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A details explanation about Taylor's and Maclaurin's series with variety of examples are included in this slide. The aim is to give the viewer the basic knowledge about the topic.
International Journal of Engineering Research and Development (IJERD)IJERD Editor
call for paper 2012, hard copy of journal, research paper publishing, where to publish research paper,
journal publishing, how to publish research paper, Call For research paper, international journal, publishing a paper, IJERD, journal of science and technology, how to get a research paper published, publishing a paper, publishing of journal, publishing of research paper, reserach and review articles, IJERD Journal, How to publish your research paper, publish research paper, open access engineering journal, Engineering journal, Mathemetics journal, Physics journal, Chemistry journal, Computer Engineering, Computer Science journal, how to submit your paper, peer reviw journal, indexed journal, reserach and review articles, engineering journal, www.ijerd.com, research journals,
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Power Series,Taylor's and Maclaurin's SeriesShubham Sharma
A details explanation about Taylor's and Maclaurin's series with variety of examples are included in this slide. The aim is to give the viewer the basic knowledge about the topic.
Partial differentiation, total differentiation, Jacobian, Taylor's expansion, stationary points,maxima & minima (Extreme values),constraint maxima & minima ( Lagrangian multiplier), differentiation of implicit functions.
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حل تمارين الكتاب
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حلول الاسألة الوزارية
اعداد الدكتور أنس ذياب خلف
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Further Results On The Basis Of Cauchy’s Proper Bound for the Zeros of Entire...IJMER
International Journal of Modern Engineering Research (IJMER) is Peer reviewed, online Journal. It serves as an international archival forum of scholarly research related to engineering and science education.
International Journal of Modern Engineering Research (IJMER) covers all the fields of engineering and science: Electrical Engineering, Mechanical Engineering, Civil Engineering, Chemical Engineering, Computer Engineering, Agricultural Engineering, Aerospace Engineering, Thermodynamics, Structural Engineering, Control Engineering, Robotics, Mechatronics, Fluid Mechanics, Nanotechnology, Simulators, Web-based Learning, Remote Laboratories, Engineering Design Methods, Education Research, Students' Satisfaction and Motivation, Global Projects, and Assessment…. And many more.
2.2 Special types of Correlation
2.3 Point Biserial Correlation rPB
2.3.1 Calculation of rPB
2.3.2 Significance Testing of rPB
2.4 Phi Coefficient (φ )
2.4.1 Significance Testing of phi (φ )
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2.9.3 Computational Alternative for Kendall’s Tau
2.9.4 Significance Testing for Kendall’s Tau
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3. C R E E M O S E N L A E X I G E N C I A
C U R S O D E Á L G E B R A
𝑅𝑒𝑐𝑜𝑛𝑜𝑐𝑒𝑟 𝑙𝑎 𝑖𝑔𝑢𝑎𝑙𝑑𝑎𝑑
𝑑𝑒 𝑓𝑢𝑛𝑐𝑖𝑜𝑛𝑒𝑠
𝑅𝑒𝑎𝑙𝑖𝑧𝑎𝑟 𝑙𝑎𝑠 𝑜𝑝𝑒𝑟𝑎𝑐𝑖𝑜𝑛𝑒𝑠
𝑐𝑜𝑛 𝑓𝑢𝑛𝑐𝑖𝑜𝑛𝑒𝑠
𝐶𝑎𝑙𝑐𝑢𝑙𝑎𝑟 𝑙𝑎 𝑐𝑜𝑚𝑝𝑜𝑠𝑖𝑐𝑖ó𝑛
𝑑𝑒 𝑓𝑢𝑛𝑐𝑖𝑜𝑛𝑒𝑠
4. C R E E M O S E N L A E X I G E N C I A
C U R S O D E Á L G E B R A
Álgebra de funciones
En este tema se verá la igualdad de
funciones, las operaciones entre funciones y
la composición de funciones.
