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Computing Values of Function and Derivatives using Taylor Series: sin(x) and cos(x) - Prof, Assignments of Mathematics

Boston College (BC)Mathematics

Prof. Daniel W. Chambers

The computation of values of the functions sin(x) and cos(x) along with their first five derivatives using taylor series. The problems involve finding the values of the function and its derivatives at specific points.

Typology: Assignments

Pre 2010

Uploaded on 08/31/2009

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888 Chapter 10 Taylor Series

Section 10.3 Taylor Polynomials

Problems for Practice

1. We compute the values of the function and its first five derivatives as follows:

()

3

4

3

1

32

3

sin

32

1

cos

32

3

sin

32

1

cos

32

x

f

fx

π

=

!"

=

#$

%&

!"

′=−=−

#$

%&

!"

′′ =−=−

#$

%&

!"

′′′ ==

#$

%&

!"

==

#$

%&

() () ()

24

3

4

33

13 3

2 2 3 4 12 3 48

xx

Px x x

ππ

!" !"

−−

!" !"

=− − − +−+

#$ #$

%& %&

%& %& .

2. We compute the values of the function and its first five derivatives as follows:

()

2

4

2

5

2

1

2

cos 0

2

sin 1

2

cos 0

2

sin 1

2

cos 0

2

x

f

fx

π

=

!"

=

#$

%&

!"

′==

#$

%&

!"

′′ =−=−

#$

%&

!"

′′′ =−=

#$

%&

!"

==

#$

%&

!"

==

#$

%&

() ()()

24

5

22

12 24

xx

Px

ππ

−−

=−+.

888 Chapter 10 Taylor Series

Section 10.3 Taylor Polynomials

Problems for Practice

1. We compute the values of the function and its first five derivatives as follows:

()

3

4

3

1

32

3

sin

32

1

cos

32

3

sin

32

1

cos

32

x

f

fx

π

=

!"

=

#$

%&

!"

′=−=−

#$

%&

!"

′′ =−=−

#$

%&

!"

′′′ ==

#$

%&

!"

==

#$

%&

() () ()

24

3

4

33

13 3

2 2 3 4 12 3 48

xx

Px x x

ππ

!" !"

−−

!" !"

=− − − +−+

#$ #$

%& %&

%& %& .

2. We compute the values of the function and its first five derivatives as follows:

()

2

4

2

5

2

1

2

cos 0

2

sin 1

2

cos 0

2

sin 1

2

cos 0

2

x

f

fx

π

=

!"

=

#$

%&

!"

′==

#$

%&

!"

′′ =−=−

#$

%&

!"

′′′ =−=

#$

%&

!"

==

#$

%&

!"

==

#$

%&

() ()()

24

5

22

12 24

xx

Px

ππ

−−

=−+.

Partial preview of the text

Download Computing Values of Function and Derivatives using Taylor Series: sin(x) and cos(x) - Prof and more Assignments Mathematics in PDF only on Docsity!

888 Chapter 10 Taylor Series

Section 10.3 Taylor Polynomials

Problems for Practice

1. We compute the values of the function and its first five derivatives as follows:

( )

( ) ( )

3

4

3

sin 3 2

cos 3 2

sin 3 2

cos 3 2

x

f

f x

π

=

# $=^ =

( )

( ) ( )

2 3 4

4

x x P x x x

! " (^)! π" − π! "! π" −π = − (^) # $ (^) # − (^) $ − + (^) # $# − (^) $ +

% & %^ &^ % &%^ &

2. We compute the values of the function and its first five derivatives as follows:

( )

( ) ( )

2 2 2 4 2 5 2

cos 0 2

sin 1 2

cos 0 2

sin 1 2

cos 0 2

x

f

f x

π

π π π π π π

=

# $=^ =

( )

( ) ( )

2 4

5

x x P x

( )

( ) ( )

3

4

3

sin 3 2

cos 3 2

x

f x

π

=

# $=^ =

( )

( ) ( )

2 3 4

4

x x P x x x

! " (^)! π" − π! "! π" −π = − (^) # $ (^) # − (^) $ − + (^) # $# − (^) $ +

% & %^ &^ % &%^ &

2. We compute the values of the function and its first five derivatives as follows:

( )

( ) ( )

2 2 2 4 2 5 2

cos 0 2

sin 1 2

cos 0 2

sin 1 2

cos 0 2

x

f

f x

π

π π π π π π

=

# $=^ =

( )

( ) ( )

2 4

5

x x P x

( )

4

0

5

0

0 16cos 2 16

0 32sin 2 0

x

f x

=

2 4 P 5 (^) x = − 1 + 2 x − 2 x 3.

4. We compute the values of the function and its first six derivatives as follows:

( )

4 5 6

1

x

f e

f f f f f f e e

2 3 4 5 6

6

x x x x x P x e e x e e e e e

5. We compute the values of the function and its first five derivatives as follows:

( )

0

4 5

0

x x x x x x

f

f x e

f f f e

=

2 3 4 5 5

P x x x x x x

890 Chapter 10 Taylor Series

6. We compute the values of the function and its first five derivatives as follows:

( )

0 0 0 4 0 5 0

0 3cos 3 3

0 9sin 3 0

0 27 cos 3 27

0 81sin 3 0

0 243cos 3 243

x

f

f x

=

3 5

5

x x P x = x − +.

7. We compute the values of the function and its first five derivatives as follows:

4 2

5 2

x

f

f x

=

Computing Values of Function and Derivatives using Taylor Series: sin(x) and cos(x) - Prof, Assignments of Mathematics

Related documents

Partial preview of the text

Download Computing Values of Function and Derivatives using Taylor Series: sin(x) and cos(x) - Prof and more Assignments Mathematics in PDF only on Docsity!

Section 10.3 Taylor Polynomials

# $=^ =

# $=^ =

# $=^ =

# $=^ =

# $=^ =

# $=^ =

f f f f f f e e