# 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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## References

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