Calculus (I)

Calculus (I)

W^EN-C^HING L^IEN

Department of Mathematics National Cheng Kung University

2008

WEN-CHINGLIEN Calculus (I)

Ch1-2: Elementary Functions

1. Notations

Definition

A ” function ” is a rule that assigns each element x ∈A exactly one element y ∈B.

f :A−→B f(x) =y

y : image(or value), A : domain of f

f(A)≡ {y :y =f(x),x ∈A}: the range of f Example 1: State the range f(x) = x²

f(x) = _x+¹₁,x 6=−1

WEN-CHINGLIEN Calculus (I)

Ch1-2: Elementary Functions

1. Notations

Definition

A ” function ” is a rule that assigns each element x ∈A exactly one element y ∈B.

f :A−→B f(x) =y

y : image(or value), A : domain of f

f(A)≡ {y :y =f(x),x ∈A}: the range of f Example 1: State the range f(x) = x²

f(x) = _x+¹₁,x 6=−1

WEN-CHINGLIEN Calculus (I)

Ch1-2: Elementary Functions

1. Notations

Definition

A ” function ” is a rule that assigns each element x ∈A exactly one element y ∈B.

f :A−→B f(x) =y

y : image(or value), A : domain of f

f(A)≡ {y :y =f(x),x ∈A}: the range of f Example 1: State the range f(x) = x²

f(x) = _x+¹₁,x 6=−1

WEN-CHINGLIEN Calculus (I)

Ch1-2: Elementary Functions

1. Notations

Definition

A ” function ” is a rule that assigns each element x ∈A exactly one element y ∈B.

f :A−→B f(x) =y

y : image(or value), A : domain of f

f(A)≡ {y :y =f(x),x ∈A}: the range of f Example 1: State the range f(x) = x²

f(x) = _x+¹₁,x 6=−1

WEN-CHINGLIEN Calculus (I)

Ch1-2: Elementary Functions

1. Notations

Definition

A ” function ” is a rule that assigns each element x ∈A exactly one element y ∈B.

f :A−→B f(x) =y

y : image(or value), A : domain of f

f(A)≡ {y :y =f(x),x ∈A}: the range of f Example 1: State the range f(x) = x²

f(x) = _x+¹₁,x 6=−1

WEN-CHINGLIEN Calculus (I)

Ch1-2: Elementary Functions

1. Notations

Definition

A ” function ” is a rule that assigns each element x ∈A exactly one element y ∈B.

f :A−→B f(x) =y

y : image(or value), A : domain of f

f(A)≡ {y :y =f(x),x ∈A}: the range of f Example 1: State the range f(x) = x²

f(x) = _x+¹₁,x 6=−1

WEN-CHINGLIEN Calculus (I)

Ch1-2: Elementary Functions

1. Notations

Definition

A ” function ” is a rule that assigns each element x ∈A exactly one element y ∈B.

f :A−→B f(x) =y

y : image(or value), A : domain of f

f(A)≡ {y :y =f(x),x ∈A}: the range of f Example 1: State the range f(x) = x²

f(x) = _x+¹₁,x 6=−1

WEN-CHINGLIEN Calculus (I)

Ch1-2: Elementary Functions

1. Notations

Definition

A ” function ” is a rule that assigns each element x ∈A exactly one element y ∈B.

f :A−→B f(x) =y

y : image(or value), A : domain of f

f(A)≡ {y :y =f(x),x ∈A}: the range of f Example 1: State the range f(x) = x²

f(x) = _x+¹₁,x 6=−1

WEN-CHINGLIEN Calculus (I)

Ch1-2: Elementary Functions

1. Notations

Definition

A ” function ” is a rule that assigns each element x ∈A exactly one element y ∈B.

f :A−→B f(x) =y

y : image(or value), A : domain of f

f(A)≡ {y :y =f(x),x ∈A}: the range of f Example 1: State the range f(x) = x²

f(x) = _x+¹₁,x 6=−1

WEN-CHINGLIEN Calculus (I)

Definition

f and g are equal ⇐⇒(1) f and g have the same domain (2) f(x) =g(x)for all x

Definition

f(x) = f(−x): even.

f(x) = −f(−x): odd.