Estas operaciones generan nuevas funciones,
entre sus usos se encuentra en el
movimiento amortiguado, generado cuando
al movimiento oscilatorio se le añade un
amortiguador para reducir su periodo. Se
presenta la gráfica de un movimiento
amortiguado.
𝑓 𝑡 = 𝐴𝑒−𝛾𝑡cos(𝑤𝑡 + 𝛼)
5. C R E E M O S E N L A E X I G E N C I A
Igualdad de funciones
C U R S O D E Á L G E B R A
Definición
Consideremos las funciones f y g bien definidas, luego
se define:
𝑓 = 𝑔 ↔ 𝐷𝑜𝑚𝑓 = 𝐷𝑜𝑚𝑔 ∧ 𝑓 𝑥 = 𝑔 𝑥
Ejemplo 1
Las funciones:
𝑓 𝑥 = (𝑥 + 1)(𝑥 − 1) ; 𝑥 ∈ ℝ
𝑔 𝑥 = 𝑥2
− 1 ; 𝑥 ∈ ℝ
Son iguales, puesto que:
• 𝐷𝑜𝑚𝑓 = 𝐷𝑜𝑚𝑔 = ℝ
• 𝑓 𝑥 = 𝑔 𝑥 = 𝑥2
− 1
Ejemplo 2
Las funciones:
𝑓 𝑥 = 𝑥𝑠𝑔𝑛(𝑥) ∧ 𝑔 𝑥 = 𝑥 ; 𝑥 ∈ ℝ
Redefiniendo, se tiene:
𝑠𝑔𝑛(𝑥) = ൞
1 ; 𝑥 > 0
¿f y g son iguales?
0 ; 𝑥 = 0
−1 ; 𝑥 < 0
→ 𝑓(𝑥) = 𝑥𝑠𝑔𝑛(𝑥) = ൞
𝑥 ; 𝑥 > 0
0 ; 𝑥 = 0
−𝑥 ; 𝑥 < 0
Entonces
𝐷𝑜𝑚𝑓 = 𝐷𝑜𝑚𝑔 = ℝ ∧ 𝑓 𝑥 = 𝑔 𝑥
∴ 𝑓 = 𝑔
; 𝑥 ∈ ℝ
𝑥𝑠𝑔𝑛 𝑥
Primero analizamos el dominio
= 𝑥
6. C R E E M O S E N L A E X I G E N C I A
C U R S O D E Á L G E B R A
Álgebra de funciones
Es el conjunto de operaciones (adición, sustracción, multiplicación y división) que se definen entre dos o más funciones.
𝐹𝑢𝑛𝑐𝑖ó𝑛 𝑁𝑜𝑡𝑎𝑐𝑖ó𝑛 𝑅𝑒𝑔𝑙𝑎 𝑑𝑒
𝑐𝑜𝑟𝑟𝑒𝑠𝑝𝑜𝑛𝑑𝑒𝑛𝑐𝑖𝑎
𝐷𝑜𝑚𝑖𝑛𝑖𝑜
𝑆𝑢𝑚𝑎 𝑓 + 𝑔 𝑓 + 𝑔 𝑥 = 𝑓 𝑥 + 𝑔(𝑥) 𝐷𝑜𝑚(𝑓) ∩ 𝐷𝑜𝑚(𝑔)
