## MAT B41 — Week 12

**You made it to the last week! You’re done!**

On Homework 5, you solved the Napkin Ring Problem. Check it out! That is super cool!

**Additional resources:**

- Khan Academy on triple integrals (Pt. 1).
- Khan Academy on triple integrals (Pt. 2).
- Kristal King on triple integrals.
- PatrickJMT on triple integrals.

**Suggested Exercises:**

- 6.1 Geometry of Maps from to : 1, 3, 6, 11
- 6.2 The Change of Variables Theorem: 3, 4, 7, 10, 11, 21, 23, 26, 28
- 6.3 Applications : 1, 3, 4, 5, 6, 11, 13, 16

**Notes:**

The notes are available here.

## MAT B41 — Week 11

**Homework 5 is due! Homework 6 (tex) is now available!**

The Mock Midterm will be Friday July 27 in SY110 from 12–3pm.

**Additional resources:**

- Khan Academy on triple integrals (Pt. 1).
- Khan Academy on triple integrals (Pt. 2).
- Kristal King on triple integrals.
- PatrickJMT on triple integrals.

**Suggested Exercises:**

- 5.4 Changing the Order of Integration: 2,3,7,9,14
- 5.5 The Triple Integral: 1,3,4,9,10,11,12,16,18,20,21

**Notes:**

## MAT B41 — Week 10

There has been a minor change to the syllabus: Homework 6 will now be assigned in Week 11 and due in Week 12.

**Additional resources:**

- Khan Academy on constrained optimization.
- Eugene Khutoryansky on Double Integrals (after five minutes the video goes beyond our course).
- PatrickJMT on Changing Order of Integration (Pt.1)
- PatrickJMT on Changing Order of Integration (Pt.2)

**Geogebra Demonstrations:**

- Cavalieri’s Principle with Triangles by Irina Boyadzhiev
- Archimede’s Cone-Sphere Theorem by Brian Sterr

**Suggested Exercises:**

- 5.1 Introduction to Double and Triple Integrals: 1,2,3,8,9,13
- 5.2 The Double Integral over a Rectangle: 1,2,3,4,5,6
- 5.3 The Double Integral over a More General Regions: 1,2,3,7,13

**Past Finals:**

*Final 2015*: Find the volume of the solid bound by the parabolic cylinder and the planes , , and .*Final 2015*: Evaluate where is the region bounded by and for .*Final 2016*: Evaluate where is triangle with vertices , , and .*Final 2016*: Evaluate .

**Notes:**

## MAT B41 — Week 9

**Homework 4 (tex) is now available.**

**Additional resources:**

- Khan Academy on constrained optimization.
- Kristal King on Lagrange multipliers.

**Suggested Exercises:**

- 3.4 Constrained Extrema and Lagrange Multipliers: 1a, 3, 4, 5, 13, 14, 15, 19, 20, 28

**Past Finals:**

*Final 2016*: Find the minimum value of on the closed triangular region in with vertices , , and .*Final 2015*: Find the maximum values of on the solid ball .*Final 2015*: Find the points on the intersection of the paraboloid and the plane that are closest and farthest from the origin.

**Notes:**

## MAT B41 — Week 8

**Parker comes back! Public lecture on Friday in IC 200 at 11am.**

**Additional resources:**

- Khan Academy on multivariate maxima and minima.
- Kristal King on local extrema.

**Suggested Exercises:**

- Course Notes: Quadratic forms and determinants
- 3.3 Extrema of Real-Valued Functions: 1,2,3,11,13,21,29,31,52

**Past Finals:**

*Final 2015*: Let . Find and classify the critical points of .*Final 2016*: Let . Find and classify the critical points of .

**Notes:**

## MAT B41 — Week 7

**Midterms are graded! New list of suggested exercises is available!**

There is a new list of suggested exercises available here.

**Suggested Exercises**

- Chapter 1 Review (p. 71): 1, 4, 5, 7, 8, 16, 18, 20
- Chapter 2 Review (p. 145): 1, 2, 3, 5, 6, 10, 15, 25

## MAT B41 — Week 6

**Term Test is this Friday. Reading week is next week!**

We have a modified class schedule this week.

**Tuesday:**Taylor Series and Review.**Thursday:**Class Survey and Work Period, everyone is invited to bring problems to practice.**Friday:**Term Test, will be held Friday afternoon.