Example 1. f(x) = |x|, g(x) =sin(x), 2. f(x) =2x +1, g(x) =cos(x) Notations

Remark1: the open interval(a,b) ={x :a<x <b} the closed interval[a,b] ={x :a≤x ≤b}

[a,b),(a,b],[a,∞),(−∞,a],(a,∞),(−∞,a) R={x :−∞<x <∞}

WEN-CHINGLIEN Calculus (I)

Definition

f and g are equal ⇐⇒(1) f and g have the same domain (2) f(x) =g(x)for all x

Definition

f(x) = f(−x): even.

f(x) = −f(−x): odd.

Example 1. f(x) = |x|, g(x) =sin(x), 2. f(x) =2x +1, g(x) =cos(x) Notations

Remark1: the open interval(a,b) ={x :a<x <b} the closed interval[a,b] ={x :a≤x ≤b}

[a,b),(a,b],[a,∞),(−∞,a],(a,∞),(−∞,a) R={x :−∞<x <∞}

WEN-CHINGLIEN Calculus (I)

Definition

f and g are equal ⇐⇒(1) f and g have the same domain (2) f(x) =g(x)for all x

Definition

f(x) = f(−x): even.

f(x) = −f(−x): odd.

Example 1. f(x) = |x|, g(x) =sin(x), 2. f(x) =2x +1, g(x) =cos(x) Notations

Remark1: the open interval(a,b) ={x :a<x <b} the closed interval[a,b] ={x :a≤x ≤b}

[a,b),(a,b],[a,∞),(−∞,a],(a,∞),(−∞,a) R={x :−∞<x <∞}

WEN-CHINGLIEN Calculus (I)

Definition

f and g are equal ⇐⇒(1) f and g have the same domain (2) f(x) =g(x)for all x

Definition

f(x) = f(−x): even.

f(x) = −f(−x): odd.

Example 1. f(x) = |x|,g(x) =sin(x), 2. f(x) =2x +1, g(x) =cos(x) Notations

Remark1: the open interval(a,b) ={x :a<x <b} the closed interval[a,b] ={x :a≤x ≤b}

[a,b),(a,b],[a,∞),(−∞,a],(a,∞),(−∞,a) R={x :−∞<x <∞}

WEN-CHINGLIEN Calculus (I)

Definition

f and g are equal ⇐⇒(1) f and g have the same domain (2) f(x) =g(x)for all x

Definition

f(x) = f(−x): even.

f(x) = −f(−x): odd.

Example 1. f(x) = |x|, g(x) =sin(x), 2. f(x) =2x +1, g(x) =cos(x) Notations

Remark1: the open interval(a,b) ={x :a<x <b} the closed interval[a,b] ={x :a≤x ≤b}

[a,b),(a,b],[a,∞),(−∞,a],(a,∞),(−∞,a) R={x :−∞<x <∞}

WEN-CHINGLIEN Calculus (I)

Definition

f and g are equal ⇐⇒(1) f and g have the same domain (2) f(x) =g(x)for all x

Definition

f(x) = f(−x): even.

f(x) = −f(−x): odd.

Example 1. f(x) = |x|, g(x) =sin(x), 2. f(x) =2x +1,g(x) =cos(x) Notations

Remark1: the open interval(a,b) ={x :a<x <b} the closed interval[a,b] ={x :a≤x ≤b}

[a,b),(a,b],[a,∞),(−∞,a],(a,∞),(−∞,a) R={x :−∞<x <∞}

WEN-CHINGLIEN Calculus (I)

Definition

f and g are equal ⇐⇒(1) f and g have the same domain (2) f(x) =g(x)for all x

Definition

f(x) = f(−x): even.

f(x) = −f(−x): odd.

Example 1. f(x) = |x|, g(x) =sin(x), 2. f(x) =2x +1, g(x) =cos(x) Notations

Remark1: the open interval(a,b) ={x :a<x <b} the closed interval[a,b] ={x :a≤x ≤b}

[a,b),(a,b],[a,∞),(−∞,a],(a,∞),(−∞,a) R={x :−∞<x <∞}

WEN-CHINGLIEN Calculus (I)

Definition

f and g are equal ⇐⇒(1) f and g have the same domain (2) f(x) =g(x)for all x

Definition

f(x) = f(−x): even.

f(x) = −f(−x): odd.

