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MAT137Y1 – LEC0501

Calculus!

Welcome to MAT137!

September 10th, 2018

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Information about this section:

Jean-Baptiste (JB) Campesato

@ campesat@math.toronto.edu u Please start the subject with “MAT137:”

Room MP202. Schedule:

D • Monday, 4pm to 5pm

• Wednesday, 4pm to 6pm Website for this section:

m http://www.math.toronto.edu/campesat/mat137.html (no need to copy the slides in your notes, they are online)

Jean-Baptiste Campesato MAT137Y1 – LEC0501 – Calculus! – Sep 10, 2018 2 / 10

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Resources:

m MAT137 website: http://uoft.me/MAT137 Visit this webpage regularly, it contains:

• The outline of the course (read it!)

• The videos!

• Announcements

• Piazza: an online forum

• Timetable for the office hours

(you can attend any office hours, not only mine)

• And more…

m Precalculus resources:

http://uoft.me/precalc u Enrol in a tutorial!

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Goals of the course

Calculus: understanding of the basic concepts and results of calculus.

Mathematical logic: being able to understand a

mathematical proof and being able to write (even a short) proof.

Problem solving: improving your problem solving skills to react when you face a problem which is new to you.

Jean-Baptiste Campesato MAT137Y1 – LEC0501 – Calculus! – Sep 10, 2018 4 / 10

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Goals of the course

Calculus: understanding of the basic concepts and results of calculus.

Mathematical logic: being able to understand a

mathematical proof and being able to write (even a short) proof.

Problem solving: improving your problem solving skills to react when you face a problem which is new to you.

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Goals of the course

Calculus: understanding of the basic concepts and results of calculus.

Mathematical logic: being able to understand a

mathematical proof and being able to write (even a short) proof.

Problem solving: improving your problem solving skills to react when you face a problem which is new to you.

Jean-Baptiste Campesato MAT137Y1 – LEC0501 – Calculus! – Sep 10, 2018 4 / 10

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Organization:

Before each lecture: watch the indicated videos.

During each lecture, I will give you the list of the videos you will have to watch before the next lecture.

You may also find this list on the webpage of the section.

uThey are prerequisites for the lecture, I assume you watched them, and I am not going to repeat their contents during the lecture.

For Wednesday (Sep 12), watch the videos:

• Sets: 1.1, 1.2

• Quantifiers: 1.3, 1.4, 1.5, 1.6

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Organization:

Before each lecture: watch the indicated videos.

During each lecture, I will give you the list of the videos you will have to watch before the next lecture.

You may also find this list on the webpage of the section.

uThey are prerequisites for the lecture, I assume you watched them, and I am not going to repeat their contents during the lecture.

For Wednesday (Sep 12), watch the videos:

• Sets: 1.1, 1.2

• Quantifiers: 1.3, 1.4, 1.5, 1.6

Jean-Baptiste Campesato MAT137Y1 – LEC0501 – Calculus! – Sep 10, 2018 5 / 10

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Organization:

Before each lecture: watch the indicated videos.

During each lecture, I will give you the list of the videos you will have to watch before the next lecture.

You may also find this list on the webpage of the section.

uThey are prerequisites for the lecture, I assume you watched them, and I am not going to repeat their contents during the lecture.

For Wednesday (Sep 12), watch the videos:

• Sets: 1.1, 1.2

• Quantifiers: 1.3, 1.4, 1.5, 1.6

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How did the students do last year?

It depends on how many problem sets they submitted:

Advice:work on the problem sets,practice makes perfect.

Jean-Baptiste Campesato MAT137Y1 – LEC0501 – Calculus! – Sep 10, 2018 6 / 10

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How did the students do last year?

It depends on how many problem sets they submitted:

Advice:work on the problem sets,practice makes perfect.

(12)

A warm up problem

Put𝑛points on a circle and then join each pair of these points by a line segment.1

In how many regions is the circle divided?

1We assume that inside the circle, three of these line segments never intersect at the same point.

Jean-Baptiste Campesato MAT137Y1 – LEC0501 – Calculus! – Sep 10, 2018 7 / 10

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A warm up problem: how many regions do we obtain?

# of points # of regions 1

2 3 4 5 6

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A warm up problem: how many regions do we obtain?

# of points # of regions 1

2 3 4 5 6

Jean-Baptiste Campesato MAT137Y1 – LEC0501 – Calculus! – Sep 10, 2018 8 / 10

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A warm up problem: how many regions do we obtain?

# of points # of regions

1 1

2 3 4 5 6

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A warm up problem: how many regions do we obtain?

# of points # of regions

1 1

2 3 4 5 6

Jean-Baptiste Campesato MAT137Y1 – LEC0501 – Calculus! – Sep 10, 2018 8 / 10

(17)

A warm up problem: how many regions do we obtain?

# of points # of regions

1 1

2 2

3 4 5 6

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A warm up problem: how many regions do we obtain?