𝐷𝑖𝑓𝑒𝑟𝑒𝑛𝑐𝑖𝑎 𝑓 − 𝑔 𝑓 − 𝑔 𝑥 = 𝑓 𝑥 − 𝑔(𝑥) 𝐷𝑜𝑚(𝑓) ∩ 𝐷𝑜𝑚(𝑔)
𝑃𝑟𝑜𝑑𝑢𝑐𝑡𝑜 𝑓. 𝑔 𝑓. 𝑔 𝑥 = 𝑓 𝑥 . 𝑔(𝑥) 𝐷𝑜𝑚(𝑓) ∩ 𝐷𝑜𝑚(𝑔)
𝐷𝑖𝑣𝑖𝑠𝑖ó𝑛 𝑓
𝑔
𝑓
𝑔
𝑥 =
𝑓 𝑥
𝑔 𝑥
𝐷𝑜𝑚 𝑓 ∩ 𝐷𝑜𝑚 𝑔 − 𝑥/𝑔 𝑥 = 0
7. C R E E M O S E N L A E X I G E N C I A
C U R S O D E Á L G E B R A
Ejemplo 1
Si 𝑓 = 1; 2 ; 2; 3 ; 3; 7 ; 5; 8
𝑔 = 1; 4 ; 2; 2 ; 3; 0 ; 4; 7
Calcule 𝑓 + 𝑔 ; 𝑓 − 𝑔 ; 𝑓. 𝑔 ;
𝑓
𝑔
Resolución
Tenemos:
𝐷𝑜𝑚 𝑓 = 1; 2; 3; 5
𝐷𝑜𝑚 𝑔 = 1; 2; 3; 4
Como:
𝐷𝑜𝑚 (𝑓 + 𝑔) = 𝐷𝑜𝑚 𝑓 ∩ 𝐷𝑜𝑚 𝑔
𝐷𝑜𝑚 (𝑓 + 𝑔) = 1; 2; 3
Luego:
(𝑓 + 𝑔)(1) = 𝑓(1) +𝑔(1)
ቄ
2
ቄ
4
= 6 → 1; 6 ∈ 𝑓 + 𝑔
(𝑓 + 𝑔)(2) = 𝑓(2) +𝑔(2)
ቄ
3
ቄ
2
= 5 → 2; 5 ∈ 𝑓 + 𝑔
(𝑓 + 𝑔)(3) = 𝑓(3) +𝑔(3)
ቄ
7
ቄ
0
= 7 → 3; 7 ∈ 𝑓 + 𝑔
Entonces:
𝑓 + 𝑔 = 1; 6 ; 2; 5 ; 3; 7
8. C R E E M O S E N L A E X I G E N C I A
Como:
𝐷𝑜𝑚 (𝑓 − 𝑔) = 𝐷𝑜𝑚 𝑓 ∩ 𝐷𝑜𝑚 𝑔
𝐷𝑜𝑚 (𝑓 − 𝑔) = 1; 2; 3
Luego:
(𝑓 − 𝑔)(1) = 𝑓(1) −𝑔(1)
ቄ
2
ቄ
4
= −2
(𝑓 − 𝑔)(2) = 𝑓(2) −𝑔(2)
ቄ
3
ቄ
2
= 1
(𝑓 − 𝑔)(3) = 𝑓(3) −𝑔(3)
ቄ
7
ቄ
0
= 7
Entonces:
𝑓 − 𝑔 = 1; −2 ; 2; 1 ; 3; 7
C U R S O D E Á L G E B R A
Como:
𝐷𝑜𝑚 (𝑓. 𝑔) = 𝐷𝑜𝑚 𝑓 ∩ 𝐷𝑜𝑚 𝑔
𝐷𝑜𝑚 (𝑓. 𝑔) = 1; 2; 3
Luego:
(𝑓. 𝑔)(1) = 𝑓(1) . 𝑔(1)
ቄ
2
ቄ
4
= 8
(𝑓. 𝑔)(2) = 𝑓(2) . 𝑔(2)
ቄ
3
ቄ
2
= 6
(𝑓. 𝑔)(3) = 𝑓(3) . 𝑔(3)
ቄ
7
ቄ
0
= 0
Entonces:
𝑓. 𝑔 = 1; 8 ; 2; 6 ; 3; 0
Como:
𝐷𝑜𝑚 (𝑓/𝑔) = 𝐷𝑜𝑚𝑓 ∩ 𝐷𝑜𝑚𝑔
