## Announcements

• Solutions to the final exam problems
• ## Class schedule

• TR 14:35–15:55, Leacock 26

 Date Topics Tu 1/6 Sequences and convergence Th 1/8 Basic techniques for computing limits. Boundedness. Tu 1/13 Monotone convergence. Telescoping series. Th 1/15 Geometric and harmonic series. Alternating series. Tu 1/20 Absolute convergence. Comparison and limit comparison tests. Ratio test. Th 1/22 Root test. Integral test. p-test. Tu 1/27 Power series. Th 1/29 Taylor and Maclaurin series. Tu 2/3 More on Taylor series. Th 2/5 Review of vector geometry. Dot and cross products. Tu 2/10 Applications of cross product. Parametric curves. Velocity. Th 2/12 Velocity. Tangent vectors. Arclength. Vector integrals. Arclength parameterization. Tu 2/17 Acceleration. Normal and binormal. Th 2/19 Curvature formulas. Osculating circle. Tu 2/24 Components of acceleration. Ellipse. Archimedean spiral. Th 2/26 Midterm exam Tu 3/10 The fundamental equations for plane curves. Torsion. Frenet-Serret equations. Th 3/12 Torsion formula. Multivariate functions. Limit. Tu 3/17 Continuity. Partial derivatives. Th 3/19 Higher order derivatives. Simple minimization and maximization problems. Tu 3/24 Weierstrass theorem on existence of maximizers. Th 3/26 Directional derivatives. Gradient. Tu 3/31 Chain rule. Classification of critical points. The Hessian. Th 4/2 The Hessian test for functions of 2 variables. Lagrange multipliers. Tu 4/7 Double integrals. Fubini's theorem. Th 4/9 Change of coordinates / Variable substitution. Polar coordinates. Surface area. Fr 4/17 Final exam (9am, Currie Gym)

## Textbooks

Please choose one of the following options.
• Any edition of any of the calculus texts by James Stewart, so long as it contains the multivariate calculus and sequences/series chapters. For example, James Stewart, Calculus: Early transcendentals, or James Stewart, Multivariate calculus: Early transcendentals will do.
• An alternative is David Massey, Worldwide multivariable calculus, although this book does not cover sequences and series, which is a small but important part of our course. This can be remedied by using the textbook Worldwide integral calculus by the same author, or relying on any of the online resources suggested below.

## Online resources

• Paul's online math notes
• Calculus textbooks by David Guichard
• Calculus textbook by Gilbert Strang
• Calculus textbooks by Jerrold Marsden and Alan Weinstein
• Vector calculus textbook by Michael Corral

## 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 midterm exam is scheduled on Thursday February 26, during the regular class time.
• It is a closed book, closed note exam. No calculators will be allowed or needed.
• The duration of the exam is 1 hour.
• The questions will be similar to the problems from Webwork assignments 1-3, and to these practice problems.

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

• The final exam is scheduled on Friday April 17, starting at 9am, in the Currie Gym.
• It is a closed book, closed note exam. No calculators will be allowed or needed.
• The exam has 5 questions, with 2-3 questions on the midterm topics, and 2-3 questions on the rest.
• You are responsible for what is covered in class, in the reading assignments, and in the relevant sections of the textbook (see below).
• It goes without saying that you are also expected to be familiar with the mathematics that you would normally learn before this course.

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):

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.