## MAT B41 — Final Exam Details

Posted in 2018 -- MAT B41 by pgadey on 2018/08/01

## Canada Math Camp — Storer Calculus

Posted in Math by pgadey on 2018/07/31

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The handout for the talk is available here:

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## MAT B41 — Week 12

Posted in 2018 -- MAT B41, Lecture Notes by pgadey on 2018/07/31

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!

Suggested Exercises:

• 6.1 Geometry of Maps from $\mathbb{R}^2$ to $\mathbb{R}^2$: 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.

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## Mock Final Exam!

Posted in 2018 -- MAT B41 by pgadey on 2018/07/28
The Mock Final is now available!

Thanks everyone, who came out and wrote today! We had about thirty people in total. The last writer finished at approximated 14:50pm. It seems like the final will take approximately three hours. Please attempt the mock final, it is the best preparation for the real final.

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## MAT B41 — Week 11

Posted in 2018 -- MAT B41, Lecture Notes by pgadey on 2018/07/24

(Archimedes Thoughtful by Domenico Fetti 1620 from Wikimedia)
Homework 5 is due! Homework 6 (tex) is now available!

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

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:

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## Homework #5 Question 4

Posted in Math by pgadey on 2018/07/20

Consider a solid ball of radius $R$. Cut a cylindrical hole, through the center of the ball, such that the remaining body has height $h$. Call this the donut $D(R,h)$. Use Cavalieri’s principle to calculate the volume of $D(R,h)$. Calculate the volumes of $D(25,6)$ and $D(50,6)$.

Several students have asked what $D(R,h)$ looks like. Here are some pictures that I found to illustrate the concept. The donut $D(R,h)$ is the region between the red sphere and blue cylinder. The golden balls below show various views of the donut. The donut should fit between the two planes $z=h/2$ and $z=-h/2$, so that it has total height $h$.

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## MAT B41 — Week 10

Posted in 2018 -- MAT B41, Lecture Notes by pgadey on 2018/07/17

(Photo by Ian Alexander) Homework #5 (tex) is available. Drop deadline next Monday.

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

Geogebra Demonstrations:

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 $B$ bound by the parabolic cylinder $x = (y-4)^2 + 3$ and the planes $z=x+2y-4$, $z=x+4y-7$, and $x+2y=11$.
• Final 2015: Evaluate $\int_D e^{x+y} dA$ where $D$ is the region bounded by $y=x-1$ and $y=12-x$ for $2 \leq y \leq 4$.
• Final 2016: Evaluate $\int_D (1-2x) dA$ where $D$ is triangle with vertices $(0,0)$, $(2,3)$, and $(5,3)$.
• Final 2016: Evaluate $\int_0^1 \int_x^{\sqrt[3]{x}} e^{x/y} dy dx$.

Notes:

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Posted in Math by pgadey on 2018/07/11

I was looking through the Geogebra site and found this lovely applet Orthographic Projection by Malin Christersson.

This is a lovely tool for investigating one of my favourite facts about hexagons:

The area maximizing orthogonal projection of a cube is the regular hexagon.

It turns out that Malin has tonnes of awesome geometry stuff online!

Awesome math art!

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## MAT B41 — Week 9

Posted in 2018 -- MAT B41, Lecture Notes by pgadey on 2018/07/10

Homework 4 (tex) is now available.

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 $f(x,y) = 4x^2y - x^3y - x^2y^2$ on the closed triangular region in $\mathbb{R}^2$ with vertices $(0,0)$, $(6,0)$, and $(0,6)$.
• Final 2015: Find the maximum values of $f(x,y,z) = x^2 - 4x + y^2 - 2y + z^2 - 4z - 1$ on the solid ball $x^2 + y^2 + z^2 \leq 9$.
• Final 2015: Find the points on the intersection of the paraboloid $z=x^2+y^2$ and the plane $x+y+z=1$ that are closest and farthest from the origin.

Notes:

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## Public Talks for UTSC

Posted in Math by pgadey on 2018/07/05

## From Colourings to Fixed Points

The notes for the talk are available here.

## Uniform Convergence

The notes for the talk are available here.

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