**Term Test Policy**

From the Course Syllabus, we have the following policy:

The Term Test will be written outside of regular lecture hours. If you cannot attend reasons of creed or religion, then you will must contact Parker as early as possible to arrange for an alternative sitting. If you miss the midterm test for medical reasons, you must contact Parker within 24 hours of the test.

You will need to send a UTSC Verification of Student Illness or Injury form:

Students who miss the midterm test will be asked to provide the Verification Form and a timetable for the next five days. You will be given only one opportunity to write the make-up test.

Parker scheduled to the midterm to avoid Ramadan, but he forgot about Eid Al-Fitr. If you want to celebrate Eid with your family, instead of writing the midterm at the usual time, please contact Parker immediately.

**The Week after Reading Week!**

Parker will be traveling to STLHE 2018 and the IBL Workshop.

Our Tuesday evening lecture will be delivered by Ivan Khatchatourian.

Our Thursday morning lecture will be delivered by Ray Grinnell.

## MAT B41 — Week 5

**Mock Midterm this Friday 12-2pm in IC 130. Midterm Test next week!**

Several people asked about Question 7 off of Homework #3. The intent of the question was for to be undefined. This is different from the textbook, where the function is given explicitly. You may write and without knowing the function .

**Additional resources:**

- Wikipedia on Clairaut’s theorem
- Khan Academy on Second Partial Derivatives
- Krista King on Second Partial Derivatives
- PatrickJMT on Second Partial Derivatives
- ThreeBlueOneBrown on Single Variable Taylor Series
- Mark Ancliff on Multivariate Taylor Series

**Suggested Exercises:**

- 3.1 Iterated Partial Derivatives: 1, 2, 3, 4, 5, 6, 7, 12, 14, 15, 16
- 3.2 Taylor’s Theorem: 1, 2, 5, 9

**Past Term Tests:**

*2016 Term Test*:

Give the 4th degree Taylor polynomial about the origin of .

*2015 Term Test*:

- Compute the 4th degree Taylor polynomial about the origin of .
- Find the linear approximation to the function at the point and use it to estimate .

**Notes:**

## MAT B41 — Week 4

**Homework 2 is due! Homework 3 (tex) is now available.**

**Midterm information:**

- Mock Term Test Friday 8 June @ 12:00-2:00 PM in IC-130
- Real Term Test Friday 15 June @ 3:00-5:00 PM in IC-130/230

All questions regarding format, question allocation, style of test, will be addressed by the Mock Term Test. I highly recommend you attend the Mock Term Test, because it will be the perfect preparation for the Real Term Test. More information will be available early next week.

In order to prepare for the Term Test, I recommend attempting the suggested exercises.

**Additional resources:**

- Khan Academy on partial derivatives.
- PatrickJMT on partial derivatives.
- KristaKing on partial derivatives.
- Eugene Khutoryansky explains gradients and partial derivatives. (Beautiful animation!)
- Educational Videos explains gradients and partial derivatives.

**Suggested Exercises:**

- 2.3 Differentiation: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 19, 22, 24, 25 (Do as much as possible.)
- 2.5 Properties of Derivatives: 2, 3, 6, 14, 17
- 2.6 Gradients and Directional Derivatives: 1, 4, 6, 10, 11, 19

**Past Term Tests:**

*2016 Term Test*: Let be given by .

- Find an equation for the tangent plane to the graph at the point .
- Find the direction of maximum increase in at the point . What is the rate of maximum increase?

*2015 Term Test*:

- Determine the rate of change in as you move from towards .
- Compute the directional derivative of at the point in the direction of the upward normal for the plane .

**Notes:**

## MAT B41 — Week 3

**Homework 2 is available. Due next week!**

**Additional resources:**

- Wikipedia on the triangle inequality.
- Wikipedia on matrix inverses.
- Khan Academy on matrix inverses.
- Khan Academy on 3×3 matrix inverses.
- Joel Feinstein on open balls and boundedness.
- Denis Potapov proves open balls are open.

**Suggested Exercises:**

- Section 1.5 n-Dimensional Euclidean Space: 1, 2, 7, 9, 10, 11, 15, 16, 17
- Section 2.2 Limits and Continuity: 1, 2, 4b, 8, 11a, 16, 18, 20, 32, 33

**Past Term Tests:**

*2016 Term Test*: Calculate the following limits, showing all your steps, or show that the limit does not exist:

Define by and when .

Find all points such that is continuous at .

*2015 Term Test*:

Define by and when .

Find all points such that is continuous at .

**Notes:**

- Parker’s notes
- Kostya’s notes — Now with colour!

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