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Formulas

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The document contains many mathematical formulas across various topics: 1) Formulas for solving equations like quadratic, logarithmic, and trigonometric equations. 2) Formulas for limits, derivatives, integrals, and operations involving exponents, logarithms, and radicals. 3) Set notation formulas defining sets of real numbers based on conditions.

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1
2

๐‘ฅ:

ฬ‚
ยฑ๐‘โˆ’ โˆš๐œ‹โˆ—โˆ

๐‘ƒ:

5

log3(๐‘ฅ+ โˆš ฬ‚
๐›ฝ)
(2๐‘ฅ๐‘ฆ)2 โˆ—(3๐‘Ž๐‘+3๐‘ฅ)3

3

โˆš

๐‘ฅ 3 โˆ—๐‘ฆ 2

=

๐‘…๐‘‡

โˆ’
๐‘ฃโˆ’๐‘

๐‘

๐‘Žโˆ—๐‘‘+๐‘โˆ—๐‘

๐‘ฅ 3 โˆ’๐‘ฆ2
โˆš2

๐‘Ž๐‘›โˆ— ๐‘

๐‘š

= ๐‘Ž ๐‘›+๐‘š

๐‘Ž ๐‘› /๐‘

๐‘

+ ๐‘=
๐‘

๐‘š

1
๐‘Ž

๐‘š

(๐‘Ž2 +๐‘ 2 +๐‘ฅ 2 +๐‘ฆ 2 )โˆ—(๐‘ฅ 3 +๐‘ 3 )

๐‘Ž3 +๐‘2

๐‘ฅ1=
๐‘

โˆ’ ๐‘=
๐‘

๐‘š

๐‘Žโˆ’ ๐‘› =

๐‘š

โˆš3๐‘ฅโˆ’2๐‘ฆ 3 โˆ’23

(๐‘ฅ ๐‘Ž โˆ’๐‘ฆ ๐‘ )โˆ—(32 +2๐‘ฅโˆ’๐‘ฆ 3 )

๐‘Ž

๐‘

๐‘›

๐‘Ž3

=

= 2โˆš

๐‘Ž+๐‘

๐‘Ž๐‘› = โˆš๐‘Ž

๐‘›

23

โˆ—

๐‘Ž๐‘ฅ 2 + ๐‘๐‘ฅ + ๐‘ = 0
๐‘Ž

๐‘Žโˆ’๐‘› =

๐‘โˆ—๐‘‘

๐‘ฃ2

1
2
๐‘ฆ2

๐‘Ž ๐‘ฅ + +๐‘ ๐‘ฅ

22 โˆ—๐‘Ž2 โˆ—๐‘ 2

ฬ…ฬ…ฬ…// โƒ‘โƒ‘โƒ‘โƒ‘ = ฬ‚ โˆฉ ๐‘’๐‘ ๐‘‘: {๐‘’}
ฬ‚
๐‘Ž๐‘ ๐‘๐‘‘ ๐‘Ž๐‘

+ ๐‘‘=
๐‘

โˆš

5๐‘ฅ 2 +3๐‘ฆ2 โˆ’๐‘Ž3 โˆ’๐‘3

โˆ€๐‘ฅ โˆˆ โ„œ: โˆƒ๐‘ฆ โˆˆ/xโ‰ฅ ๐‘œ โ‹ ๐‘ฆ โˆˆ ๐‘ฅ

๐‘Ž

๐‘Ž

โˆ’๐‘+โˆš๐‘2 4๐‘Ž๐‘
2

๐‘Ž

๐‘Žโˆ’๐‘
๐‘

๐‘ฅ2=

โˆ’๐‘โˆ’โˆš๐‘ 2 4๐‘Ž๐‘

๐‘

๐‘Žโˆ— ๐‘ =

2๐‘Ž
๐‘Žโˆ—๐‘+๐‘
๐‘

1
๐‘›
โˆš๐‘Ž ๐‘š

= ๐‘Ž ๐‘›โˆ’๐‘š

๐œ‹

๐ท = {๐‘ฅ โˆˆ โ„/โˆƒ๐‘“(๐‘ฅ)}

๐ท = โ„ โˆ’ {(2๐พ + 1) โˆ— 2 ; ๐พ โˆˆ ๐‘}

๐ท = โ„ โˆ’ {๐พ๐œ‹; โˆˆ ๐‘}

๐‘ฅ > ๐‘Ž โŸน ๐‘“(๐‘ฅ) โ‰ค ๐‘“(๐‘Ž)