𝐷𝑜𝑚 (𝑓/𝑔) = 1; 2
Luego:
(𝑓/𝑔)(1) = 𝑓(1) /𝑔(1)
ቄ
2
ቄ
4
= 1/2
(𝑓/𝑔)(2) = 𝑓(2) /𝑔(2)
ቄ
3
ቄ
2
= 3/2
Entonces:
𝑓
𝑔
= 1;
1
2
; 2;
3
2
− 𝑥/𝑔 𝑥 = 0
1; 2; 3
൞
3
9. C R E E M O S E N L A E X I G E N C I A
C U R S O D E Á L G E B R A
𝑭𝒖𝒏𝒄𝒊ó𝒏 𝒑𝒐𝒕𝒆𝒏𝒄𝒊𝒂
Dada la función 𝑓, se define la función potencia 𝑓𝑛
como:
𝑓𝑛
= 𝑓. 𝑓. 𝑓 … 𝑓
൞
𝑛 𝑣𝑒𝑐𝑒𝑠
; 𝑛 ∈ ℤ+
∧ 𝑛 ≥ 2
Donde:
𝑓𝑛
൝
𝐷𝑜𝑚 (𝑓𝑛
) = 𝐷𝑜𝑚𝑓
𝑓𝑛
(𝑥) = 𝑓 𝑥 𝑛
𝑶𝒃𝒔𝒆𝒓𝒗𝒂𝒄𝒊ó𝒏:
∀𝑘 ∈ ℝ , se tiene que:
Ejemplo 2
Sean 𝑓 𝑦 𝑔 funciones tales que:
𝑓 𝑥 = 𝑥2
− 𝑥 − 1 ; 𝑔 = −1; 1 ; 0; 2 ; 1; 3
Encuentre 𝑓2
+ 2𝑔
Resolución
Los dominios son:
𝐷𝑜𝑚(𝑓) = ℝ ∧ 𝐷𝑜𝑚(𝑔) = −1 ; 0 ; 1
→ 𝐷𝑜𝑚(𝑓2
+ 2𝑔) = 𝐷𝑜𝑚(𝑓2
) ∩ 𝐷𝑜𝑚(2𝑔)
= 𝐷𝑜𝑚(𝑓) ∩ 𝐷𝑜𝑚(𝑔) = −1 ; 0 ; 1
(𝑓2
+2𝑔)(−1) = 𝑓2
(−1) + 2𝑔(−1) = 3
(𝑓2
+2𝑔)(0) = 𝑓2
(0) + 2𝑔(0)
(𝑓2
+2𝑔)(1) = 𝑓2
(1) + 2𝑔(1)
→ 𝑓2
+ 2𝑔 = −1; 3 ; 0; 5 ; 1; 7
𝑘𝑓 ൝
𝐷𝑜𝑚 𝑘𝑓 = 𝐷𝑜𝑚𝑓
𝑘𝑓 (𝑥) = 𝑘 ∙ 𝑓 𝑥
= (1)2 +2(1)
= 5
= (−1)2 +2(2)
= 7
= (−1)2 +2(3)
10. C R E E M O S E N L A E X I G E N C I A
C U R S O D E Á L G E B R A
Ejemplo 3
Sean 𝑓 𝑦 𝑔 funciones tales que:
𝑓 𝑥 = 2𝑥 − 1 ; 𝑥 ∈ ۦ−6; ሿ
5
𝑔 𝑥 = 𝑥2
+ 1 ; 𝑥 ∈ ሾ−4; ۧ
2
Halle su rango y grafique 𝑓 + 𝑔
Resolución
𝐷𝑜𝑚(𝑓 + 𝑔) = 𝐷𝑜𝑚(𝑓) ∩ 𝐷𝑜𝑚(𝑔)
−6 5
𝐷𝑜𝑚(𝑓)
−4 2
𝐷𝑜𝑚(𝑔)
𝐷𝑜𝑚(𝑓) ∩ 𝐷𝑜𝑚(𝑔) = ሾ−4; ۧ
2
𝐷𝑜𝑚(𝑓 + 𝑔) = ሾ−4; ۧ
2
Además:
(𝑓 + 𝑔) 𝑥 = 𝑓 𝑥 + 𝑔(𝑥)