Example 1. f(x) = |x|, g(x) =sin(x), 2. f(x) =2x +1, g(x) =cos(x) Notations

Remark1: the open interval(a,b) ={x :a<x <b} the closed interval[a,b] ={x :a≤x ≤b}

[a,b),(a,b],[a,∞),(−∞,a],(a,∞),(−∞,a) R={x :−∞<x <∞}

WEN-CHINGLIEN Calculus (I)

Definition

f and g are equal ⇐⇒(1) f and g have the same domain (2) f(x) =g(x)for all x

Definition

f(x) = f(−x): even.

f(x) = −f(−x): odd.

Example 1. f(x) = |x|, g(x) =sin(x), 2. f(x) =2x +1, g(x) =cos(x) Notations

Remark1: the open interval(a,b) ={x :a<x <b} the closed interval[a,b] ={x :a≤x ≤b}

[a,b),(a,b],[a,∞),(−∞,a],(a,∞),(−∞,a) R={x :−∞<x <∞}

WEN-CHINGLIEN Calculus (I)

Definition

f and g are equal ⇐⇒(1) f and g have the same domain (2) f(x) =g(x)for all x

Definition

f(x) = f(−x): even.

f(x) = −f(−x): odd.

Example 1. f(x) = |x|, g(x) =sin(x), 2. f(x) =2x +1, g(x) =cos(x) Notations

Remark1: the open interval(a,b) ={x :a<x <b} the closed interval[a,b] ={x :a≤x ≤b}

[a,b),(a,b],[a,∞),(−∞,a],(a,∞),(−∞,a) R={x :−∞<x <∞}

WEN-CHINGLIEN Calculus (I)

Definition

f and g are equal ⇐⇒(1) f and g have the same domain (2) f(x) =g(x)for all x

Definition

f(x) = f(−x): even.

f(x) = −f(−x): odd.

Example 1. f(x) = |x|, g(x) =sin(x), 2. f(x) =2x +1, g(x) =cos(x) Notations

Remark1: the open interval(a,b) ={x :a<x <b} the closed interval[a,b] ={x :a≤x ≤b}

[a,b),(a,b],[a,∞),(−∞,a],(a,∞),(−∞,a) R={x :−∞<x <∞}

WEN-CHINGLIEN Calculus (I)

2. Introduction to Elementary Functions

Polynomial Functions

f(x) =a0+a1x + +a2x²+a3x³+· · ·+anxⁿ n : integer≥0

if an6=0 , n=the degree of the polynomial function Rational Functions

f(x) = p(x)

q(x),q(x)6=0,p,q :polynomial Ex1: y = _x−2^x ,x 6=2

WEN-CHINGLIEN Calculus (I)

2. Introduction to Elementary Functions

Polynomial Functions

f(x) =a0+a1x + +a2x²+a3x³+· · ·+anxⁿ n : integer≥0

if an6=0 , n=the degree of the polynomial function Rational Functions

f(x) = p(x)

q(x),q(x)6=0,p,q :polynomial Ex1: y = _x−2^x ,x 6=2

WEN-CHINGLIEN Calculus (I)

2. Introduction to Elementary Functions

Polynomial Functions

f(x) =a0+a1x + +a2x²+a3x³+· · ·+anxⁿ n : integer≥0

if an6=0 , n=the degree of the polynomial function Rational Functions

f(x) = p(x)

q(x),q(x)6=0,p,q :polynomial Ex1: y = _x−2^x ,x 6=2

WEN-CHINGLIEN Calculus (I)

Ex2: y = 1

x²−1,x 6=1,−1 Ex3: Prove that √

x is not a rational function.

WEN-CHINGLIEN Calculus (I)

Ex2: y = 1

x²−1,x 6=1,−1 Ex3: Prove that √

x is not a rational function.

