Memorial University of Newfoundland Webpage
Memorial University of Newfoundland Webpage

Mathematics 2000: Calculus III (Winter 2020)

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

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  • This course has now concluded. Best of luck with your future studies.

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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.
  • January 8th: Finite and infinite sequences, notation for sequences, describing a sequence with a formula (Section 1.1)
  • January 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)
  • January 13th: Common sequences (constant sequences, p-sequences, geometric sequences), operations on sequences, finding a formula for a sequence, alternating sequences, factorials (Section 1.1)
  • January 15th: Sequences involving factorial and factorial-like expressions, recursively-defined sequences (Section 1.1); limits of sequences, limits of constant sequences (Section 1.2)
  • January 17th: Limits of p-sequences and geometric sequences, using basic properties to evaluate limits of sequences, rewriting sequences to find the limit, the Evaluation Theorem and L'Hôpital's Rule (Section 1.2); 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)
  • January 20th: The Squeeze Theorem, the Absolute Sequence Theorem (and its proof), monotonic (increasing/decreasing) sequences (Section 1.2)
  • January 22nd: Monotonic tails, bounded sequences, the Bounded Monotonic Sequence Theorem (Section 1.2); infinite series (Section 1.3)
  • January 24th: Continuity for functions of two variables (Section 2.2); partial derivatives, higher-order partial derivatives, Clairault's Theorem (Section 2.3)
  • January 29th: The sequence of partial sums, the Divergence Test (and the proof of its contrapositive) (Section 1.3)
  • January 31st: Partial differentiation equations (Section 2.3); functional dependence for multivariable functions (Section 2.4)
  • February 3rd: Telescoping series, constant series, p-series, geometric series (including the proof of their convergence properties) (Section 1.3)
  • February 5th: 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)
  • February 7th: The General Chain Rule, application to implicit differentiation, implicitly-defined functions of two variables (Section 2.4); relative extrema and critical points for functions of two variables (Section 2.5)
  • February 10th: Series that satisfy the requirements of the Integral Test for x >= N, estimating the remainder using the Integral Test (Section 1.4)
  • February 14th: The Second Derivatives Test, absolute extrema for functions of two variables (Section 2.5); partial integrals (Section 2.6)
  • February 24th: More about remainder estimates (Section 1.4); the Direct Comparison Test (and its proof), identifying test series (Section 1.5)
  • February 26th: The Limit Comparison Test (and the proof of part 1), determining the appropriate comparison test (Section 1.5); alternating series (Section 1.6)
  • February 28th: Iterated integrals, double integrals over rectangles, Fubini's Theorem, volumes under surfaces (Section 2.6)
  • March 2nd: The Alternating Series Test, estimating the remainder using the Alternating Series Test, absolute series, absolute and conditional convergence, the Absolute Series Test (and its proof) (Section 1.6)
  • March 4th: The Ratio Test, the Root Test (Section 1.6)
  • March 6th: Double integrals over general regions, Type I integrals for vertically simple regons, Type II integrals for horizontally simple regions (Section 2.7)
  • March 9th: Using the Root Test (Section 1.6); strategies for determining whether a series converges (Section 1.7); power series (Section 1.8)
  • March 11th: Radius of convergence, interval of convergence, shortcut for finding the radius of convergence (Section 1.8)
  • March 13th: Reversing the order of integration, double integrals over regions with circular symmetry (Section 2.7)
  • March 16th: Special cases of the shortcut for finding the radius of convergence (Section 1.8)
  • March 23rd: Representing functions as power series, using the geometric series to find power series representations, term-by-term differentiation and integration (Section 1.9)
  • March 25th: Using term-by-term differentiation and integration to find power series representations (Section 1.9); the Taylor coefficient, Taylor series, Maclaurin series (Section 1.10)
  • March 27th: More about double integrals over regions with circular symmetry (Section 2.7); the polar coordinate system, converting between Cartesian and polar coordinates, simple graphs in polar coordinates (Section 2.8)
  • March 30th: Taylor and Maclaurin series, common Maclaurin series, using common Maclaurin series to find power series representations, the Taylor polynomial (Section 1.10)
  • April 1st: Applications of Taylor and Maclaurin series (Section 1.10); definition of complex numbers, equality of complex numbers, complex arithmetic, conjugate, modulus (Section 1.11)
  • April 3rd: More complicated graphs in polar coordinates, double integrals in polar coordinates (Sections 2.8, 2.9)
  • April 6th: Euler's formula and the polar form of complex numbers, DeMoivre's Theorem (Section 1.11)
  • April 8th: Converting double integrals in Cartesian coordinates to double integrals in polar coordinates, double integrals in polar coordinates with non-constant bounds (Section 2.9)