International Journal of Research in Information Technology (IJRIT)

www.ijrit.com

ISSN 2001-5569

The Derivatives of Some Functions 1

1

Chii-Huei Yu

Assistant Professor, Department of Management and Information, Nan Jeon University of Science and Technology, Tainan City, Taiwan 1

[email protected]

Abstract This study uses the mathematical software Maple for the auxiliary tool to evaluate the derivatives of two types of functions. We can obtain the Fourier series expansions of any order derivatives of these two types of functions by using geometric series and differentiation term by term theorem. In addition, we provide two examples to do calculation practically. The research methods adopted in this study involved finding solutions through manual calculations and verifying these solutions by using Maple. This type of research method not only allows the discovery of calculation errors, but also helps modify the original directions of thinking from manual and Maple calculations. For this reason, Maple provides insights and guidance regarding problem-solving methods.

Keywords: derivatives, Fourier series expansions, geometric series, differentiation term by term theorem, Maple.

1. Introduction As information technology advances, whether computers can become comparable with human brains to perform abstract tasks, such as abstract art similar to the paintings of Picasso and musical compositions similar to those of Beethoven, is a natural question. Currently, this appears unattainable. In addition, whether computers can solve abstract and difficult mathematical problems and develop abstract mathematical theories such as those of mathematicians also appears unfeasible. Nevertheless, in seeking for alternatives, we can study what assistance mathematical software can provide. This study introduces how to conduct mathematical research using the mathematical software Maple. The main reasons of using Maple in this study are its simple instructions and ease of use, which enable beginners to learn the operating techniques in a short period. By employing the powerful computing capabilities of Maple, difficult problems can be easily solved. Even when Maple cannot determine the solution, problem-solving hints can be identified and inferred from the approximate values calculated and solutions to similar problems, as determined by Maple. For this reason, Maple can provide insights into scientific research. Inquiring through an online support system provided by Maple or browsing the Maple website (www.maplesoft.com) can facilitate further understanding of Maple and might provide unexpected insights. For the instructions and operations of Maple, [1-7] can be adopted as references.

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In calculus and engineering mathematics curricula, finding f ( n ) ( c) ( the n -th order derivative value of function f ( x ) at x = c ), in general, necessary goes through two procedures: Firstly determining f ( n ) ( x ) ( the n -th order derivative of f ( x ) ), and secondly substituting x = c to f ( n ) ( x ) . When evaluating the higher order derivative values of a function (i.e. n is large), these two procedures will make us face with increasingly complex calculations. Therefore, to obtain the answers through manual calculations is not an easy thing. In this paper, we mainly study the differential problem of the following two types of functions

f ( x ) = tan −1[b tan( λx + β )]

(1)

g ( x ) = tan −1[b cot( λ x + β )]

(2)

where b, λ , β are real numbers, b ≠ 0 . We can obtain the Fourier series expansions of any order derivatives of these two types of functions by using geometric series and differentiation term by term theorem ; these are the major results in this study (i.e., Theorems 1, 2), and hence greatly reduce the difficulty of calculating their higher order derivative values. As for the study of related differential problems can refer to [8-16]. On the other hand, we propose two functions to determine there any order derivatives and some higher order derivative values practically. The research methods adopted in this study involved finding solutions through manual calculations and verifying these solutions by using Maple. This type of research method not only allows the discovery of calculation errors, but also helps modify the original directions of thinking from manual and Maple calculations. For this reason, Maple provides insights and guidance regarding problem-solving methods.

2. Main Results Firstly, we introduce a notation and two formulas used in this study.

2.1 Notation. Let z = a + ib be a complex number, where i = − 1 , a, b are real numbers. We denote a the real part of z by Re( z ) , and b the imaginary part of z by Im( z ) .

2.2 Euler's formula.

e iy = cos y + i sin y , where y is any real number. 2.3 DeMoivre's formula.

(cos y + i sin y ) n = cos ny + i sin ny , where n is any integer, y is any real number. Next, we introduce two important theorems used in this paper.