(𝑓 + 𝑔) 𝑥 = 2𝑥 − 1 + 𝑥2
+ 1
(𝑓 + 𝑔) 𝑥 = 𝑥2
+ 2𝑥
Luego:
(𝑓 + 𝑔) 𝑥 = 𝑥2
+ 2𝑥 ; 𝑥 ∈ ሾ−4; ۧ
2
Calculamos el rango
(𝑓 + 𝑔) 𝑥 = 𝑥2
+ 2𝑥 +1 −1
൞
(𝑥 + 1)2
(𝑓 + 𝑔) 𝑥 = (𝑥 + 1)2
−1
Como:
𝑥 ∈ ሾ−4; ۧ
2
−4 ≤ 𝑥 < 2
+1
−3 ≤ 𝑥 + 1< 3
( )2
0 ≤ (𝑥 + 1)2 ≤ 9
−1
−1 ≤ (𝑥 + 1)2
−1 ≤ 8
𝑅𝑎𝑛(𝑓 + 𝑔) = −1; 8
Su gráfica es
𝑋
𝑌
−1
−1
−4
8
2
0
−2
11. C R E E M O S E N L A E X I G E N C I A
Composición de funciones
C U R S O D E Á L G E B R A
Definición
Sean f y g son dos funciones, se denota y define su
composición (f compuesta con g) por :
𝑓𝑜𝑔 ൝
𝐷𝑜𝑚(𝑓𝑜𝑔) = 𝑥 ∈ 𝐷𝑜𝑚 𝑔 / 𝑔 𝑥 ∈ 𝐷𝑜𝑚𝑓
𝑓𝑜𝑔 𝑥 = 𝑓(𝑔 𝑥 )
En forma gráfica, tenemos:
𝑔 𝑓
𝐷𝑜𝑚𝑔 𝑅𝑎𝑛𝑔 𝐷𝑜𝑚𝑓 𝑅𝑎𝑛𝑓
𝑓𝑜𝑔
Ejemplo 1
Si 𝑓 = 1; 2 ; 3; 5 ; 4; 1 ; 7; 0
𝑔 = 0; 1 ; 1; 2 ; 2; 4 ; 4; 7
Calcule 𝑓𝑜𝑔
𝑔
0 1
1 2
2 4
4 7
𝑓
2
3 5
1
0
𝑓𝑜𝑔
𝑓𝑜𝑔 = 0; 2 ; 2; 1 ; 4; 0
𝑥 𝑔 𝑥 𝑓(𝑔 𝑥 )
12. C R E E M O S E N L A E X I G E N C I A
Además
C U R S O D E Á L G E B R A
Si 𝑓 = 1; 2 ; 3; 5 ; 4; 1 ; 7; 0
𝑔 = 0; 1 ; 1; 2 ; 2; 4 ; 4; 7
Halle 𝑔𝑜𝑓
𝑓
1 2
3 5
4 1
7 0
𝑔
4
4 7
2
1
𝑔𝑜𝑓
𝑔𝑜𝑓 = 1; 4 ; 4; 2 ; 7; 1
𝐺𝑒𝑛𝑒𝑟𝑎𝑙𝑚𝑒𝑛𝑡𝑒 𝑓𝑜𝑔 ≠ 𝑔𝑜𝑓
Ejemplo 2
Sean 𝑓 𝑦 𝑔 funciones tales que:
𝑓 𝑥 = −5𝑥 + 3 ; 𝑥 ∈ ۦ−4; ሿ
6
𝑔 𝑥 = 2𝑥 ; 𝑥 ∈ ሾ1; ۧ
4
Halle 𝑓𝑜𝑔
Resolución
𝐷𝑜𝑚(𝑓𝑜𝑔) = 𝑥 ∈ 𝐷𝑜𝑚 𝑔 / 𝑔 𝑥 ∈ 𝐷𝑜𝑚𝑓
𝑥 ∈ ሾ1; ۧ