โˆซ ๐‘‘๐‘ฅ = ๐‘ฅ + ๐‘

lim ๐‘“(๐‘ฅ) = โˆž โŸบ โˆ€๐พ โˆˆ โ„+ โˆƒ๐›ฟ(๐พ) > 0/0 < |๐‘ฅ โˆ’ ๐‘ฅ0 | < ๐›ฟ โŸน ๐‘“(๐‘ฅ) > ๐พ
๐‘ฅโ†’๐‘Ž

๐‘›
๐‘Ž ๐‘›+1 โˆ’ ๐‘ ๐‘›โˆ’1 = (๐‘Ž + ๐‘) โˆ‘ ๐‘˜=0 ๐‘Ž ๐‘˜ ๐‘ ๐‘›โˆ’๐‘˜

log ๐‘Ž ๐‘ฅ = ๐‘ฆ โ‡’ ๐‘Ž ๐‘ฆ = ๐‘ฅ

log ๐‘Ž
๐‘›

log ๐‘Ž

๐‘›

๐‘‘๐‘ฅ = โˆซ ๐ถ(๐‘ฅ)๐‘‘๐‘ฅ + โˆซ

log

log ๐‘Ž ๐‘ฅ = log ๐‘

( โˆš ๐‘ฅ) = ๐‘› log ๐‘Ž ๐‘ฅ
๐‘›

๐‘ƒ(๐‘ฅ)
๐‘„(๐‘ฅ)

(๐‘ฅ โˆ— ๐‘ฆ) = log ๐‘Ž ๐‘ฅ + log ๐‘Ž ๐‘ฆ

1

๐‘›

โˆซ

๐‘

๐‘›

๐‘Ž โˆš๐‘˜ + ๐‘ โˆš๐‘˜ + ๐‘ โˆš๐‘˜ = (๐‘Ž + ๐‘ + ๐‘) โˆš๐‘˜

๐‘›

๐‘ฅ
๐‘Ž

๐‘š
โˆšโˆš๐‘Ž=

๐‘›โˆ’๐‘š

โˆš๐‘Ž

๐‘…(๐‘ฅ)
๐‘„(๐‘ฅ)