2.4 Geometric series. ∞ 1 = ∑ ( −u ) k , where u is a complex number, u < 1 . 1 + u k =0

2.5 Differentiation term by term theorem ([17]). If, for all non-negative integer k , the functions g k : ( a, b) → R satisfy the following three conditions：(i) there exists a point x0 ∈ ( a, b) such that interval (a , b) , (iii)

∞

∞

∑ g k ( x0 )

k =0

is convergent, (ii) all functions g k (x ) are differentiable on open ∞

d ∑ dx g k ( x ) is uniformly convergent on (a, b) . Then ∑ g k ( x ) is uniformly convergent and k =0 k =0

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differentiable on (a , b) . Moreover, its derivative

∞ d d ∞ gk ( x) = ∑ gk ( x) . ∑ dx k =0 k =0 dx

We define tan −1 ( ∞) = π / 2 and tan −1 ( −∞) = −π / 2 . The following is the first major result in this study, we obtain the Fourier series expansions of any order derivatives of the function (1).

2.6 Theorem 1. Suppose b, λ , β are real numbers, b ≠ 0 , and n is a positive integer, n ≥ 2 . Let the domain of

f ( x ) = tan −1[b tan( λx + β )] be ( −∞, ∞) . Case (1). If b > 0 , then the n -th order derivative of f ( x ) , ∞ nπ 1− b f ( n ) ( x ) = (2λ ) n ∑ k n −1 − sin2k (λx + β ) + 2 1+ b k =1

(3)

∞ nπ 1+ b f ( n ) ( x ) = −(2λ )n ∑ k n −1 − sin2k (λx + β ) + 2 1− b k =1

(4)

k

for all x ∈ R . Case (2). If b < 0 , then

k

for all x ∈ R .

2.6.1 Proof. Case (1). If b > 0 , the case of b = 1 is trivial, so we may assume b ≠ 1 . Let p =

q=

1 2

1 2

(1 − b) . Because the derivative of f (x) , f ′(x)

=

=

=

=

d tan−1[b tan(λx + β )] dx 1 1 + b tan (λx + β ) 2

2

⋅ bλ sec2 (λx + β )

bλ cos (λx + β ) + b 2 sin 2 (λx + β ) 2

2 bλ (1 + b ) + (1 − b 2 ) cos 2 ( λ x + β ) 2

2b =λ⋅ 2 p + 2 pq cos 2 ( λ x + β ) + q 2 2 p 2 + 2 pq cos 2 ( λ x + β ) = λ ⋅−1+ 2 p + 2 pq cos 2 ( λ x + β ) + q 2

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(1 + b) ,

2 p [ p + q cos 2 ( λ x + β )] = λ ⋅ − 1 + 2 2 [ p + q cos 2 ( λ x + β )] + q sin 2 ( λ x + β )

= λ ⋅ − 1 +

2p q

2 + sin 2 ( λ x + β )

p + cos 2 ( λ x + β ) q

p + cos 2 ( λ x + β ) q

2

p + e − i 2 (λx + β ) 2p q = λ ⋅ − 1 + ⋅ Re q p p + e i 2 ( λ x + β ) + e − i 2 ( λ x + β ) q q

(by Euler's formula) 1 = λ ⋅ − 1 + 2 Re qz 1+ p

i 2(λx + β ) (where z = e )

∞ qz k = λ ⋅ − 1 + 2 Re ∑ − k = 0 p

(because − qz = q < 1 , we can use geometric series) p

p

k

∞

q = − λ + 2 λ ⋅ ∑ − cos 2 k ( λ x + β ) k =0 p (by DeMoivre's formula and Euler's formula) ∞

k

q = λ + 2 λ ⋅ ∑ − cos 2 k ( λ x + β ) p k = 1

(5)

It follows that f (x)

= tan−1[b tan(λx + β )] ∞ q k = ∫ λ + 2 λ ⋅ ∑ − cos 2 k ( λ x + β ) dx k =1 p

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= λx +

k

∞

1 q ∑ k − p sin 2 k (λx + β ) + K k =1

(6)

where K is a constant. Let λx + β = 0 , we obtain K = β , and hence f (x)

= tan−1[b tan(λx + β )] ∞

k

1 q = λ x + β + ∑ − sin 2 k ( λ x + β ) k =1 k p ∞

k

1 1− b = λx + β + ∑ − sin 2 k ( λ x + β ) k =1 k 1 + b

(7)

By differentiation term by term theorem, differentiating n -times with respect to x on both sides of (7), we obtain the n -th order derivative of f ( x ) ,

f (n) ( x)

=

k ∞ 1 d 1− b λ x + β + ∑ − + sin 2 k ( λ x β ) dx n k =1 k 1 + b

∞ nπ 1− b = (2λ ) n ∑ k n −1 − sin2k (λx + β ) + 2 1+ b k =1 k

for all n ≥ 2 , and x ∈ R . Case (2). If b < 0 . Because f (x)

= tan−1[b tan(λx + β )] = − tan−1[−b tan(λx + β )]

= −(λx + β ) −

∞

k

1 1+ b ∑ k − 1 − b sin 2k (λ x + β ) k =1

(8)

Also using differentiation term by term theorem, differentiating n -times with respect to x on both sides of (8), we have the n -th order derivative of f ( x ) , ∞ nπ 1+ b f ( n ) ( x ) = −(2λ )n ∑ k n −1 − sin2k (λx + β ) + 1 − b 2 k =1 k

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for all n ≥ 2 , and x ∈ R .