WEN-CHINGLIEN Calculus (I)

Exponential Functions

f(x) =a^x,x ∈R,a:base, >0,6=1 Example1: f(x) = e^x

e : the natural exponential base , e=2.718· · · Application: Radioactive Decay—

Carbon (C¹⁴)

- to determine the absolute age of fossils - radioactive, decay into N¹⁴

the amount of C¹⁴ at time t=W(t) t =0:W(0) =w0

According to the law of physics

W(t) = w0e^−λt,t ≥0, λ:the decay rate

WEN-CHINGLIEN Calculus (I)

Exponential Functions

f(x) =a^x,x ∈R,a:base, >0,6=1 Example1: f(x) = e^x

e : the natural exponential base , e=2.718· · · Application: Radioactive Decay—

Carbon (C¹⁴)

- to determine the absolute age of fossils - radioactive, decay into N¹⁴

the amount of C¹⁴ at time t=W(t) t =0:W(0) =w0

According to the law of physics

W(t) = w0e^−λt,t ≥0, λ:the decay rate

WEN-CHINGLIEN Calculus (I)

Exponential Functions

f(x) =a^x,x ∈R,a:base, >0,6=1 Example1: f(x) = e^x

e : the natural exponential base , e=2.718· · · Application: Radioactive Decay—

Carbon (C¹⁴)

- to determine the absolute age of fossils - radioactive, decay into N¹⁴

the amount of C¹⁴ at time t=W(t) t =0:W(0) =w0

According to the law of physics

W(t) = w0e^−λt,t ≥0, λ:the decay rate

WEN-CHINGLIEN Calculus (I)

Exponential Functions

f(x) =a^x,x ∈R,a:base, >0,6=1 Example1: f(x) = e^x

e : the natural exponential base , e=2.718· · · Application: Radioactive Decay—

Carbon (C¹⁴)

- to determine the absolute age of fossils - radioactive, decay into N¹⁴

the amount of C¹⁴ at time t=W(t) t =0:W(0) =w0

According to the law of physics

W(t) = w0e^−λt,t ≥0, λ:the decay rate

WEN-CHINGLIEN Calculus (I)

Exponential Functions

f(x) =a^x,x ∈R,a:base, >0,6=1 Example1: f(x) = e^x

e : the natural exponential base , e=2.718· · · Application: Radioactive Decay—

Carbon (C¹⁴)

- to determine the absolute age of fossils - radioactive, decay into N¹⁴

the amount of C¹⁴ at time t=W(t) t =0:W(0) =w0

According to the law of physics

W(t) = w0e^−λt,t ≥0, λ:the decay rate

WEN-CHINGLIEN Calculus (I)

Exponential Functions

f(x) =a^x,x ∈R,a:base, >0,6=1 Example1: f(x) = e^x

e : the natural exponential base , e=2.718· · · Application: Radioactive Decay—

Carbon (C¹⁴)

- to determine the absolute age of fossils - radioactive, decay into N¹⁴

the amount of C¹⁴ at time t=W(t) t =0:W(0) =w0

According to the law of physics

W(t) = w0e^−λt,t ≥0, λ:the decay rate

WEN-CHINGLIEN Calculus (I)

Exponential Functions

f(x) =a^x,x ∈R,a:base, >0,6=1 Example1: f(x) = e^x

e : the natural exponential base , e=2.718· · · Application: Radioactive Decay—

Carbon (C¹⁴)

- to determine the absolute age of fossils - radioactive, decay into N¹⁴

the amount of C¹⁴ at time t=W(t) t =0:W(0) =w0

According to the law of physics

W(t) = w0e^−λt,t ≥0, λ:the decay rate

WEN-CHINGLIEN Calculus (I)

Def: Half-life(Th) of the material = the length of time that it takes for half of the material to decay.

W(Th) = ¹₂w0=w0e^−λTh

⇒2=e^λTh

⇒ln 2=λTh

⇒Th = ^{ln 2}λ (Th=5730 years for C¹⁴)

WEN-CHINGLIEN Calculus (I)

Def: Half-life(Th) of the material = the length of time that it takes for half of the material to decay.

W(Th) = ¹₂w0=w0e^−λTh

⇒2=e^λTh

⇒ln 2=λTh

⇒Th = ^{ln 2}λ (Th=5730 years for C¹⁴)

WEN-CHINGLIEN Calculus (I)

Def: Half-life(Th) of the material = the length of time that it takes for half of the material to decay.