๐‘‘๐‘ฅ

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Formulas

  • 1. Formulas: 1 2 ๐‘ฅ: ฬ‚ ยฑ๐‘โˆ’ โˆš๐œ‹โˆ—โˆ ๐‘ƒ: 5 log3(๐‘ฅ+ โˆš ฬ‚ ๐›ฝ) (2๐‘ฅ๐‘ฆ)2 โˆ—(3๐‘Ž๐‘+3๐‘ฅ)3 3 โˆš ๐‘ฅ 3 โˆ—๐‘ฆ 2 = ๐‘…๐‘‡ โˆ’ ๐‘ฃโˆ’๐‘ ๐‘ ๐‘Žโˆ—๐‘‘+๐‘โˆ—๐‘ ๐‘ฅ 3 โˆ’๐‘ฆ2 โˆš2 ๐‘Ž๐‘›โˆ— ๐‘ ๐‘š = ๐‘Ž ๐‘›+๐‘š ๐‘Ž ๐‘› /๐‘ ๐‘ + ๐‘= ๐‘ ๐‘š 1 ๐‘Ž ๐‘š (๐‘Ž2 +๐‘ 2 +๐‘ฅ 2 +๐‘ฆ 2 )โˆ—(๐‘ฅ 3 +๐‘ 3 ) ๐‘Ž3 +๐‘2 ๐‘ฅ1= ๐‘ โˆ’ ๐‘= ๐‘ ๐‘š ๐‘Žโˆ’ ๐‘› = ๐‘š โˆš3๐‘ฅโˆ’2๐‘ฆ 3 โˆ’23 (๐‘ฅ ๐‘Ž โˆ’๐‘ฆ ๐‘ )โˆ—(32 +2๐‘ฅโˆ’๐‘ฆ 3 ) ๐‘Ž ๐‘ ๐‘› ๐‘Ž3 = = 2โˆš ๐‘Ž+๐‘ ๐‘Ž๐‘› = โˆš๐‘Ž ๐‘› 23 โˆ— ๐‘Ž๐‘ฅ 2 + ๐‘๐‘ฅ + ๐‘ = 0 ๐‘Ž ๐‘Žโˆ’๐‘› = ๐‘โˆ—๐‘‘ ๐‘ฃ2 1 2 ๐‘ฆ2 ๐‘Ž ๐‘ฅ + +๐‘ ๐‘ฅ 22 โˆ—๐‘Ž2 โˆ—๐‘ 2 ฬ…ฬ…ฬ…// โƒ‘โƒ‘โƒ‘โƒ‘ = ฬ‚ โˆฉ ๐‘’๐‘ ๐‘‘: {๐‘’} ฬ‚ ๐‘Ž๐‘ ๐‘๐‘‘ ๐‘Ž๐‘ + ๐‘‘= ๐‘ โˆš 5๐‘ฅ 2 +3๐‘ฆ2 โˆ’๐‘Ž3 โˆ’๐‘3 โˆ€๐‘ฅ โˆˆ โ„œ: โˆƒ๐‘ฆ โˆˆ/xโ‰ฅ ๐‘œ โ‹ ๐‘ฆ โˆˆ ๐‘ฅ ๐‘Ž ๐‘Ž โˆ’๐‘+โˆš๐‘2 4๐‘Ž๐‘ 2 ๐‘Ž ๐‘Žโˆ’๐‘ ๐‘ ๐‘ฅ2= โˆ’๐‘โˆ’โˆš๐‘ 2 4๐‘Ž๐‘ ๐‘ ๐‘Žโˆ— ๐‘ = 2๐‘Ž ๐‘Žโˆ—๐‘+๐‘ ๐‘ 1 ๐‘› โˆš๐‘Ž ๐‘š = ๐‘Ž ๐‘›โˆ’๐‘š ๐œ‹ ๐ท = {๐‘ฅ โˆˆ โ„/โˆƒ๐‘“(๐‘ฅ)} ๐ท = โ„ โˆ’ {(2๐พ + 1) โˆ— 2 ; ๐พ โˆˆ ๐‘} ๐ท = โ„ โˆ’ {๐พ๐œ‹; โˆˆ ๐‘} ๐‘ฅ > ๐‘Ž โŸน ๐‘“(๐‘ฅ) โ‰ค ๐‘“(๐‘Ž) โˆซ ๐‘‘๐‘ฅ = ๐‘ฅ + ๐‘ lim ๐‘“(๐‘ฅ) = โˆž โŸบ โˆ€๐พ โˆˆ โ„+ โˆƒ๐›ฟ(๐พ) > 0/0 < |๐‘ฅ โˆ’ ๐‘ฅ0 | < ๐›ฟ โŸน ๐‘“(๐‘ฅ) > ๐พ ๐‘ฅโ†’๐‘Ž ๐‘› ๐‘Ž ๐‘›+1 โˆ’ ๐‘ ๐‘›โˆ’1 = (๐‘Ž + ๐‘) โˆ‘ ๐‘˜=0 ๐‘Ž ๐‘˜ ๐‘ ๐‘›โˆ’๐‘˜ log ๐‘Ž ๐‘ฅ = ๐‘ฆ โ‡’ ๐‘Ž ๐‘ฆ = ๐‘ฅ log ๐‘Ž ๐‘› log ๐‘Ž ๐‘› ๐‘‘๐‘ฅ = โˆซ ๐ถ(๐‘ฅ)๐‘‘๐‘ฅ + โˆซ log log ๐‘Ž ๐‘ฅ = log ๐‘ ( โˆš ๐‘ฅ) = ๐‘› log ๐‘Ž ๐‘ฅ ๐‘› ๐‘ƒ(๐‘ฅ) ๐‘„(๐‘ฅ) (๐‘ฅ โˆ— ๐‘ฆ) = log ๐‘Ž ๐‘ฅ + log ๐‘Ž ๐‘ฆ 1 ๐‘› โˆซ ๐‘ ๐‘› ๐‘Ž โˆš๐‘˜ + ๐‘ โˆš๐‘˜ + ๐‘ โˆš๐‘˜ = (๐‘Ž + ๐‘ + ๐‘) โˆš๐‘˜ ๐‘› ๐‘ฅ ๐‘Ž ๐‘š โˆšโˆš๐‘Ž= ๐‘›โˆ’๐‘š โˆš๐‘Ž ๐‘…(๐‘ฅ) ๐‘„(๐‘ฅ) ๐‘‘๐‘ฅ