■

Next, we determine the Fourier series expansions of any order derivatives of the function (2). 2.7 Theorem 2. If the assumptions are the same as Theorem 1, and suppose the domain of

g ( x ) = tan −1[b cot( λ x + β )] is ( −∞, ∞) . Case (1). If b > 0 , then the n -th order derivative of g ( x ) , ∞ nπ 1− b g ( n ) ( x ) = (−2λ )n ∑ ( −1) k k n −1 − sin− 2k (λx + β ) + 2 1+ b k =1 k

(9)

for all n ≥ 2 , and x ∈ R . Case (2). If b < 0 , then ∞ nπ 1+ b g ( n ) ( x ) = −( −2λ )n ∑ (−1)k k n −1 − sin− 2k (λx + β ) + 2 1− b k =1 k

(10)

for all n ≥ 2 , and x ∈ R .

2.7.1 Proof. Case (1). If b > 0 . Because g( x)

= tan −1[b cot( λx + β )] = tan −1[b tan( − λ x − β + π / 2 )]

π ∞ 1 1− b π = − λx − β + + ∑ − sin 2 k − λ x − β + 2 k =1 k 1 + b 2 k

(11)

(using (7)) By differentiation term by term theorem, differentiating n -times with respect to x on both sides of (11), we obtain the n -th order derivative of g ( x ) , ∞ π nπ 1− b g ( n ) ( x ) = ( −2λ ) n ∑ k n −1 − sin2k − λx − β + + 1 + b 2 2 k =1 k

∞ nπ 1− b = (−2λ )n ∑ ( −1) k k n −1 − sin− 2k (λx + β ) + 2 1+ b k =1 k

for all n ≥ 2 , and x ∈ R . Case (2). If b < 0 . Because g( x)

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= tan −1[b cot( λx + β )] = − tan −1[ − b tan( − λ x − β + π / 2)]

π ∞ 1 1+ b π = − − λx − β + − ∑ − sin 2 k − λ x − β + 2 k =1 k 1 − b 2 k

(12)

(using (11)) Also using differentiation term by term theorem, differentiating n -times with respect to x on both sides of (12), we can evaluate the n -th order derivative of g ( x ) , ∞ π nπ 1+ b g ( n ) ( x ) = −( −2λ )n ∑ k n −1 − sin2k − λx − β + + 2 2 1− b k =1 k

∞ nπ 1+ b = −( −2λ )n ∑ (−1)k k n −1 − sin− 2k (λx + β ) + 2 1− b k =1 k

for all n ≥ 2 , and x ∈ R .

■

3. Examples In the following, aimed at the differential problem of the two types of functions in this study, we provide two functions and use Theorems 1, 2 to determine the Fourier series expansions of their any order derivatives. In addition, we employ Maple to calculate the approximations of some higher order derivative values and their solutions for verifying our answers.

3.1 Example 1. Suppose the domain of the function

7 π f ( x ) = tan −1 tan 6 x − 3 5

(13)

is ( −∞, ∞ ) (the case of b = 7 / 5, λ = 6, β = −π / 3 in Theorem 1). By Case (1) of Theorem 1, we obtain any n -th order derivative of f ( x ) , ∞ π nπ 1 f ( n ) ( x ) = 12 n ⋅ ∑ k n −1 sin 2k 6 x − + 3 2 6 k =1 k

(14)

for all n ≥ 2 , and x ∈ R . Thus, we can determine 8-th order derivative value of f ( x ) at x =

π 4

,

∞ 1 kπ π f (8 ) = 128 ⋅ ∑ k 7 sin 4 3 k =1 6 k

(15)

Next, we use Maple to verify the correctness of (15). >f:=x->arctan(7/5*tan(6*x-Pi/3));

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>evalf(([email protected]@8)(f)(Pi/4),22);

>evalf(12^8*sum(k^7*(1/6)^k*sin(k*Pi/3),k=1..infinity),22);

