2008 W -C L Calculus(I)

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Calculus (I)

WEN-CHING LIEN

Department of Mathematics National Cheng Kung University

2008

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HW 1:

1.5: 30,42,46,54 1.6: 84

1.7: 48, 53,54,55 S 1: Prove that

x is not a rational function.

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Ch1-1: Introduction

1. The real number system

(1) Natural numbers N= {1,2,3, . . .} (2) Integers

Z= {0,±1,±2, . . .} (3) Rational numbers

Q= {x :x = pq, where p,q are integers and q 6=0} (4) Irrarional numbers

= {real numbers that are not rational numbers} ex:√

2, π, . . .

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Ch1-1: Introduction

1. The real number system

(1) Natural numbers N= {1,2,3, . . .} (2) Integers

Z= {0,±1,±2, . . .} (3) Rational numbers

Q= {x :x = pq, where p,q are integers and q 6=0} (4) Irrarional numbers

= {real numbers that are not rational numbers} ex:√

2, π, . . .

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Ch1-1: Introduction

1. The real number system

(1) Natural numbers N= {1,2,3, . . .} (2) Integers

Z= {0,±1,±2, . . .} (3) Rational numbers

Q= {x :x = pq, where p,q are integers and q 6=0} (4) Irrarional numbers

= {real numbers that are not rational numbers} ex:√

2, π, . . .

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Ch1-1: Introduction

1. The real number system

(1) Natural numbers N= {1,2,3, . . .} (2) Integers

Z= {0,±1,±2, . . .} (3) Rational numbers

Q= {x :x = pq, where p,q are integers and q 6=0} (4) Irrarional numbers

= {real numbers that are not rational numbers} ex:√

2, π, . . .

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Ch1-1: Introduction

1. The real number system

(1) Natural numbers N= {1,2,3, . . .} (2) Integers

Z= {0,±1,±2, . . .} (3) Rational numbers

Q= {x :x = pq, where p,q are integers and q 6=0} (4) Irrarional numbers

= {real numbers that are not rational numbers} ex:√

2, π, . . .

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Ch1-1: Introduction

1. The real number system

(1) Natural numbers N= {1,2,3, . . .} (2) Integers

Z= {0,±1,±2, . . .} (3) Rational numbers

Q= {x :x = pq, where p,q are integers and q 6=0} (4) Irrarional numbers

= {real numbers that are not rational numbers} ex:√

2, π, . . .

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2. Intervals

(a,b], [a,b), (a,b), [a,b], (−∞,b], (a,∞) open , closed

R= (−∞,∞) =the set of all real numbers

3 Boundedness:

LetS be a set of real numbers.

(1)S is bounded above if∃M s.t. xM ,x ∈ S. M is called an upper bound of S.

(2)S is bounded below if∃M s.t. xM ,x ∈ S. M is called a lower bound ofS.

(3)S is bounded if bounded above and below.

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2. Intervals

(a,b], [a,b), (a,b), [a,b], (−∞,b], (a,∞) open , closed

R= (−∞,∞) =the set of all real numbers

3 Boundedness:

LetS be a set of real numbers.

(1)S is bounded above if∃M s.t. xM ,x ∈ S. M is called an upper bound of S.

(2)S is bounded below if∃M s.t. xM ,x ∈ S. M is called a lower bound ofS.

(3)S is bounded if bounded above and below.

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2. Intervals

(a,b], [a,b), (a,b), [a,b], (−∞,b], (a,∞) open , closed

R= (−∞,∞) =the set of all real numbers

3 Boundedness:

LetS be a set of real numbers.

(1)S is bounded above if∃M s.t. xM ,x ∈ S. M is called an upper bound of S.

(2)S is bounded below if∃M s.t. xM ,x ∈ S. M is called a lower bound ofS.

(3)S is bounded if bounded above and below.

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2. Intervals

(a,b], [a,b), (a,b), [a,b], (−∞,b], (a,∞) open , closed

R= (−∞,∞) =the set of all real numbers

3 Boundedness:

LetS be a set of real numbers.

(1)S is bounded above if∃M s.t. xM ,x ∈ S. M is called an upper bound of S.

(2)S is bounded below if∃M s.t. xM ,x ∈ S. M is called a lower bound ofS.

(3)S is bounded if bounded above and below.

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2. Intervals

(a,b], [a,b), (a,b), [a,b], (−∞,b], (a,∞) open , closed

R= (−∞,∞) =the set of all real numbers

3 Boundedness:

LetS be a set of real numbers.

(1)S is bounded above if∃M s.t. xM ,x ∈ S. M is called an upper bound of S.

(2)S is bounded below if∃M s.t. xM ,x ∈ S. M is called a lower bound ofS.

(3)S is bounded if bounded above and below.

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2. Intervals

(a,b], [a,b), (a,b), [a,b], (−∞,b], (a,∞) open , closed

R= (−∞,∞) =the set of all real numbers

3 Boundedness:

LetS be a set of real numbers.

(1)S is bounded above if∃M s.t. xM ,x ∈ S. M is called an upper bound of S.

(2)S is bounded below if∃M s.t. xM ,x ∈ S. M is called a lower bound ofS.

(3)S is bounded if bounded above and below.

