Memorial University of Newfoundland Webpage
Memorial University of Newfoundland Webpage

Mathematics 2000: Calculus III (Fall 2018)

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]

.


News back to top
  • This course has now concluded. Best of luck with your future studies.

General Information back to top

Lecture Topics back to top
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)

Worksheets, Assignments and Tests back to top

Solutions back to top