3.2 Example 2. Suppose the domain of the function

7π g ( x ) = tan −1 − 5 cot 2 x + 4

(16)

is ( −∞, ∞ ) (the case of b = − 5 , λ = 2, β = 7π / 4 in Theorem 2). By Case (2) of Theorem 2, we obtain any n -th order derivative of g (x ) ,

g

(n )

∞

k n −1

( x ) = −( −4) ⋅ ∑ ( −1) k n

k =1

k

5 − 1 7π nπ sin− 2k 2 x + + 5 +1 4 2

(17)

for all n ≥ 2 , and x ∈ R . Therefore, we can evaluate 9-th order derivative value of g (x ) at x = −

g

5π , 4 k

∞ 5π (3k + 1)π k 8 5 − 1 9 sin − = 4 ⋅ ∑ ( −1) k 2 4 k =1 5 +1

(9)

(18)

We also use Maple to verify the correctness of (18). >g:=x->arctan(-sqrt(5)*cot(2*x+7*Pi/4));

>evalf(([email protected]@9)(g)(-5*Pi/4),22);

>evalf(4^9*sum((-1)^k*k^8*((sqrt(5)-1)/(sqrt(5)+1))^k*sin((3*k+1)*Pi/2),k=1..infinity),22);

4. Conclusions From the above discussion, we know the geometric series and the differentiation term by term theorem play significant roles in the theoretical inferences of this study. In fact, the applications of these two theorems are extensive, and can be used to easily solve many difficult problems; we endeavor to conduct further studies on related applications. On the other hand, Maple also plays a vital assistive role in problem-solving. In the future, we will extend the research topic to other calculus and engineering mathematics problems and solve these problems by using

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Maple. These results will be used as teaching materials for Maple on education and research to enhance the connotations of calculus and engineering mathematics.

5. References [1] C. T. J. Dodson and E. A. Gonzalez, Experiments in Mathematics Using Maple, New York: Springer-Verlag, 1995. [2] D. Richards, Advanced Mathematical Methods with Maple, New York: Cambridge University Press, 2002. [3] M. L. Abell and J. P. Braselton, Maple by Example, 3rd ed., New York: Elsevier Academic Press, 2005. [4] R. J. Stroeker and J. F. Kaashoek, Discovering Mathematics with Maple : An Interactive Exploration for Mathematicians, Engineers and Econometricians, Basel: Birkhauser Verlag, 1999. [5] F. Garvan, The Maple Book, London: Chapman & Hall/CRC, 2001. [6] J. S. Robertson, Engineering Mathematics with Maple, New York: McGraw-Hill, 1996. [7] C. Tocci and S. G. Adams, Applied Maple for Engineers and Scientists, Boston: Artech House, 1996. [8] C. -H. Yu, “A study on the differential problem,” International Journal of Research in Aeronautical and Mechanical Engineering, vol. 1, issue. 3, pp. 52-57, July 2013. [9] C. -H. Yu, “ The differential problem of two types of functions,” International Journal of Computer Science and Mobile Computing , vol. 2, issue. 7, pp. 137-145, July 2013. [10] C. -H. Yu, “ Evaluating the derivatives of two types of functions,” International Journal of Computer Science and Mobile Applications, vol. 1, issue. 2, pp. 1-8, August 2013. [11]C. -H. Yu, “ Evaluating the derivatives of trigonometric functions with Maple,” International Journal of Research in Computer Applications and Robotics, vol. 1, issue. 4, pp. 23-28, July 2013. [12] C.-H. Yu, “Application of Maple on solving the differential problem of rational functions,” Applied Mechanics and Materials, in press. [13] C. -H. Yu, “ A study on some differential problems with Maple,” Proceedings of 6th IEEE/International Conference on Advanced Infocomm Technology, National United University, Taiwan, no. 00291, July 2013. [14] C. -H. Yu, “ Application of Maple on solving some differential problems,” Proceedings of IIE Asian Conference 2013, National Taiwan University of Science and Technology, Taiwan, vol. 1, pp. 585-592, July 2013. [15] C. -H. Yu, “The differential problem of four types of functions,” Journal of Kang-Ning, vol. 14, in press. [16] C. -H. Yu, “ A study on the differential problem of some trigonometric functions, ” Journal of Jen-Teh, vol. 10, pp. 27-34, June 2013. [17] T. M. Apostol, Mathematical Analysis, 2nd ed., Boston: Addison-Wesley Publishing Co., Inc., p230, 1975.

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