# MATH 2310 3.00AF (Fall 2010)Calculus of several variables, with applications

WebAssign has your final exam score (and see below under "final exam" for a stem and leaf plot of all final exam scores). WebAssign also has a final score (out of 100) which is the adjusted score on which your final grade is based. I don't seem to be able to enter actual letter grades into WebAssign, but you can easily convert the adjusted final score into a letter grade:
A+ (90-100), A (80-89), B+ (75-79), B (70-74), C+ (65-69), C (60-64), D+ (55-59), D (49-54), E (40-48), F (0-39).
The adjusted final score is computed by first applying the formula given in the course outline, and then adjusting it up slightly. I didn't have to adjust it much - the first midterm was low, but the second midterm and the assignments were high, and the final exam about right. It balanced out so that the final adjustment was only about .8 at the A+ range, growing to about 1.5 in the D range.

# Final Exam

The final exam covers the whole course. But since you have been tested on 2/3 of the course already, the exam will be weighted slightly more to the material covered since the second midterm.

The descriptions of the material covered up to the first two midterms (given below) still apply. You can use the assignments and practice problems given earlier to go over that material. I have reset the number of tries allowed by WebAssign, for the first two practice problem sets, in case anyone wants to redo some of those questions.

I have also posted a third set of practice problems, covering the material since the second midterm. In other words:

• Section 15.5: applications to areas, volumes, mass, centres of mass, and probability. You are not responsible for moments of inertia or radii of gyration.
• Section 15.6: triple integrals. In other words, Riemann sums, iterated integrals, applications to volumes, mass, centres of mass.
• Section 15.7: cylindrical coordinates. In other words, a geometric understanding of this coordinate system, changing triple integrals from rectangular to cylindrical coordinates, and evaluating these integrals.
• Section 15.8: spherical coordinates. In other words, a geometric understanding of this coordinate system, changing triple integrals from rectangular to spherical coordinates, and evaluating these integrals.
• Section 16.1: elementary geometry of vector fields (only in the plane).
• Section 16.2: definition and evaluation of line integrals of vector fields (only in the plane). In other words, the last 3 pages of this section, but in the plane rather than in space.
• Section 16.3: fundamental theorem for gradient vector fields in the plane (Theorem 2). Verifying that a vector field is conservative (Theorem 6). Evaluating line integrals by figuring out how to write the vector field as a gradient.
For the last 3 sections, I am expecting you to be able to do fairly elementary calculations (ie the kind of thing you'll find in the practice problem set). A more in depth exploration of line integrals of vector fields is left to future courses (ie MATH 3010)

# Assignments

## Practice problems on Sections 16.1 - 16.3

There are problems from these sections in the set of practice problems for the final exam (see below). But since you will be doing these during the exam period, they are for practice only - you are not submitting any questions from these sections for grades.

## Assignment 7

Is due Monday December 6. See WebAssign. It covers sections 15.5 through 15.8

## Assignment 6

Is due Friday November 19. See WebAssign. It covers section 15.4, which you're responsible for on the midterm.

## Assignment 5

Is due Monday November 15. See WebAssign. It covers sections 15.1 through 15.3, which you're responsible for on the midterm.

## Assignment 4

Is due Friday November 5. See WebAssign.

## Assignment 3

Is due Wednesday October 20. See WebAssign.

## Assignment 2

Is due Friday October 1. The first quiz will cover material from assignments 1 and 2.

I told you in class that you were not responsible for curvature. But if you want to practice what we did on curvature, some problems you could try are Section 13.3, numbers 32, 36, 37.

Some of you have had questions about integrating some of the functions that arise when computing arc length. A sad fact of life is that it is only in your MATH 1310 class that integrals always work out nicely. In real life, you need to often use numerical integration. This is actually really simple to do, and I've built you a little excel file here to remind you how to do this.

## Assignment 1

Due Monday September 27

There are 20 (short) problems on vectors, lines, and planes. This is the minimum you need to do to learn the material. If you find them difficult, pick similar problems from the text and do more, for practice.

Normally you'll have a week for assignments, but I've given you a second weekend in case setting up webassign causes delays. Try not to leave it to the last minute, because you will have a second assignment due the end of that week. Both assignment 1 and assignment 2 are needed for the first quiz. Don't bother with parts of questions that say "sketch and hand in".

Webassign should give everyone slightly different numbers to work with. But the problems are based on the following problems from the text. If you can't access webassign the first weekend, but want to work on problems, try the following (but be aware that you will have to redo them with the "correct" numbers once your webassign account is active):

• Section 12.1 numbers 4, 6, 8
• Section 12.2 numbers 8, 20, 24
• Section 12.3 numbers 9, 18, 24, 25, 38
• Section 12.4 numbers 4, 20, 30
• Section 12.5 numbers 4, 10. 24, 28, 34, 44

# Quizzes

## Quiz 3

The 3rd quiz is based on material from sections 15.5 through 15.8. In section 15.5 you are only responsible for the applications involving areas, volumes, mass, centres of mass, and probability. Not moments of inertia, or the radius of gyration.
Solutions: version A, version B.

## Quiz 2

Solutions: version A, version B.

The 2nd quiz is based on material from sections 14.7 and 14.8

## Quiz 1

Solutions: version A, version B.

# Midterms

## Midterm II

Solutions

Covers sections 14.7-14.8 and 15.1-15.4; In other words: local maxima, minima, and saddles, interior critical points, and the 2nd derivative test (for functions of 2 variables). Boundary critical points, via parametrization or Lagrange multipliers (for functions of 2 variables). For functions or 3 variables, we mainly looked at constrained optimization (ie Lagrange multipliers or via parametrization) for either 1 constraint or 2 constraints. We also covered double integrals via Riemann sums, and then iterated integrals for Type I and Type II regions. We treated both rectangular and polar coordinates. The material from Midterm I will not be explicitly tested, but it may come up in problems because of the cumulative nature of the material. About the only topic from the above sections that I didn't choose to cover is the material on average values (from section 15.1). I am not testing sections 10.3 and 10.4 specifically, but you may want to review that material to help with section 15.4

You may not use calculators (and you don't need them) or notes. You will have 50 minutes. You are not responsible for reproducing proofs given in class. On the other hand, I'm not promising that all problems will be numerical calculations - you may have to manipulate concepts.

There are practice problems on WebAssign. People asked for practice problems they could try that WebAssign so I've created a list of such problems. It is not intended that anyone do all of them. Nor will any of these questions count towards your grade. But if you want to practice a particular topic, you should be able to find questions related to that topic among the list. Do however many you feel necessary.

## Midterm I

Solutions

The material for the first midterm is what you have practised on the first 3 assignments. Specifically:

• Section 12.1
• Section 12.2
• Section 12.3 (no direction cosines)
• Section 12.4 (no torque)
• Section 12.5
• Section 12.6 (not so much the names and properties of the specific quadric surfaces, but the general approach to understanding surfaces as in Section 14.1)
• Section 13.1
• Section 13.2
• Section 13.3 (including unit tangent vectors, but not curvature, torsion, or binormal vectors)
• Section 13.4 (but not decomposing acceleration into a tangential component. You are not expected to memorize Kepler's or Hooke's laws).
• Section 14.1
• Section 14.2 (but I am not going to ask you epsilon-delta questions)
• Section 14.3 (no partial differential equations or Cobb-Douglas functions)
• Section 14.4
• Section 14.5 (no tree diagrams, or chain rule with more than 3 variables)
• Section 14.6