# of points # of regions

1 1

2 2

3 4 5 6

Jean-Baptiste Campesato MAT137Y1 – LEC0501 – Calculus! – Sep 10, 2018 8 / 10

(19)

A warm up problem: how many regions do we obtain?

# of points # of regions

1 1

2 2

3 4

4 5 6

(20)

A warm up problem: how many regions do we obtain?

# of points # of regions

1 1

2 2

3 4

4 5 6

Jean-Baptiste Campesato MAT137Y1 – LEC0501 – Calculus! – Sep 10, 2018 8 / 10

(21)

A warm up problem: how many regions do we obtain?

# of points # of regions

1 1

2 2

3 4

4 8

5 6

Any idea for a general formula?

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A warm up problem: how many regions do we obtain?

# of points # of regions

1 1

2 2

3 4

4 8

5 6

Any idea for a general formula?

Jean-Baptiste Campesato MAT137Y1 – LEC0501 – Calculus! – Sep 10, 2018 8 / 10

(23)

A warm up problem: how many regions do we obtain?

# of points # of regions

1 1

2 2

3 4

4 8

5 16

6

Any idea for a general formula?

Does it still hold?

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A warm up problem: how many regions do we obtain?

# of points # of regions

1 1

2 2

3 4

4 8

5 16

6

Any idea for a general formula?

Does it still hold?

Jean-Baptiste Campesato MAT137Y1 – LEC0501 – Calculus! – Sep 10, 2018 8 / 10

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A warm up problem: how many regions do we obtain?

# of points # of regions

1 1

2 2

3 4

4 8

5 16

6 32

Any idea for a general formula?

Does it still hold?

(26)

A warm up problem: how many regions do we obtain?

# of points # of regions

1 1

2 2

3 4

4 8

5 16

6 32

Any idea for a general formula?

Does it still hold?

Hmmm…Let’s count again! Just to be sure…

Jean-Baptiste Campesato MAT137Y1 – LEC0501 – Calculus! – Sep 10, 2018 8 / 10

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A warm up problem: how many regions do we obtain?

# of points # of regions

1 1

2 2

3 4

4 8

5 16

6 32/// 31

Any idea for a general formula?

Does it still hold?

(28)

A warm up problem: how many regions do we obtain?

# of points # of regions

1 1

2 2

3 4

4 8

5 16

6 32/// 31

Actually, the formula is not2𝑛−1but

( 𝑛 − 1

0 )+ (

𝑛 − 1 1 )+

( 𝑛 − 1

2 )+ (

𝑛 − 1 3 )+

( 𝑛 − 1

4 )= 𝑛

24(𝑛3−6𝑛2+23𝑛−18)+1

Jean-Baptiste Campesato MAT137Y1 – LEC0501 – Calculus! – Sep 10, 2018 8 / 10

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A warm up problem: how many regions do we obtain?

Conclusion

Examples are great to help you to understand new things, to find new ideas, to guess a formula…

But they are not enough!

We needproofsto be sure that a result is true in general.

uWarning: I didn’t say that examples are useless.

They are very important. I use them a lot when I want to understand a mathematical notion which is new to me.

When you face a new mathematical result, Istronglyencourage you to “challenge” it on examples.

It may help you to better understand it!

(30)

A warm up problem: how many regions do we obtain?

Conclusion

Examples are great to help you to understand new things, to find new ideas, to guess a formula…

But they are not enough!

We needproofsto be sure that a result is true in general.

uWarning: I didn’t say that examples are useless.

They are very important. I use them a lot when I want to understand a mathematical notion which is new to me.

When you face a new mathematical result, Istronglyencourage you to “challenge” it on examples.

It may help you to better understand it!

Jean-Baptiste Campesato MAT137Y1 – LEC0501 – Calculus! – Sep 10, 2018 9 / 10

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Another warm up problem

Which of the following statements are equivalent to this one:

“No two students in this class are not on fire.”

1 “For any pair of students in this class, at least one of them is on fire.”

2 “At least two students in this class are on fire.”

3 “All student in this class, except at most one, are on fire.”

4 “At least two students in this class are not on fire.”

5 “If I choose two students in this class and one of them is not on fire, then the other one is on fire.”

ĠJust in case, notice the location of the closest extinguisher…

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Another warm up problem

Which of the following statements are equivalent to this one:

“No two students in this class are not on fire.”

1 “For any pair of students in this class, at least one of them is on fire.”

2 “At least two students in this class are on fire.”

3 “All student in this class, except at most one, are on fire.”

4 “At least two students in this class are not on fire.”

5 “If I choose two students in this class and one of them is not on fire, then the other one is on fire.”

ĠJust in case, notice the location of the closest extinguisher…

Jean-Baptiste Campesato MAT137Y1 – LEC0501 – Calculus! – Sep 10, 2018 10 / 10

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