W(Th) = ¹₂w0=w0e^−λTh

⇒2=e^λTh

⇒ln 2=λTh

⇒Th = ^{ln 2}λ (Th=5730 years for C¹⁴)

WEN-CHINGLIEN Calculus (I)

Def: Half-life(Th) of the material = the length of time that it takes for half of the material to decay.

W(Th) = ¹₂w0=w0e^−λTh

⇒2=e^λTh

⇒ln 2=λTh

⇒Th = ^{ln 2}λ (Th=5730 years for C¹⁴)

WEN-CHINGLIEN Calculus (I)

Def: Half-life(Th) of the material = the length of time that it takes for half of the material to decay.

W(Th) = ¹₂w0=w0e^−λTh

⇒2=e^λTh

⇒ln 2=λTh

⇒Th = ^{ln 2}λ (Th=5730 years for C¹⁴)

WEN-CHINGLIEN Calculus (I)

Def: Half-life(Th) of the material = the length of time that it takes for half of the material to decay.

W(Th) = ¹₂w0=w0e^−λTh

⇒2=e^λTh

⇒ln 2=λTh

⇒Th = ^{ln 2}λ (Th=5730 years for C¹⁴)

WEN-CHINGLIEN Calculus (I)

Inverse Functions Definition

f(x)is a one-to-one function if

x16=x2⇒f(x1)6=f(x2) orf(x1) =f(x2)⇒x1=x2

Horizontal line test Ex: f(x) =x²,x³, . . .

WEN-CHINGLIEN Calculus (I)

Inverse Functions Definition

f(x)is a one-to-one function if

x16=x2⇒f(x1)6=f(x2) orf(x1) =f(x2)⇒x1=x2

Horizontal line test Ex: f(x) =x²,x³, . . .

WEN-CHINGLIEN Calculus (I)

Inverse Functions Definition

f(x)is a one-to-one function if

x16=x2⇒f(x1)6=f(x2) orf(x1) =f(x2)⇒x1=x2

Horizontal line test Ex: f(x) =x²,x³, . . .

WEN-CHINGLIEN Calculus (I)

Inverse Functions Definition

f(x)is a one-to-one function if

x16=x2⇒f(x1)6=f(x2) orf(x1) =f(x2)⇒x1=x2

Horizontal line test Ex: f(x) =x²,x³, . . .

WEN-CHINGLIEN Calculus (I)

Definition

Let f be a one-to-one function f :A−→B The inverse function f⁻¹is defined as

f⁻¹(y) =x ify =f(x) which has domain f(A)and range A Ex: f(x) =x²+1,x ≥0,⇒f⁻¹=?

Ex: f(x) =e^x,⇒f⁻¹(x)≡ln x Ex: f(x) =a^x,⇒f⁻¹(x)≡log_ax

WEN-CHINGLIEN Calculus (I)

Definition

Let f be a one-to-one function f :A−→B The inverse function f⁻¹is defined as

f⁻¹(y) =x ify =f(x) which has domain f(A)and range A Ex: f(x) =x²+1,x ≥0,⇒f⁻¹=?

Ex: f(x) =e^x,⇒f⁻¹(x)≡ln x Ex: f(x) =a^x,⇒f⁻¹(x)≡log_ax

WEN-CHINGLIEN Calculus (I)

Definition

Let f be a one-to-one function f :A−→B The inverse function f⁻¹is defined as

f⁻¹(y) =x ify =f(x) which has domain f(A)and range A Ex: f(x) =x²+1,x ≥0,⇒f⁻¹=?

Ex: f(x) =e^x,⇒f⁻¹(x)≡ln x Ex: f(x) =a^x,⇒f⁻¹(x)≡log_ax

WEN-CHINGLIEN Calculus (I)

Definition

Let f be a one-to-one function f :A−→B The inverse function f⁻¹is defined as

f⁻¹(y) =x ify =f(x) which has domain f(A)and range A Ex: f(x) =x²+1,x ≥0,⇒f⁻¹=?

Ex: f(x) =e^x,⇒f⁻¹(x)≡ln x Ex: f(x) =a^x,⇒f⁻¹(x)≡log_ax

WEN-CHINGLIEN Calculus (I)

Definition

Let f be a one-to-one function f :A−→B The inverse function f⁻¹is defined as

f⁻¹(y) =x ify =f(x) which has domain f(A)and range A Ex: f(x) =x²+1,x ≥0,⇒f⁻¹=?

