Parker Glynn-Adey

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.

Tagged with: exam, mat b41, mock exam

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MAT B41 — Week 11

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

Domenico-Fetti_Archimedes_1620

(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.

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:

Tagged with: mat b41, 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 $D(50,6)$ .

donut

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 $h$ .

SphericalRingSolid_800 napring2

Tagged with: geometry, graphics

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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.

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 $B$ 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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Malin Christersson’s Cube Toy

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!

Hyperbolic tiler!
Pythagora’s Tree!
A whole Tumblr full of great animations!

Awesome math art!

Tagged with: art, links, math

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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.

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 $f(x,y) = 4x^2y - x^3y - x^2y^2$ 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.