2008 W -C L Calculus(I)

Calculus (I)

W^EN-C^HING L^IEN

Department of Mathematics National Cheng Kung University

2008

14.1 Elementary Examples

Definition (1)

D ⊂Rⁿ.A real-valued function f on D assigns a real numbers to each element in D

f :D→R

(x1, . . . ,xn) 7−→f(x1, . . . ,xn) D: the domain

{f(x1, . . . ,xn) : (x1, . . . ,xn) ∈D}: the range of f

14.1 Elementary Examples

Definition (1)

D ⊂Rⁿ.A real-valued function f on D assigns a real numbers to each element in D

f :D→R

(x1, . . . ,xn) 7−→f(x1, . . . ,xn) D: the domain

{f(x1, . . . ,xn) : (x1, . . . ,xn) ∈D}: the range of f

14.1 Elementary Examples

Definition (1)

D ⊂Rⁿ.A real-valued function f on D assigns a real numbers to each element in D

f :D→R

(x1, . . . ,xn) 7−→f(x1, . . . ,xn) D: the domain

{f(x1, . . . ,xn) : (x1, . . . ,xn) ∈D}: the range of f

14.1 Elementary Examples

Definition (1)

D ⊂Rⁿ.A real-valued function f on D assigns a real numbers to each element in D

f :D→R

(x1, . . . ,xn) 7−→f(x1, . . . ,xn) D: the domain

{f(x1, . . . ,xn) : (x1, . . . ,xn) ∈D}: the range of f

14.1 Elementary Examples

Definition (1)

D ⊂Rⁿ.A real-valued function f on D assigns a real numbers to each element in D

f :D→R

(x1, . . . ,xn) 7−→f(x1, . . . ,xn) D: the domain

{f(x1, . . . ,xn) : (x1, . . . ,xn) ∈D}: the range of f

14.1 Elementary Examples

Definition (1)

D ⊂Rⁿ.A real-valued function f on D assigns a real numbers to each element in D

f :D→R

(x1, . . . ,xn) 7−→f(x1, . . . ,xn) D: the domain

{f(x1, . . . ,xn) : (x1, . . . ,xn) ∈D}: the range of f

14.1 Elementary Examples

Definition (1)

D ⊂Rⁿ.A real-valued function f on D assigns a real numbers to each element in D

f :D→R

(x1, . . . ,xn) 7−→f(x1, . . . ,xn) D: the domain

{f(x1, . . . ,xn) : (x1, . . . ,xn) ∈D}: the range of f

Definition (2)

S = {(x,y,z) : z =f(x,y), (x,y) ∈D}: the graph of f

Definition (2)

S = {(x,y,z) : z =f(x,y), (x,y) ∈D}: the graph of f

Definition (3) f :D ⊂R² →R

The level curve of f ={(x,y) :f(x,y) =c}, c is a given constant.

Definition (3) f :D ⊂R² →R

The level curve of f ={(x,y) :f(x,y) =c}, c is a given constant.

Definition (3) f :D ⊂R² →R

The level curve of f ={(x,y) :f(x,y) =c}, c is a given constant.

Example 1:

1 f(x,y) = x²+y² 2 f(x,y) = e⁻(^x2+y2) 3 f(x,y) = sin x·sin y

domain? range?

Example 1:

1 f(x,y) = x²+y² 2 f(x,y) = e⁻(^x2+y2) 3 f(x,y) = sin x·sin y

domain? range?

Example 1:

1 f(x,y) = x²+y² 2 f(x,y) = e⁻(^x2+y2) 3 f(x,y) = sin x·sin y

domain? range?

Example 1:

1 f(x,y) = x²+y² 2 f(x,y) = e⁻(^x2+y2) 3 f(x,y) = sin x·sin y

domain? range?

Example 1:

1 f(x,y) = x²+y² 2 f(x,y) = e⁻(^x2+y2) 3 f(x,y) = sin x·sin y

domain? range?

Example 2:

1 the graph of f(x,y) =x²+y² 2 the graph of f(x,y) =e⁻(^x2+y2)

Example 2:

1 the graph of f(x,y) =x²+y² 2 the graph of f(x,y) =e⁻(^x2+y2)

Example 2:

1 the graph of f(x,y) =x²+y² 2 the graph of f(x,y) =e⁻(^x2+y2)

Example 3:

Determine the equation of the level curve f =c with the possible values of c

Example 3:

Determine the equation of the level curve f =c with the possible values of c

Thank you.