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2. Intervals

(a,b], [a,b), (a,b), [a,b], (−∞,b], (a,∞) open , closed

R= (−∞,∞) =the set of all real numbers

3 Boundedness:

LetS be a set of real numbers.

(1)S is bounded above if∃M s.t. xM ,x ∈ S. M is called an upper bound of S.

(2)S is bounded below if∃M s.t. xM ,x ∈ S. M is called a lower bound ofS.

(3)S is bounded if bounded above and below.

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2. Intervals

(a,b], [a,b), (a,b), [a,b], (−∞,b], (a,∞) open , closed

R= (−∞,∞) =the set of all real numbers

3 Boundedness:

LetS be a set of real numbers.

(1)S is bounded above if∃M s.t. xM ,x ∈ S. M is called an upper bound of S.

(2)S is bounded below if∃M s.t. xM ,x ∈ S. M is called a lower bound ofS.

(3)S is bounded if bounded above and below.

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2. Intervals

(a,b], [a,b), (a,b), [a,b], (−∞,b], (a,∞) open , closed

R= (−∞,∞) =the set of all real numbers

3 Boundedness:

LetS be a set of real numbers.

(1)S is bounded above if∃M s.t. xM ,x ∈ S. M is called an upper bound of S.

(2)S is bounded below if∃M s.t. xM ,x ∈ S. M is called a lower bound ofS.

(3)S is bounded if bounded above and below.

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The continuum of numbers:

Integers=⇒Rational Numbers

(rational operations+, −, ×, ÷) Consider the real line L:

We have that every point Pof L is either a rational point of the form pq on lies between two successive rational point pq and p+q1

⇒ |P − pq| ≤ 1q

⇒ We can choose 1q as small as we want.

⇒ For any pointPof L, we can find rational points while are arbitrarily close toP.

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The continuum of numbers:

Integers=⇒Rational Numbers

(rational operations+, −, ×, ÷) Consider the real line L:

We have that every point Pof L is either a rational point of the form pq on lies between two successive rational point pq and p+q1

⇒ |P − pq| ≤ 1q

⇒ We can choose 1q as small as we want.

⇒ For any pointPof L, we can find rational points while are arbitrarily close toP.

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The continuum of numbers:

Integers=⇒Rational Numbers

(rational operations+, −, ×, ÷) Consider the real line L:

We have that every point Pof L is either a rational point of the form pq on lies between two successive rational point pq and p+q1

⇒ |P − pq| ≤ 1q

⇒ We can choose 1q as small as we want.

⇒ For any pointPof L, we can find rational points while are arbitrarily close toP.

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The continuum of numbers:

Integers=⇒Rational Numbers

(rational operations+, −, ×, ÷) Consider the real line L:

We have that every point Pof L is either a rational point of the form pq on lies between two successive rational point pq and p+q1

⇒ |P − pq| ≤ 1q

⇒ We can choose 1q as small as we want.

⇒ For any pointPof L, we can find rational points while are arbitrarily close toP.

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The continuum of numbers:

Integers=⇒Rational Numbers

(rational operations+, −, ×, ÷) Consider the real line L:

We have that every point Pof L is either a rational point of the form pq on lies between two successive rational point pq and p+q1

⇒ |P − pq| ≤ 1q

⇒ We can choose 1q as small as we want.

⇒ For any pointPof L, we can find rational points while are arbitrarily close toP.

(23)

The continuum of numbers:

Integers=⇒Rational Numbers

(rational operations+, −, ×, ÷) Consider the real line L:

We have that every point Pof L is either a rational point of the form pq on lies between two successive rational point pq and p+q1

⇒ |P − pq| ≤ 1q

⇒ We can choose 1q as small as we want.

⇒ For any pointPof L, we can find rational points while are arbitrarily close toP.

(24)

The continuum of numbers:

Integers=⇒Rational Numbers

(rational operations+, −, ×, ÷) Consider the real line L:

We have that every point Pof L is either a rational point of the form pq on lies between two successive rational point pq and p+q1

⇒ |P − pq| ≤ 1q

⇒ We can choose 1q as small as we want.

⇒ For any pointPof L, we can find rational points while are arbitrarily close toP.

(25)

The continuum of numbers:

Integers=⇒Rational Numbers

(rational operations+, −, ×, ÷) Consider the real line L:

We have that every point Pof L is either a rational point of the form pq on lies between two successive rational point pq and p+q1

⇒ |P − pq| ≤ 1q

⇒ We can choose 1q as small as we want.

⇒ For any pointPof L, we can find rational points while are arbitrarily close toP.

(26)

The continuum of numbers:

Integers=⇒Rational Numbers

(rational operations+, −, ×, ÷) Consider the real line L:

We have that every point Pof L is either a rational point of the form pq on lies between two successive rational point pq and p+q1

⇒ |P − pq| ≤ 1q

⇒ We can choose 1q as small as we want.

⇒ For any pointPof L, we can find rational points while are arbitrarily close toP.

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Theorem (Density)

The rational points are dense on the real line.

Corollary

Between any two distinct rational points, there are infinity many other rational points.

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Theorem (Density)

The rational points are dense on the real line.

Corollary

Between any two distinct rational points, there are infinity many other rational points.

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Thank you.

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