4); relative extrema and critical points for functions
of two variables (Section 2.5)
- October 5th: The sequence of partial sums, the Divergence Test (and
the proof of its contrapositive) (Section 1.3)
- October 10th: The Second Derivatives Test, absolute extrema for
functions of two variables, optimisation problems (Section
2.5)
- October 12th: Telescoping series, constant series, p-series, geometric
series (including the proof of their convergence properties)
(Section 1.3)
- October 17th: Basic properties of convergent series, using geometric
series to write a repeating decimal as a ratio of two integers (Section
1.3); the Integral Test (Section 1.4)
- October 19th: Series that satisfy the requirements of the Integral
Test for x >= N, estimating the remainder using the Integral Test
(Section 1.4)
- October 22nd: More about remainder estimates (Section 1.4); the
Direct Comparison Test (and the proof of part 1), the Limit
Comparison Test (Section 1.5)
- October 24th: The proof of the Limit Comparison Test part 1,
determining appropriate test series (Section 1.5); alternating
series, the Alternating Series Test (Section 1.6)
- October 26th: Partial integrals, iterated integrals, volumes under
surfaces, double integrals over rectangles (Section 2.6)
- October 29th: Estimating the remainder using the Alternating Series
Test, absolute series, absolute and conditional convergence, the Absolute
Series Test (and its proof) (Section 1.6)
- October 31st: Fubini's Theorem (Section 2.6); Type 1 and Type 2
regions, double integrals over general regions (Section 2.7)
- November 2nd: The Ratio Test, the Root Test (Section 1.6);
strategies for determining whether a series converges (Section
1.7)
- November 5th: Reversing the order of integration, double integrals
over regions with circular symmetry (Section 2.7)
- November 7th: Series of functions, power series, radius of
convergence, interval of convergence (Section 1.8)
- November 14th: The polar coordinate system, converting between
Cartesian and polar coordinates, simple graphs in polar coordinates
(Section 2.8); double integrals in polar coordinates (Section
2.9)
- November 16th: Shortcut for finding the radius of
convergence (Section 1.8); representing functions as power series,
using the geometric series to find power series representations
(Section 1.9)
- November 19th: Converting double integrals in Cartesian coordinates to
double integrals in polar coordinates (Section 2.9)
- November 21st: Double integrals in polar coordinates with non-constant
bounds (Section 2.9); term-by-term differentiation and integration
(Section 1.9)
- November 23rd: Using differentiation and integration to find power
series representations (Section 1.9); the Taylor coefficient
(Section 1.10)
- November 26th: Taylor and Maclaurin series, common Maclaurin series,
using common Maclaurin series to find power series representations, the
Taylor polynomial (Section 1.10)
- November 28th: Remainder estimates for Taylor series, the Taylor
remainder, the Taylor inequality (Section 1.10); definition of
complex numbers, equality of complex numbers (Section 1.11)
- November 30th: Arithmetic of complex numbers, conjugate, modulus,
Euler's formula and the polar form of complex numbers, DeMoivre's Theorem
(Section 1.11)
- Assignment 0
(due Friday, September 14th)
- Assignment 1
(due Friday, September 21st)
- Assignment 2
(due Friday, September 28th)
- Assignment 3
(due Friday, October 5th)
- Assignment 4
(due Wednesday, October 17th)
- Assignment 5
(due Wednesday, October 24th)
- Assignment 6
(due Wednesday, October 31st)
- Assignment 7
(due Friday, November 23rd)
- Assignment 8
(due Friday, November 23rd; note correction to the last boundary curve
in Question 4)
- Worksheet for
Section 1.1
- Worksheet for
Section 1.2
- Worksheet for
Section 1.3
- Worksheet for
Section 1.4
- Worksheet for
Section 1.5
- Worksheet for
Section 1.6
- Worksheet for
Section 1.7
- Worksheet for
Section 1.8
- Worksheet for
Section 1.9
- Worksheet for
Section 1.10
- Worksheet for
Section 1.11
- Worksheet for
Section 2.1
- Worksheet for
Section 2.2
- Worksheet for
Section 2.3
- Worksheet for
Section 2.4
- Worksheet for
Section 2.5
- Worksheet for
Section 2.6
- Worksheet for
Section 2.7
- Worksheet for
Sections 2.8 & 2.9
- Test 1
(written Monday, October 15th)
- Test 2
(written Friday, November 9th)
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