4 ∧ 2𝑥 ∈ ۦ−4; ሿ
6
→ 1 ≤ 𝑥 < 4 ∧ −4 < 2𝑥 ≤ 6
→ 1 ≤ 𝑥 < 4 ∧ −2 < 𝑥 ≤ 3 ⇒ 1 ≤ 𝑥 ≤ 3
→ 𝐷𝑜𝑚(𝑓𝑜𝑔)= ሾ1; ሿ
3
(𝑓𝑜𝑔) 𝑥 = 𝑓(𝑔 𝑥 ) = −5(𝑔 𝑥 ) + 3 = −5(2𝑥) + 3
→ (𝑓𝑜𝑔) 𝑥 = −10𝑥 + 3
13. C R E E M O S E N L A E X I G E N C I A
C U R S O D E Á L G E B R A
Propiedades
1) Las función composición es asociativo
𝑓 𝑜 𝑔 𝑜 ℎ = 𝑓 𝑜 (𝑔 𝑜 ℎ)
2) Las función composición es distributiva
a) 𝑓 + 𝑔 𝑜 ℎ = 𝑓 𝑜 ℎ + 𝑔 𝑜 ℎ
b) 𝑓 − 𝑔 𝑜 ℎ = 𝑓 𝑜 ℎ − 𝑔 𝑜 ℎ
c) 𝑓. 𝑔 𝑜 ℎ = (𝑓 𝑜 ℎ ). (𝑔 𝑜 ℎ)
d) 𝑓
𝑔
𝑜 ℎ =
𝑓 𝑜 ℎ
𝑔 𝑜 ℎ
¡Cuidado!
ℎ 𝑜 𝑓 + 𝑔 ≠ ℎ 𝑜 𝑓 + ℎ 𝑜 𝑔
ℎ 𝑜 𝑓 − 𝑔 ≠ ℎ 𝑜 𝑓 − (ℎ 𝑜 𝑔)
ℎ 𝑜 𝑓. 𝑔 ≠ ℎ 𝑜 𝑓 . (ℎ 𝑜 𝑔)
ℎ 𝑜
𝑓
𝑔
≠
ℎ 𝑜 𝑓
ℎ 𝑜 𝑔
3) Para toda función 𝑓 se cumple:
𝑓 𝑜 𝐼 = 𝑓 = 𝐼 𝑜 𝑓
donde I es la función identidad
14. C R E E M O S E N L A E X I G E N C I A
C U R S O D E Á L G E B R A
Observación:
Si conocemos las gráficas de 𝑓 y 𝑔, se pueden encontrar
el bosquejo de la gráfica de la función suma 𝑓 + 𝑔.
Ejemplo
Grafique ℎ 𝑥 = 𝑥 − 2 + 4 − 𝑥
Resolución
Tenemos ℎ 𝑥 = 𝑥 − 2 + 4 − 𝑥
൝
𝑓 𝑥
൝
𝑔 𝑥
𝑋
𝑌
𝑋
𝑌
2
𝑓
4
𝑔
Hallando el dominio de ℎ(𝑥)
𝑥 − 2 ≥ 0 → 𝑥 ≥ 2
4 − 𝑥 ≥ 0 → 𝑥 ≤ 4
൝ 2 ≤ 𝑥 ≤ 4
→ 𝐷𝑜𝑚ℎ = ሾ 2; ሿ
4
Su gráfica es:
𝑋
𝑌
2
𝑓
4
𝑔
𝑎
𝑓(𝑎)
𝑔(𝑎)
𝑓(𝑎)
𝑓 𝑎 + 𝑔(𝑎)
𝑏 3
2
2 𝑓 + 𝑔
15. w w w . a c a d e m i a c e s a r v a l l e j o . e d u . p e