Ex: f(x) =e^x,⇒f⁻¹(x)≡ln x Ex: f(x) =a^x,⇒f⁻¹(x)≡log_ax

WEN-CHINGLIEN Calculus (I)

Applications

Radioactive isotopes such as Carbon 14(C¹⁴) is used to determine the age of fossils or minerals.

Ex1:

Suppose that samples of wood found in an archeological excavation site contain about 23%as much C¹⁴ as living plant material. Determine when the wood was cut.

0.23= W(t) w0

=e^−λt

⇒λt =ln(0.23)⁻¹

λ = ln 2

5730 forC¹⁴

⇒t = 5730

ln 2 ln(0.23)⁻¹≈12150years ago.

WEN-CHINGLIEN Calculus (I)

Applications

Radioactive isotopes such as Carbon 14(C¹⁴) is used to determine the age of fossils or minerals.

Ex1:

Suppose that samples of wood found in an archeological excavation site contain about 23%as much C¹⁴ as living plant material. Determine when the wood was cut.

0.23= W(t) w0

=e^−λt

⇒λt =ln(0.23)⁻¹

λ = ln 2

5730 forC¹⁴

⇒t = 5730

ln 2 ln(0.23)⁻¹≈12150years ago.

WEN-CHINGLIEN Calculus (I)

Applications

Radioactive isotopes such as Carbon 14(C¹⁴) is used to determine the age of fossils or minerals.

Ex1:

Suppose that samples of wood found in an archeological excavation site contain about 23%as much C¹⁴ as living plant material. Determine when the wood was cut.

0.23= W(t) w0

=e^−λt

⇒λt =ln(0.23)⁻¹

λ = ln 2

5730 forC¹⁴

⇒t = 5730

ln 2 ln(0.23)⁻¹≈12150years ago.

WEN-CHINGLIEN Calculus (I)

Applications

Radioactive isotopes such as Carbon 14(C¹⁴) is used to determine the age of fossils or minerals.

Ex1:

Suppose that samples of wood found in an archeological excavation site contain about 23%as much C¹⁴ as living plant material. Determine when the wood was cut.

0.23= W(t) w0

=e^−λt

⇒λt =ln(0.23)⁻¹

λ = ln 2

5730 forC¹⁴

⇒t = 5730

ln 2 ln(0.23)⁻¹≈12150years ago.

WEN-CHINGLIEN Calculus (I)

Applications

Radioactive isotopes such as Carbon 14(C¹⁴) is used to determine the age of fossils or minerals.

Ex1:

Suppose that samples of wood found in an archeological excavation site contain about 23%as much C¹⁴ as living plant material. Determine when the wood was cut.

0.23= W(t) w0

=e^−λt

⇒λt =ln(0.23)⁻¹

λ = ln 2

5730 forC¹⁴

⇒t = 5730

ln 2 ln(0.23)⁻¹≈12150years ago.

WEN-CHINGLIEN Calculus (I)

Applications

Radioactive isotopes such as Carbon 14(C¹⁴) is used to determine the age of fossils or minerals.

Ex1:

Suppose that samples of wood found in an archeological excavation site contain about 23%as much C¹⁴ as living plant material. Determine when the wood was cut.

0.23= W(t) w0

=e^−λt

⇒λt =ln(0.23)⁻¹

λ = ln 2

5730 forC¹⁴

⇒t = 5730

ln 2 ln(0.23)⁻¹≈12150years ago.

WEN-CHINGLIEN Calculus (I)

Applications

Radioactive isotopes such as Carbon 14(C¹⁴) is used to determine the age of fossils or minerals.

Ex1:

Suppose that samples of wood found in an archeological excavation site contain about 23%as much C¹⁴ as living plant material. Determine when the wood was cut.

0.23= W(t) w0

=e^−λt

⇒λt =ln(0.23)⁻¹

λ = ln 2

5730 forC¹⁴

⇒t = 5730

ln 2 ln(0.23)⁻¹≈12150years ago.

WEN-CHINGLIEN Calculus (I)

Thank you.

WEN-CHINGLIEN Calculus (I)