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Feg 1034 Calculus & Analysis 1

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Autor:   •  December 6, 2017  •  Research Paper  •  1,195 Words (5 Pages)  •  10 Views

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FEG 1034        Calculus 1        Chapter 02[pic 1]

[pic 2]

Chapter 2

Function and limit

[pic 3]

FEG 1034 Calculus & Analysis 1


FEG 1034        Calculus 1        Chapter 05[pic 4]

[pic 5]

What thing will come out in Final

[pic 6]

2.1 Limit

  • Vertical Asymptotes

  • Horizontal Asymptotes

2.2 Inverse Function

  • Characteristic of inverse function

  • Horizontal test
  • Second derivative test

2.3 Derivative of inverse function

FEG 1034 Calculus & Analysis 1        2


FEG 1034        Calculus 1        Chapter 02[pic 7]

[pic 8]

2.1 Limit

[pic 9]

= 3−1 and        =  2 +        + 1

[pic 10][pic 11][pic 12]

 −1

[pic 13]

Both graph are the same, but just that can’t include = 1, which is not continuous when = 1.

[pic 14]

FEG 1034 Calculus & Analysis 1        3


FEG 1034        Calculus 1        Chapter 02[pic 15]

[pic 16]

2.1.1 Vertical Asymptotes

[pic 17]

How to determine the vertical asymptotes?

  • Check the existence of denominator (must be simplify)

  • Check the value to make denominator become zero, hence make the function become undefined.

For this example, the vertical asymptotes is in        = 2.

FEG 1034 Calculus & Analysis 1        4


FEG 1034        Calculus 1        Chapter 02[pic 18]

[pic 19]

Example 1 – find vertical asymptotes

Find the vertical asymptotes for the functions.

a).        = 1

b).        = 1

[pic 20]

FEG 1034 Calculus & Analysis 1        5


FEG 1034        Calculus 1        Chapter 02[pic 21]

[pic 22]

2.1.2 Horizontal Asymptotes

[pic 23]

How to determine the horizontal asymptotes?

  • If the limit at infinite is defined, then there is the horizontal asymptotes.

=

For this example,

3

lim

= −0

3

= 0

 →−∞  −2

−∞

lim        3        = 0

[pic 24]

 →+∞  −2


3

[pic 25][pic 26][pic 27][pic 28][pic 29]

+∞ = 0


horizontal asymptotes

at        =  .

FEG 1034 Calculus & Analysis 1        6


FEG 1034        Calculus 1        Chapter 02[pic 30]

[pic 31]

Example 2 – find horizontal asymptotes

[pic 32]

Find the horizontal asymptotes for the functions.

2 + 1

2

1

1

=

= 1 +

= 1 −

+

2 − 1

2 − 1

+ 1

− 1

=

lim

2

+ 1

= 1 +

lim

1

+

1

= 1

− 1

+ 1

− 1

 →−∞  2

 →−∞

lim

2

+ 1

= 1 +

lim

1

+

1

= 1

− 1

+ 1

− 1

 →+∞  2

 →+∞

[pic 33][pic 34]

...

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