Textbooks

Online resources

Course outline

Instructor: Dr. Gantumur Tsogtgerel

Prerequisite: MATH 141 (Calculus 2)

Corequisite: MATH 133 (Linear algebra and geometry), or familiarity with vector geometry

Restriction: Not open to students who have taken CEGEP course 201-303 or MATH 150, MATH 151 or MATH 227

Calendar description: Taylor series, Taylor's theorem in one and several variables. Review of vector geometry. Partial differentiation, directional derivative. Extrema of functions of 2 or 3 variables. Parametric curves and arc length. Polar and spherical coordinates. Multiple integrals.

Homework: Two written assignments and several Webwork assignments.

Exams: A midterm exam and a final exam.

Grading: Webwork 15% + Written assignment 15% + Midterm 15% + Final 55%

Midterm exam

The following is a list of skills and knowledge you are expected to master. (It is basically everything we will have studied by the time, but an explicit list might help you direct your study better.)

Sequences and series:

• The limits of $r^n$, $\sqrt[n]{n}$, $\sqrt[n]{n!}$, $\frac{r^n}{n!}$, $\frac{n^n}{n!}$, (where $r$ is a fixed real number)
• Basic techniques for computing limits
• Application of the monotone convergence theorem
• The n-th term test, Leibniz test, estimating the error in alternating series
• Comparison test, limit comparison test, ratio test, root test, integral test, p-test
• Understanding the proof of integral test

Power/Taylor/Maclaurin series:

• Maclaurin series of $\frac1{1-x}$, $\frac1{(1-x)^2}$, $\log(1+x)$, $\arctan x$, $e^x$, $\sin x$, $\cos x$
• Integrating and differentiating power series
• Finding the radius of convergence of a power series by using the ratio test
• Computing the Taylor series of a function by repeated differentiation
• Estimating the remainder term for Taylor polynomials

Dot and cross products:

• Length of a vector, angle between 2 vectors
• Projection of one vector onto another
• Area of a parallelogram, volume of a parallelepiped

Space and plane curves:

• Velocity, acceleration, length, unit tangent, arclength parameterization
• Unit normal, curvature, osculating circle, centre of curvature
• Osculating plane, binormal vector (for space curves)

Final exam

In grading each of the problems, the following rules will be respected (Sorry for the negative tone; I assure you my intention is positive).

• Only a complete and correct solution with clear explanations will receive a full mark.
• Careless errors and other simple mistakes will result in at least 20% deduction from your mark.
• The following errors will be considered as serious, and will result in at least 50% deduction from your mark.
  • Incorrect application of the chain rule, the product rule, or the quotient rule for differentiation
  • Incorrect application of the n-th term test

The exam is intended to test how you recognize the relevant techniques for the particular problem at hand, whether you can apply the learned techniques to solve concrete problems, and whether you can effectively communicate your solution. In what follows, we list some practice problems and other study material, that might be helpful in directing your preparation.

Midterm topics (sequences, series, Taylor series, space/plane curves):

• Practice problems for the midterm and hints to their solution
• Midterm questions with pointers to the practice problem set
• Solutions to selected problems from the midterm exam
• Feedback for the midterm
• Written assignment 1
• Webwork assignments 1–3
• Problems suggested by students (2 of these are in the exam). Thank you all for your contributions!
• Corrections to some of the solutions. Please notify me if you find more errors.

Minimization/maximization problems, and multiple integrals:

• Solutions to the problems from Written assignment 2
• Webwork assignments 4–5
• From Stewart's book:
  • §14.7: 30, 31, 33, 34, 40–44, 46, 53
  • §14.8: 3, 4, 11, 12, 20, 21, 27, 28
  • §15.2: 5, 8, 9, 11, 12, 19, 20, 27, 28
  • §15.3: 5, 6, 15, 16, 20, 21, 49–54
  • §15.4: 7–14, 24–27, 29–32
• Note: You may not have time to solve every single problem from this list, so try solving first a couple of problems from each group of similar problems.

For reference, the relevant sections of the textbook (Stewart) are:

• §11.1–§11.10, §12.1–§12.4, §13.1–§13.4, §14.1–§14.8, §15.2–§15.4
• Note: We did not cover §12.5 in class, but equations of lines and planes occur occasionally in other sections, so it may be a good idea to read it (although there isn't really much to it).
• Note: Kepler's laws from §13.4 are not covered, and will not be tested in the exam.
• Note: Linear approximations and differentials from §14.4 will not be tested in the exam.
• Note: Some knowledge from §15.5–§15.7 might be needed in Webwork assignment 5, but these sections will not be tested in the exam.

For further reference, the following are some notable deviations from the textbook.

• We have covered the Frenet-Serret equations and the torsion formula. While the former will not be in the exam, the latter is testable.
• We have learned how to rigorously establish the existence of maximizers/minimizers by the Weierstrass theorem. In maximization/minimization problems, you will be expected to give full justifications as to why your answer indeed provides the maximum/minimum of the function under consideration. See Written assignment 2 solutions for model examples.