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# 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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## 14.1 Elementary Examples

Definition (1)

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

f :DR

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

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

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## 14.1 Elementary Examples

Definition (1)

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

f :DR

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

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

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## 14.1 Elementary Examples

Definition (1)

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

f :DR

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

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

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## 14.1 Elementary Examples

Definition (1)

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

f :DR

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

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

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## 14.1 Elementary Examples

Definition (1)

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

f :DR

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

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

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## 14.1 Elementary Examples

Definition (1)

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

f :DR

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

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

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## 14.1 Elementary Examples

Definition (1)

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

f :DR

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

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

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Definition (2)

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

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Definition (2)

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

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Definition (3) f :DR2R

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

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Definition (3) f :DR2R

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

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Definition (3) f :DR2R

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

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### 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?

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### 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?

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### 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?

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### 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?

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### 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?

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### Example 2:

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

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### Example 2:

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

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### Example 2:

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

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### Example 3:

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

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### Example 3:

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

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## Thank you.

Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung

W EN -C HING L IEN Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung