Algebra Cheat Sheet By The WeSolveThem.com Team
Algebra Cheat Sheet By The WeSolveThem.com Team
Table of Contents General Symbols and Notations ........................................................................................................................................................... 4 Types of numbers ....................................................................................................................................................................................... 5 Properties ...................................................................................................................................................................................................... 5 Meanings ........................................................................................................................................................................................................ 5 Complementation of sets ........................................................................................................................................................................ 5 Set Laws ......................................................................................................................................................................................................... 5 De Morgan’s Laws ...................................................................................................................................................................................... 6 Number of Elements in a Set ................................................................................................................................................................. 6 Axioms ............................................................................................................................................................................................................ 6 Arithmetic ...................................................................................................................................................................................................... 7 Exponents ...................................................................................................................................................................................................... 7 Radicals .......................................................................................................................................................................................................... 7 Complex Numbers ..................................................................................................................................................................................... 7 Adding and Subtracting Fractions ...................................................................................................................................................... 8 Logarithmic ................................................................................................................................................................................................... 8 Log “Base” Notation ................................................................................................................................................................................... 8 Log “Natural” Notation ............................................................................................................................................................................ 8 *Factoring ...................................................................................................................................................................................................... 9 Note: .................................................................................................................................................................................................................. 9 Long Division ............................................................................................................................................................................................... 9 Complete The Square ............................................................................................................................................................................... 9 Example 1: Solving for x (Formula 1) ............................................................................................................................................. 10 Example 2: Solving for x (Formula 2) ............................................................................................................................................. 10 Compositions ............................................................................................................................................................................................ 10 Functions .................................................................................................................................................................................................... 11 Vertical Line Test ..................................................................................................................................................................................... 11 Even/Odd Function ................................................................................................................................................................................. 11 Average Rate of Change ........................................................................................................................................................................ 11 Secant Line .................................................................................................................................................................................................. 11 Difference Quotient ................................................................................................................................................................................. 11 Distance Formula .................................................................................................................................................................................... 11 Midpoint Formula ................................................................................................................................................................................... 12 Quadratic Formula .................................................................................................................................................................................. 12 Proof: ............................................................................................................................................................................................................. 12 Discriminant: ............................................................................................................................................................................................. 12 Graphing a Line ........................................................................................................................................................................................ 12 Point Slope Form: ..................................................................................................................................................................................... 13 Slope Intercept Form: ............................................................................................................................................................................. 13 Standard or General Form ................................................................................................................................................................... 13 Parallel Line (equal slopes) ................................................................................................................................................................. 13 Perpendicular Line (product of slopes are -1) ............................................................................................................................. 14 *Domain Restrictions ............................................................................................................................................................................ 14 Polynomial .................................................................................................................................................................................................. 14 Fraction ........................................................................................................................................................................................................ 14 Radical, if n is even .................................................................................................................................................................................. 14 Radical, if n is odd .................................................................................................................................................................................... 14 Fraction with Radical in denominator ........................................................................................................................................... 14 Natural Log ................................................................................................................................................................................................. 14 Exponential ................................................................................................................................................................................................. 14 Inverse Functions .................................................................................................................................................................................... 14 Asymptotes, Holes and Graphs .......................................................................................................................................................... 14
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Hole in a Graph ......................................................................................................................................................................................... 15 Three General Cases for Horizontal Asymptotes ........................................................................................................................ 15 Ex. 1 Horizontal and Vertical .............................................................................................................................................................. 15 Ex. 2 Oblique ............................................................................................................................................................................................... 15 Ex. 3 Horizontal and Vertical .............................................................................................................................................................. 15 Inequalities ................................................................................................................................................................................................ 16 Interest Formulas .................................................................................................................................................................................... 16 Physics Formulas .................................................................................................................................................................................... 16 Symmetry ................................................................................................................................................................................................... 16 By Point ........................................................................................................................................................................................................ 16 Testing .......................................................................................................................................................................................................... 16 Variations (Proportionality) .............................................................................................................................................................. 16 Common Graphs and Formulas ........................................................................................................................................................ 17 Equation of a Line .................................................................................................................................................................................... 19 Equation of Parabola ............................................................................................................................................................................. 19 Equation of Circle ..................................................................................................................................................................................... 19 Equation of Ellipse ................................................................................................................................................................................... 19 Equation of Hyperbola (1) ................................................................................................................................................................... 19 Equation of Hyperbola (2) ................................................................................................................................................................... 19 Areas .............................................................................................................................................................................................................. 20 Surface Areas ............................................................................................................................................................................................. 20 Volumes ........................................................................................................................................................................................................ 20 Business Functions ................................................................................................................................................................................... 20 Average Rate of Change of f and Slope of Secant Line ............................................................................................................. 21 Difference Quotient ................................................................................................................................................................................. 21 Functions .................................................................................................................................................................................................... 21 Graph Shifts and Compressions ........................................................................................................................................................ 21 Systems of equations ............................................................................................................................................................................. 22 Rank of matrix and pivots ................................................................................................................................................................... 22 Determinate’s of a (2x2) matrix ....................................................................................................................................................... 23 Determinate of a (3x3) and higher matrices ............................................................................................................................... 23 Cofactor Expansion ................................................................................................................................................................................. 23
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General Symbols and Notations Symbol = ≠ ± ∓ 𝑖𝑓𝑓, ⇔ ⇒ < ≤ ≥ > × ∗ or ∙ … … … … … … … … ∞ 𝛥 ∑ 𝜃
…
…
Meaning Equal Not equal Plus or Minus Minus or Plus If and only if Implies Less than Less than equal Greater than equal Greater than Times Multiplication Multiplication Multiplication Multiplication Exponential Multiplication Infinity Displacement or change of Summation
𝑓 𝑥 𝑓 𝑥, 𝑦 ∈ ∀ ∴ ∵ ≡ , , ⊂ ⊆ , …,… ∪
Theta – reserved for angles Function of 𝑥 Function of 𝑥 and 𝑦 In or element of For all Therefore Because Equivalent Open interval Closed interval Proper Subset Subset (equal) Half open/closed Set of numbers Union
∩ ℝ 𝑃 𝑥! , 𝑦!
Intersection Real numbers Point
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Example 0 = 0 1 ≠ 0 𝑥 = ±𝑎 ⇒ 𝑥 = 𝑎 𝑜𝑟 𝑥 = −𝑎 𝑥 = ∓𝑎 ⇒ 𝑥 = −𝑎 𝑜𝑟 𝑥 = 𝑎 𝑝 ⇒ 𝑞 and 𝑞 ⇒ 𝑝 then 𝑝 ⇔ 𝑞 𝑝 ⇒ 𝑞 𝑥 − 𝑎 < 0 ⇒ 𝑥 < 𝑎 𝑥 − 𝑎 ≤ 0 ⇒ 𝑥 ≤ 𝑎 𝑥 − 𝑎 ≥ 0 ⇒ 𝑥 ≥ 𝑎 𝑥 − 𝑎 > 0 ⇒ 𝑥 > 𝑎 2×3 = 6 2 ∗ 3 = 6 or 2 ⋅ 3 = 6 2 3 = 6 2 3 = 6 2 3 = 6 2 3 !!! 3 − 2 = 6 ! 1 = 6 Never ends 𝛥𝑥 = 𝑥 − 𝑥! !
𝑎! 𝑥 ! = 𝑎! 𝑥 ! + 𝑎! 𝑥 ! + 𝑎! 𝑥 ! !!!
𝜋 = 45° 4 𝑓 𝑥 = 𝑥 ! + ⋯ 𝑓 𝑥, 𝑦 = 𝑥𝑦 ! + ⋯ 𝑥 ∈ 𝑎, 𝑏 means 𝑎 ≤ 𝑥 < 𝑏 ∀! (𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥) 𝑥 − 𝑎 = 0 ⇔ 𝑥 = 𝑎 ∴ 𝑥 = 𝑎 ∵ 𝑥 − 𝑎 = 0, 𝑥 = 𝑎 −2, 3 ≡ −2 < 𝑥 < 3 −2, 3 ≡ −2 < 𝑥 < 3 2, 3 ≡ 2 ≤ 𝑥 ≤ 3 𝐴 ⊂ 𝐵 ⇒ 𝐵 ⊄ 𝐴 𝐴 ⊆ 𝐵 ⇒ 𝐴 = 𝐵 1, 4 ≡ 1 ≤ 𝑥 < 4 1,3,5,7 𝐷 = −∞, 0 ∪ 0, ∞ 1,2,3 ∪ 3,4,5 = 1, 2, 3, 4, 5 1,2,3 ∩ 3,4,5 = 3 𝐷 = −∞, ∞ 1, 𝑓 1 𝜃=
Types of numbers
Integers … , −3, −2, −1,0, 1,2,3, …
Rational ! , 𝑏 ≠ 0 and ! 𝑎, 𝑏 are integers
Irrational A number that cannot be expressed as a fraction e.g. 𝜋
Complex 𝑥 = 𝑎 + 𝑏𝑖 where 𝑎 and 𝑏 are any number
Properties Reflexive 𝑎 = 𝑎
Symmetric 𝑎 = 𝑏 then 𝑏 = 𝑎
Transitive 𝑎 = 𝑏 and 𝑏 = 𝑐 then𝑎 = 𝑐
Substitution 𝑎 = 𝑏 then b can replace 𝑎
Meanings Both 𝐴 and 𝐵 have the same elements Subset: If every element of a set 𝐴 is in 𝐵 Proper Subset: If every element in A is also in B but 𝐴 ≠ 𝐵: Intersection: The elements that are both in 𝐴 and in 𝐵 Union: All elements from 𝐴 and 𝐵 are in 𝐴 union 𝐵 Compliment: If 𝐴 ⊂ 𝑈, and 𝑈 is the universal set
𝐴 = 𝐵 𝐴 ⊆ 𝐵 ⇒ 𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴 ⇔ 𝐴 = 𝐵 𝐴 ⊂ 𝐵 𝐴∩𝐵 = 𝑥 𝑥 ∈𝐴∧𝑥 ∈𝐵 𝐴∪𝐵 = 𝑥 𝑥 ∈𝐴∨𝑥 ∈𝐵∨𝑥 ∈𝐴∩𝐵 𝐴 = 𝐴! = 𝑥 𝑥 ∈ 𝑈 ∧ 𝑥 ∉ 𝐴
Complementation of sets a. 𝑈 ! = ∅
b. ∅! = 𝑈
c.
𝐴!
!
= 𝐴
d. 𝐴 ∪ 𝐴! = 𝑈
e. 𝐴 ∩ 𝐴! = ∅
Set Laws 𝐴 ∪ 𝐵 = 𝐵 ∪ 𝐴 𝐴 ∩ 𝐵 = 𝐵 ∩ 𝐴 𝐴∪ 𝐴∩ 𝐴∪ 𝐴∩
𝐵 ∪ 𝐶 = 𝐴 ∪ 𝐵 ∪ 𝐶 𝐵 ∩ 𝐶 = 𝐴 ∩ 𝐵 ∩ 𝐶 𝐵∩𝐶 = 𝐴∪𝐵 ∩ 𝐴∪𝐶 𝐵∪𝐶 = 𝐴∩𝐵 ∪ 𝐴∩𝐶
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Commutative law for union Commutative law for intersection Associative law for union Associative law for intersection Distributive law for union Distributive law for intersection
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De Morgan’s Laws
i.
𝐴∪𝐵
!
= 𝐴! ∩ 𝐵!
ii.
𝐴∩𝐵
!
= 𝐴! ∪ 𝐵!
Number of Elements in a Set Note: 𝐴 ∧ 𝐵 are finite sets 𝑛 𝐴∪𝐵 =𝑛 𝐴 +𝑛 𝐵 −𝑛 𝐴∩𝐵 𝑛 𝐴∩𝐵 =𝑛 𝐴 +𝑛 𝐵 −𝑛 𝐴∪𝐵 𝑛 𝐴 ∪ 𝐵 ∪ 𝐶 = 𝑛 𝐴 + 𝑛 𝐵 + 𝑛 𝐶 − 𝑛 𝐴 ∩ 𝐵 − 𝑛 𝐴 ∩ 𝐶 − 𝑛 𝐵 ∩ 𝐶 + 𝑛(𝐴 ∩ 𝐵 ∩ 𝐶)
Axioms Substitution Principle Commutative – Addition Commutative – Multiplication Associativity – Addition Associativity – Multiplication Reflexive Symmetric Transitive Distribution Property Cancellation Property Identity – Addition Additive Inverse Identity – Multiplication Multiplicative Property – Zero Multiplicative Property for -1 Multiplicative Inverse 6
If 𝑎 = 𝑏, then 𝑎 can be substituted for 𝑏 𝑎 + 𝑏 = 𝑏 + 𝑎 𝑎𝑏 = 𝑏𝑎 𝑎 + 𝑏 + 𝑐 = 𝑎 + 𝑏 + 𝑐 𝑎 𝑏𝑐 = 𝑎𝑏 𝑐 𝑎 = 𝑎 If 𝑎 = 𝑏 then 𝑏 = 𝑎 If 𝑎 = 𝑏 and 𝑏 = 𝑐 then 𝑎 = 𝑐 𝑎 𝑏 + 𝑐 = 𝑎𝑏 + 𝑎𝑐 and 𝑎 + 𝑏 𝑐 = 𝑎𝑐 + 𝑏𝑐 − −𝑎 = 𝑎 𝑎 + 0 = 𝑎 and 0 + 𝑎 = 𝑎 𝑎 + −𝑎 = 0 and – 𝑎 + 𝑎 = 0 𝑎 1 = 𝑎 and 1 𝑎 = 𝑎 𝑎 0 = 0 and 0 𝑎 = 0 𝑎 −1 = −𝑎 and −1 𝑎 = −𝑎 𝑎 𝑎!! = 1 and 𝑎!! 𝑎 = 1
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Arithmetic
𝑎𝑏 ± 𝑎𝑐 = 𝑎 𝑏 ± 𝑐 = 𝑏 ± 𝑐 𝑎
𝑎 𝑏 = 𝑎 𝑐 𝑏𝑐
𝑎 𝑐 𝑎𝑑 ± 𝑏𝑐 ± = 𝑏 𝑑 𝑏𝑑
𝑎𝑏 + 𝑎𝑐 = 𝑏 + 𝑐, 𝑎 ≠ 0 𝑎 𝑎±𝑏 𝑎 𝑏 = ± 𝑐 𝑐 𝑐
𝑎−𝑏 𝑏−𝑎 = 𝑐−𝑑 𝑑−𝑐 𝑎 𝑎 𝑐 𝑎𝑐 = ∙ = 𝑏 1 𝑏 𝑏 𝑐
𝑎
𝑖𝑓 𝑎 ± 𝑏 = 0 𝑡ℎ𝑒𝑛 𝑎 = ∓𝑏
𝑏 𝑎𝑏 = 𝑐 𝑐
𝑎 𝑏 = 𝑎 ∙ 𝑑 = 𝑎𝑑 𝑐 𝑏 𝑐 𝑏𝑐 𝑑
Exponents 𝑎! = 𝑎
𝑎! = 1
1 = 𝑎! 𝑎!! 𝑎 ! 𝑎! = ! 𝑏 𝑏 𝑎! ! = 𝑎!" = 𝑎!" = 𝑎! !
! !
=𝑎
𝑎 𝑏
=
𝑎 𝑎
!!
1 𝑎!
𝑎!! = !!!
𝑎! = 𝑎!!! ! 𝑎
𝑎!! 𝑏 ! = 𝑏 !! 𝑎!
𝑎!
! !
! !
!
= 𝑎! = 𝑎!
Radicals
𝑎=
!
𝑎=
!
!
! !
𝑎! = 𝑎 !
!
𝑎=
!"
𝑎=
!
! 𝑎!"
𝑎 = 𝑏
!
𝑎
!
𝑏
!
=
𝑥! = 𝑥 ,
𝑎! ! 𝑏!
𝑎 = 𝑏
! !
!
!
!
𝑎! = 𝑎 !
𝑎! = 𝑎 , 𝑛 𝑖𝑠 𝑒𝑣𝑒𝑛
−∞ < 𝑥 < ∞
𝑎! = 𝑎, 𝑛 𝑖𝑠 𝑜𝑑𝑑
𝑥
!
= 𝑥,
𝑥 ≥ 0
Complex Numbers 𝑥 = 𝑎 ± 𝑖𝑏 𝑖 = −1
Conjugate 𝑥 = 𝑎 ∓ 𝑏𝑖 !
𝑎 + 𝑏𝑖 𝑐 + 𝑑𝑖 = 𝑎𝑐 − 𝑏𝑑 + 𝑎𝑑 + 𝑏𝑐 𝑖
𝑖 = −1
𝑥𝑥 = 𝑎! + 𝑏 !
−𝑎 = 𝑖 𝑎, 𝑎 ≥ 0
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Adding and Subtracting Fractions
𝑎 𝑐 𝑎𝑑 ± 𝑏𝑐 ± = 𝑏 𝑑 𝑏𝑑
𝑔 𝑥 ℎ 𝑥 𝑔 𝑥 𝑟 𝑥 ± 𝑓 𝑥 ℎ 𝑥 ± = 𝑓 𝑥 𝑟 𝑥 𝑓 𝑥 𝑟 𝑥
Logarithmic
Log “Base” Notation Note: log 𝑥 = log!" 𝑥 or it may be log 𝑥 = ln 𝑥 = log ! 𝑥; log x is the general notation for ln x but in some books or calculators log x = log!" x and vice-versa. ln 𝑏 𝑦 = log ! 𝑥 ⇒ 𝑥 = 𝑏 ! 𝑒 = 2.718281828 … = log ! 𝑏 ln 𝑎 log ! 𝑎 = 1 log ! 1 = 0 log ! 𝑎 ! = 𝑥 ! log ! 𝑥 = ln 𝑥 log ! 𝑥𝑦 = log ! 𝑥 + log ! 𝑦 log ! 𝑥 = 𝑏 log ! 𝑥 ∞ ∞ 𝑥 1 𝑎! 𝑡 ! log ! = log ! 𝑥 − log ! 𝑦 !" 𝑒= 𝑒 = 𝑦 𝑛! 𝑛! !!! !!! Log “Natural” Notation *It is unlikely that the notation involving “log” will be used throughout the course; you may see it in the beginning of the course, as a review of some sort but that should be about all you’ll see. The “ln 𝑢” notation will be the standard as it is easier to manipulate. ln 𝑏 𝑦 = ln 𝑥 ⇒ 𝑥 = 𝑒 ! 𝑦 = 𝑒 ! ⇒ 𝑥 = ln 𝑦 log ! 𝑏 = ln 𝑎 ! ln 𝑎 = undefined, 𝑎 ≤ 0 ln 1 = 0 1 𝑒= 𝑛! !!!
ln 𝑒 ! = 𝑥 ⇒ 𝑒 !" ! = 𝑥 ln 𝑥𝑦 = ln 𝑥 + ln 𝑦 Domains:
ln 𝑒 ! = 1 ⇒ 𝑒 !" ! = 1 𝑥 ln = ln 𝑥 − ln 𝑦 𝑦 ln 𝑥 , 𝐷 = 0, ∞
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ln 𝑥 ! = 𝑏 ln 𝑥 ln 𝑥 !! = ln
1 = − ln 𝑥 𝑥
ln 𝑥 , 𝐷 = 𝑥 𝑥 > 0, 𝑥 < 0
*Factoring
𝑥 ! + 𝑥 ! = 𝑥 ! 1 + 𝑥 !!! = 𝑥 ! 𝑥 !!! + 1 𝑥 ! + 2𝑎𝑥 + 𝑎! = 𝑥 + 𝑎 ! 𝑥 ! + 3𝑎𝑥 ! + 3𝑎! 𝑥 + 𝑎! = 𝑥 + 𝑎 ! 𝑥 ! + 𝑎! = (𝑥 + 𝑎)(𝑥 ! − 𝑎𝑥 + 𝑎! ) 𝑥 + 𝑎 ! = 𝑥 ! + 3𝑎𝑥 ! + 3𝑎! 𝑥 + 𝑎! Note: *Common mistake students make when solving for x:
𝑥! − 𝑥 = 0
𝑥 ! − 𝑎! = 𝑥 + 𝑎 𝑥 − 𝑎 𝑥 ! + 𝑎 + 𝑏 𝑥 + 𝑎𝑏 = (𝑥 + 𝑎)(𝑥 + 𝑏) 𝑥 ! − 3𝑎𝑥 ! + 3𝑎! 𝑥 − 𝑎! = 𝑥 − 𝑎 ! 𝑥 ! − 𝑎! = 𝑥 − 𝑎 𝑎! + 𝑎𝑥 + 𝑥 ! 𝑥 − 𝑎 ! = 𝑥 ! − 3𝑎𝑥 ! + 3𝑎! 𝑥 − 𝑎!
The solution of 𝑥 = 0 was lost, thus:
⇒
𝑥! = 𝑥
⇒
𝑥 = 1 ⇒
𝑥! − 𝑥 = 0 𝑥 𝑥 − 1 = 0 ⇔ 𝑥 = 0 𝑜𝑟 𝑥 = 1
Long Division (quotient)(divisor)+(remainder)=dividend P=Divisor
Q=Dividend
R=Quotient
Complete The Square
𝑦 = 𝑎𝑥 ! + 𝑏𝑥 + 𝑐
𝑏 𝑏 𝑏 ! 𝑏 ! = 𝑎 𝑥! + 𝑥 + 𝑐 = 𝑎 𝑥! + 𝑥 + − + 𝑐 𝑎 𝑎 2𝑎 2𝑎 𝑏 𝑏 ! 𝑏 ! 𝑏 ! 𝑏! = 𝑎 𝑥! + 𝑥 + −𝑎 +𝑐 =𝑎 𝑥+ −𝑎 + 𝑐 𝑎 2𝑎 2𝑎 2𝑎 4𝑎! 𝑏 ! 𝑏! =𝑎 𝑥+ − +𝑐 2𝑎 4𝑎 𝑏 ! 𝑏! ∴ 𝑦=𝑎 𝑥+ +𝑐− 2𝑎 4𝑎
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Example 1: Solving for x (Formula 1) 𝑎𝑥 ! + 𝑏𝑥 = 0 𝑏 0 𝑏 ⇒ 𝑥! + 𝑥 = ⇒ 𝑥 ! + 𝑥 + 0 = 0 𝑎 𝑎 𝑎 𝑏 𝑏 ! 𝑏 ! ! ⇒ 𝑥 + 𝑥+ − =0 ⇒ 𝑎 2𝑎 2𝑎 ⇒
𝑥+
𝑏 2𝑎
!
∴
𝑥=0
or
=
𝑏! 4𝑎!
⇒
𝑥+
𝑏 𝑏 𝑥 + 𝑥+ 𝑎 2𝑎
𝑏 𝑏! =± 2𝑎 4𝑎!
!
!
⇒
𝑥=−
𝑏 ! = 2𝑎 𝑏 𝑏 ± 2𝑎 2𝑎
𝑏 𝑥=− 𝑎
Example 2: Solving for x (Formula 2) ! 𝑎𝑥 + 𝑏𝑥 + 𝑐 = 0 𝑏 𝑐 0 𝑏 𝑐 𝑏 ! 𝑏 ! ⇒ 𝑥! + 𝑥 + = ⇒ 𝑥! + 𝑥 + + − = 0 𝑎 𝑎 𝑎 𝑎 𝑎 2𝑎 2𝑎 𝑏 𝑏 ! 𝑏! 𝑐 𝑏 ! 𝑏 ! − 4𝑎𝑐 ! ⇒ 𝑥 + 𝑥+ = ! !− ⇒ 𝑥+ = 𝑎 2𝑎 2 𝑎 𝑎 2𝑎 4𝑎! 𝑏 𝑏 ! − 4𝑎𝑐 𝑏 𝑏 ! − 4𝑎𝑐 ⇒ 𝑥+ =± ⇒ 𝑥=− ± 2𝑎 2𝑎 2𝑎 2𝑎 −𝑏 ± 𝑏 ! − 4𝑎𝑐 ∴ 𝑥= 2𝑎
Compositions
𝑓∘𝑔 𝑥 =𝑓 𝑔 𝑥
𝑓∙𝑔 𝑥 =𝑓 𝑥 𝑔 𝑥
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𝑓±𝑔 𝑥 =𝑓 𝑥 ±𝑔 𝑥 𝑓 𝑓 𝑥 𝑥 = , 𝑔 𝑔 𝑥
𝑔 𝑥 ≠ 0
Functions
Vertical Line Test 𝑓 𝑥 is a function if it passes the vertical line test i.e. if you draw a vertical line anywhere on the graph, and the graph of 𝑓 only crosses it once. Even/Odd Function Even: 𝑓 −𝑥 = 𝑓 𝑥 Odd: 𝑓 −𝑥 = −𝑓 𝑥 (symmetric with respect to 𝑦-axis) (symmetric with respect to origin) Average Rate of Change 𝛥𝑦 𝑓 𝑥 − 𝑓 𝑥! = , 𝑥 ≠ 𝑥! 𝛥𝑥 𝑥 − 𝑥! Secant Line ! ! !! ! The slope of the secant line is the same as the average rate of change i.e. 𝑚 = !!! ! you then take one of the two points and plug the it into 𝑦 − 𝑦! = Difference Quotient 𝑚=
Distance Formula
! ! !! !! !!!!
!
𝑥 − 𝑥! and simplify.
𝑓 𝑥 + 𝛥𝑥 − 𝑓 𝑥 𝑓 𝑥+ℎ −𝑓 𝑥 = 𝛥𝑥 ℎ
Distance between two points on a number line 𝑃 𝑥! = 𝑃 𝑥! = 𝑃 𝑎 , 𝑄 = 𝑄 𝑥 = 𝑄 𝑥! = 𝑄 𝑏 𝑑 𝑃, 𝑄 = 𝑥 − 𝑥! ! = 𝑥 − 𝑥! = 𝑏−𝑎 ! = 𝑏−𝑎 = 𝑥! − 𝑥! ! = 𝑥! − 𝑥!
Distance between two points in a Cartesian coordinate system i.e. x vs. y graph 𝑃 𝑥! , 𝑦! , 𝑄 𝑥, 𝑦 𝑑 𝑃, 𝑄 = 𝑥 − 𝑥! ! + 𝑦 − 𝑦! !
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Midpoint Formula 𝑃 𝑥! , 𝑦!
&
𝑄 𝑥! , 𝑦!
𝑚 𝑃, 𝑄 =
Quadratic Formula !
𝑎𝑥 + 𝑏𝑥 + 𝑐 = 0
⇔
Proof: ! 𝑎𝑥 + 𝑏𝑥 + 𝑐 = 0 ⇒
⇒
𝑥! + 𝑥! 𝑦! + 𝑦! , 2 2
−𝑏 ± 𝑏 ! − 4𝑎𝑐 𝑥= 2𝑎
𝑏 𝑐 0 𝑏 𝑐 𝑏 ! 𝑏 ! ! ! 𝑥 + 𝑥+ = ⇒ 𝑥 + 𝑥+ + − = 0 𝑎 𝑎 𝑎 𝑎 𝑎 2𝑎 2𝑎 𝑏 𝑏 ! 𝑏! 𝑐 𝑏 ! 𝑏 ! − 4𝑎𝑐 ! 𝑥 + 𝑥+ = ! !− ⇒ 𝑥+ = 𝑎 2𝑎 2 𝑎 𝑎 2𝑎 4𝑎!
⇒ ∴
𝑥+
𝑏 𝑏 ! − 4𝑎𝑐 =± 2𝑎 2𝑎
⇒
𝑥=−
𝑏 𝑏 ! − 4𝑎𝑐 ± 2𝑎 2𝑎
−𝑏 ± 𝑏 ! − 4𝑎𝑐 𝑥= 2𝑎
Discriminant:
i) Two real solutions if 𝑏 ! − 4𝑎𝑐 > 0
ii) Repeated solutions if 𝑏 ! − 4𝑎𝑐 = 0 iii) Two complex solutions 𝑖𝑓𝑏 ! − 4𝑎𝑐 < 0
Graphing a Line
From the form 𝑦 = 𝑚𝑥 + 𝑏 you can easily graph a line by identifying two points and then connecting them. ! ! The equation will more generally appear as 𝑦 = ±! 𝑥 + 𝑏 where 𝑚 = ±!, 𝑐 is the rise and ± 𝑑 is the run (𝑐 always goes up and 𝑑 goes either left or right.)
The first point is 𝑃! 0, 𝑏 The second point is 𝑃! (±𝑑, 𝑏 + 𝑐)
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Plot these two points and connect a line through them. Point Slope Form: 𝑦 − 𝑦! = 𝑚 𝑥 − 𝑥! 𝛥𝑦 𝑚 = 𝑠𝑙𝑜𝑝𝑒 = 𝛥𝑥 𝛥𝑦 𝑦 − 𝑦! ⇒ 𝑚= = 𝛥𝑥 𝑥 − 𝑥! 𝑦 − 𝑦! ⇒ 𝑚= 𝑥 − 𝑥! ⇒ 𝑥 − 𝑥! 𝑚 = 𝑦 − 𝑦! ∴ 𝑦 − 𝑦! = 𝑚(𝑥 − 𝑥! ) Slope Intercept Form: 𝑦 = 𝑚𝑥 + 𝑏 𝛥𝑦 𝛥𝑦 𝑦 − 𝑦! 𝑚 = 𝑠𝑙𝑜𝑝𝑒 = ⇒ 𝑚= = 𝛥𝑥 𝛥𝑥 𝑥 − 𝑥! 𝑦 − 𝑦! ⇒ 𝑚= 𝑥 − 𝑥! ⇒ 𝑥 − 𝑥! 𝑚 = 𝑦 − 𝑦! ⇒ 𝑚𝑥 − 𝑚𝑥! = 𝑦 − 𝑦! ⇒ 𝑦 = 𝑚𝑥 − 𝑚𝑥! + 𝑦! ⇒ 𝑦 = 𝑚𝑥 + 𝑦! − 𝑚𝑥! ⇒ 𝑦 = 𝑚𝑥 + 𝑦! − 𝑚𝑥! , setting 𝑏 = 𝑦! − 𝑚𝑥! ∴ 𝑦 = 𝑚𝑥 + 𝑏 Standard or General Form 𝐴𝑥 + 𝐵𝑦 = 𝐶 Parallel Line (equal slopes) 𝑦! = 𝑚𝑥 + 𝑏! ∥ 𝑦! = 𝑚𝑥 + 𝑏!
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Perpendicular Line (product of slopes are -1) 𝑦! = 𝑚𝑥 + 𝑏! ⊥ 𝑦! = −
*Domain Restrictions
1 𝑥 + 𝑏! 𝑚
For the following, 𝑓 𝑥 , 𝑔 𝑥 , ℎ 𝑥 are assumed to be continuous for all real numbers. 𝑥 = 𝑎! 𝑥 ! ± 𝑎! 𝑥 !!! ± 𝑎! 𝑥 !!! ± ⋯ ± 𝑎! 𝑥 !!! Polynomial No Restrictions 𝑓 𝑥 Fraction 𝒈 𝒙 ≠ 𝟎 ℎ 𝑥 = 𝑔 𝑥 ! Radical, if n is even 𝒈 𝒙 ≥ 𝟎 𝑓 𝑥 = 𝑔(𝑥) ! Radical, if n is odd No Restrictions 𝑓 𝑥 = 𝑔(𝑥) 𝑓 𝑥 Fraction with Radical in 𝐈𝐟 𝐧 𝐢𝐬 𝐞𝐯𝐞𝐧 𝒈 𝒙 ℎ 𝑥 =! denominator > 𝟎 𝐢𝐟 𝐧 𝐢𝐬 𝐨𝐝𝐝 𝒈 𝒙 𝑔 𝑥 ≠ 𝟎 Natural Log 𝑓 𝑥 = ln 𝑔 𝑥 𝒈 𝒙 > 𝟎 ! ! Exponential 𝐍𝐨 𝐑𝐞𝐬𝐭𝐫𝐢𝐜𝐭𝐢𝐨𝐧𝐬 ℎ(𝑥) = 𝑓 𝑥
Inverse Functions 𝑦=𝑓 𝑥 ⇒ 𝑥 = 𝑓 𝑦 !! = 𝑓 𝑓 !! 𝑥 If 𝑓 𝑥 is one-to-one it has an inverse The domain of 𝑓 𝑥 is the range of 𝑓 !! 𝑥 The range of 𝑓 𝑥 is the domain of 𝑓 !! 𝑥 𝑦 = 𝑓 𝑥 ⇒ 𝑥 = 𝑓 𝑦 ⇒ 𝑦 = 𝑓 !! 𝑥
Asymptotes, Holes and Graphs
An asymptote occurs where the function is getting infinitely close to a line on the graph but never touches the line. Horizontal asymptotes may cross the line from time-to-time; it is the end behavior we are concerned with. There are three types of asymptotes: Horizontal, Vertical and Oblique. Oblique asymptotes, will most likely, not be used in your calculus course but vertical and horizontal will be used frequently in order to graph functions. 14 copyright © wesolvethem.com | WESOLVETHEM LLC
Hole in a Graph
𝑥! − 4 𝑓 𝑥 = 𝑥−2
⇒
𝑥≠2
Three General Cases for Horizontal Asymptotes Since there are so many conditions and situations for asymptotes and the methods learned in algebra are so minimal to what is used in calculus, we will come back to this later. Case 1 𝑥 ! + 𝑥 !!! + ⋯ 𝑛 > 𝑚 ⇒ 𝐻𝐴: 𝑦 = 0 𝑓 𝑥 = ! !!! 𝑥 +𝑥 +⋯ Case 2 𝑥 ! + 𝑥 !!! + ⋯ 𝑛 < 𝑚 ⇒ 𝐻𝐴: 𝑛𝑜𝑛𝑒 𝑓 𝑥 = ! !!! 𝑥 +𝑥 +⋯ 𝑎 Case 3 a𝑥 ! + 𝑥 !!! + ⋯ 𝑛 = 𝑚 ⇒ 𝐻𝐴: 𝑦 = 𝑓 𝑥 = 𝑏 b𝑥 ! + 𝑥 !!! + ⋯ Ex. 1 Horizontal 𝑥! + 𝑥 + 1 𝑓 𝑥 = and Vertical 𝑥! + 𝑥! + 𝑥 + 1 𝐻𝐴: 𝑦 = 0, 𝑉𝐴: 𝑥 = −2
Ex. 2 Oblique
𝑓 𝑥 =
𝑥! + 𝑥! + 𝑥 + 1 𝑥! + 𝑥 + 1
𝑁𝑜 𝐻𝐴,
𝑂𝐴:
𝐻𝐴:
3 𝑦= , 2
𝑉𝐴:
𝑦 = 𝑥
3𝑥 ! + 𝑥 𝑓 𝑥 = ! 2𝑥 + 1
Ex. 3 Horizontal and Vertical
1 𝑥 =−! 2
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Inequalities
𝑓 𝑥 𝑓 𝑥
< 𝑎 ⇒ −𝑎 < 𝑓 𝑥 < 𝑎 or 𝑓 𝑥 < 𝑎 and 𝑓 𝑥 > −𝑎 ≤ 𝑎 ⇒ −𝑎 ≤ 𝑓 𝑥 ≤ 𝑎 or 𝑓 𝑥 ≤ 𝑎 and 𝑓 𝑥 ≥ −𝑎
Interest Formulas
𝐴 = 𝐴! 𝑒 !"
𝑃 = 𝑃!
𝑟 12
𝑟 1 − 1 + 12
!!
L=Loan 𝑃 = Monthly Payment 𝑟 = Interest rate for annual 𝑡 = Loan length in months
Physics Formulas
(rate)(time)=distance 𝑟𝑡 = 𝑑
Symmetry
By Point 𝑥-axis For every point 𝑥, 𝑦 there is a 𝑥, −𝑦 𝑦-axis For every point 𝑥, 𝑦 there is a −𝑥, 𝑦 origin For every point 𝑥, 𝑦 there is a −𝑥, −𝑦 Testing 𝑥-axis: Replace each 𝑦 with a – 𝑦, if the same equation results, it is symmetric. 𝑦-axis: Replace each 𝑥 with a – 𝑥, if the same equation results, it is symmetric. Origin: Replace each 𝑥, 𝑦 with a −𝑥, – 𝑦, if the same equation results, it is symmetric.
Variations (Proportionality) 𝑘 is the constant of proportionality
𝑦 is proportional to x: y= 𝑘𝑥
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𝑦 is inversely proportional to ! 𝑥: 𝑦 = !
Common Graphs and Formulas 𝑦 = 𝑥!
𝑦 = 𝑥!
_________________________________________________________________________________________________________ 1 𝑦= 𝑥 𝑦= 𝑥
_________________________________________________________________________________________________________ ! 𝑦=𝑒 𝑦 = ln 𝑥
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𝑥! + 𝑦! = 1
𝑥 ! − 𝑦 ! = 1
_________________________________________________________________________________________________________ 𝑦! − 𝑥! = 1 𝑦 = 𝑥! − 1
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Equation of a Line 𝑠𝑙𝑜𝑝𝑒 = 𝑚 =
𝑦 = 𝑚𝑥 + 𝑏 𝑦! − 𝑦! = 𝑚 𝑥! − 𝑥! 𝐴𝑥 + 𝐵𝑦 = 𝐶 ! 𝑦 = 𝑎𝑥 + 𝑏𝑥 + 𝑐 𝑦 = 𝑎 𝑥 − ℎ ! + 𝑘 ! ! 𝑥 + 𝑦 + 𝑎𝑥 + 𝑏𝑦 + 𝑐 = 0 ⇒ ! 𝑥 − ℎ + 𝑦 − 𝑘 ! = 𝑟!
𝑦! − 𝑦! 𝑥! − 𝑥!
Equation of Parabola !
!
Vertex: ℎ, 𝑘 = − !! , 𝑓 − !!
Equation of Circle Center: ℎ, 𝑘 Radius: 𝑟 Equation of Ellipse
𝑥−ℎ 𝑎!
!
Right Point: ℎ + 𝑎, 𝑘
𝑦−𝑘 + 𝑏!
!
= 1
Left Point: ℎ − 𝑎, 𝑘 Top Point: ℎ, 𝑘 + 𝑏
Bottom Point: ℎ, 𝑘 − 𝑏 Equation of Hyperbola (1) Center: ℎ, 𝑘 ! Slope: ± !
𝑥−ℎ 𝑎!
!
−
𝑦−𝑘 𝑏!
!
= 1
!
Asymptotes: 𝑦 = ± ! 𝑥 − ℎ + 𝑘 Vertices: ℎ + 𝑎, 𝑘 , ℎ − 𝑎, 𝑘 Equation of Hyperbola (2) Center: ℎ, 𝑘 ! Slope: ± !
𝑦−𝑘 𝑎!
!
𝑥−ℎ − 𝑏!
!
= 1
!
Asymptotes: 𝑦 = ± ! 𝑥 − ℎ + 𝑘 Vertices: ℎ, 𝑘 + 𝑏 , ℎ, 𝑘 − 𝑏
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Areas Square: 𝐴 = 𝐿! = 𝑊 ! Rectangle: 𝐴 = 𝐿 ∙ 𝑊 Circle: 𝐴 = 𝜋 ∙ 𝑟 ! ! ! Ellipse: 𝐴 = 𝜋 ∙ 𝑎𝑏 Triangle: 𝐴 = ! 𝑏 ∙ ℎ Trapezoid: 𝐴 = ! 𝑎 + 𝑏 ∙ ℎ !" Parallelogram: 𝑏 ∙ ℎ Rhombus: 𝐴 = ! , 𝑝 and 𝑞 are the diagonals Surface Areas Cube: 𝐴! = 6𝐿! = 6𝑊 ! Box: 𝐴! = 2(𝐿𝑊 + 𝑊𝐻 + 𝐻𝐿) Sphere: 𝐴! = 4𝜋𝑟 ! Cone: 𝐴! = 𝜋𝑟 𝑟 + ℎ! + 𝑟 ! Cylinder: 2𝜋𝑟ℎ + 2𝜋𝑟 ! Volumes ! Cube: 𝑉 = 𝐿! = 𝑊 ! Box: 𝑉 = 𝐿 ∙ 𝑊 ∙ 𝐻 Sphere: 𝑉 = ! 𝜋 ∙ 𝑟 ! ! ! Cone: 𝑉 = ! 𝜋 ∙ 𝑟 ! ℎ Ellipsoid: 𝑉 = ! 𝜋 ∙ 𝑎𝑏𝑐, 𝑎, 𝑏, 𝑐 are the radii Business Functions Cost Function 𝐶 𝑥 Revenue Function 𝑅 𝑥 Profit Function 𝑃 𝑥 = 𝑅 𝑥 − 𝐶 𝑥 Marginal Cost Function 𝐶 ! 𝑥 ! Marginal Revenue Function 𝑅 𝑥 Marginal Profit Function 𝑃! 𝑥 = 𝑅! 𝑥 = 𝐶! 𝑥 𝐶 𝑥 Average Cost Function 𝐶 𝑥 = 𝑥 ! ! ! ! ! ! !! ! Average Revenue Function 𝑅 𝑥 = ! Average Profit Function 𝑃 𝑥 = ! = !
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Average Rate of Change of 𝒇 and Slope of Secant Line
!" !"
=
! ! !! ! !!!
= 𝑚!"#$%& from 𝑃! (𝑎, 𝑓 𝑎 ) and 𝑃! 𝑏, 𝑓 𝑏
Difference Quotient ! !"!!! !! !! ! 𝑚!"#$%& = , 𝛥𝑥 = ℎ ⇒ 𝑚!"#$%& = !"
!!! !! ! !
, ℎ ≠ 0
Functions Constant Function Identity Function Square Function Cube Function Square Root Function Cube Root Function Reciprocal Function Absolute Value Function Greatest Integer Function Piecewise Function Power Function Ration Function
𝑦 = 𝑐 𝑦 = 𝑥 𝑦 = 𝑥! 𝑦 = 𝑥! 𝑦 = 𝑥 ! 𝑦 = 𝑥 1 𝑦= 𝑥 𝑦= 𝑥 𝑦 = int 𝑥 ∗ 𝑔 𝑥 , 𝑥 ∈ 𝐷! 𝑓 𝑥 = ℎ 𝑥 , 𝑥 ∈ 𝐷! 𝑦 = 𝑎𝑥 ! 𝑔 𝑥 𝑓 𝑥 = , ℎ 𝑥 ≠ 0 ℎ 𝑥
Graph Shifts and Compressions Vertically up for 𝑓 𝑥 Vertically down for 𝑓 𝑥 Horizontally left for 𝑓 𝑥 Horizontally right for 𝑓 𝑥 𝑎𝑓 𝑥 multiply each y-coordinate by a Vertically Compressed: 0 < 𝑎 < 1 Vertically Stretched: 𝑎 > 1 ! 𝑓 𝑎𝑥 Multiply each x-coordinate by ! Horizontal Compression: 𝑎 > 1 Horizontal Stretch: 0 < 𝑎 < 1 Reflection about x-axis Reflection about y-axis
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𝑓 𝑥 + 𝑘 𝑓 𝑥 − 𝑘 𝑓 𝑥+ℎ 𝑓 𝑥−ℎ 𝑥, 𝑎𝑦 1 𝑥, 𝑦 𝑎 −𝑓 𝑥 𝑓 −𝑥
21
Systems of equations
𝑎𝑥 + 𝑏𝑦 = 𝑒 ⇒ 𝑐𝑥 + 𝑑𝑦 = 𝑓
𝑎𝑥 + 𝑏𝑦 + 𝑐𝑧 = 𝑑 𝑒𝑥 + 𝑓𝑦 + 𝑔𝑦 = ℎ 𝑖𝑥 + 𝑗𝑦 + 𝑘𝑧 = 𝑙
⇒
𝑎 𝑐 𝑎 𝑒 𝑖
𝑒 𝑏 𝑥 = 𝑓 𝑦 𝑑 𝑏 𝑓 𝑗
𝑐 𝑥 𝑑 𝑔 𝑦 = ℎ 𝑘 𝑧 𝑙
𝑎 The Coefficient Matrix = 𝑒 𝑖
𝑎 𝑐
⇒
𝑏𝑒 𝑑𝑓
⇒ 𝑏 𝑓 𝑗
𝑎 𝑒 𝑖
𝑏 𝑓 𝑗
𝑐 𝑑 𝑔ℎ 𝑘 𝑙
𝑐 𝑔 𝑘
Rank of matrix and pivots
𝟏 , 1
𝟏 0
𝟏 0 0
𝟏 , 0
𝟏
0 , 1
0 0 , 𝟏
𝑟𝑎𝑛𝑘 𝐴! = 2 𝑟𝑎𝑛𝑘 𝐴! = 2 𝑟𝑎𝑛𝑘 𝐴! = 2
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𝟏 1 1 𝟏 1 1 𝟏 0 0
1 1,
𝑟𝑎𝑛𝑘 𝐴!" = 1
𝑟𝑎𝑛𝑘 𝐴!! = 1
1 1 1 1 , 1 1 1 1 1 −𝟏 , 1 1 1 𝟏 0
𝑟𝑎𝑛𝑘 𝐴! = 1
𝑟𝑎𝑛𝑘 𝐴! = 1
𝟏 0 , 0
𝑟𝑎𝑛𝑘 𝐴! = 1
0 , 𝟏
0 𝟏
𝑟𝑎𝑛𝑘 𝐴! = 1
1,
𝟏
𝟏 1 , 1 𝟏 0
𝑟𝑎𝑛𝑘 𝐴! = 1
𝑟𝑎𝑛𝑘 𝐴! = 1
1 1,
𝟏
1,
𝟏
1 1 , 𝟏
𝑟𝑎𝑛𝑘 𝐴!" = 1
𝑟𝑎𝑛𝑘 𝐴!" = 2
𝑟𝑎𝑛𝑘 𝐴!" = 3
Determinate’s of a (2x2) matrix Various ways to check determinant (2x2): 𝑎 𝐴= 𝑐
𝑏 𝑑
𝑎 ⇒ det 𝐴 = 𝐴 = 𝑐
𝐴=
𝑎 0
𝑏 𝑐
⇒ det 𝐴 = 𝐴 =
𝑎 0
𝑏 = 𝑎 𝑐 𝑐
𝐴=
𝑎 0
0 𝑏
⇒ det 𝐴 = 𝐴 =
𝑎 0
0 = 𝑎 𝑏 𝑏
𝑏 = 𝑎 𝑑 − 𝑏 𝑐 𝑑
Determinate of a (3x3) and higher matrices Cofactor Expansion Note:
𝑎 𝑑 𝑔
𝑎 𝑐
𝑎 𝑏 𝑒 𝑓 𝑖 𝑗 𝑚 𝑛
𝑐 𝑔 𝑘 𝑜
𝑏 𝑒 ℎ
𝑑 𝑓 ℎ = +𝑎 𝑗 𝑙 𝑛 𝑝
𝑏 = 𝑎 𝑑 − 𝑏 𝑐 𝑑
𝑐 𝑓 = +𝑎 𝑒 ℎ 𝑖 𝑔 𝑘 𝑜
𝑑 𝑓 −𝑏 𝑔 𝑖
ℎ 𝑒 𝑙 −𝑏 𝑖 𝑝 𝑚
𝑔 𝑘 0
𝑓 𝑑 +𝑐 𝑔 𝑖
ℎ 𝑒 𝑙 +𝑐 𝑖 𝑝 𝑚
𝑓 𝑗 𝑛
𝑒 ℎ ℎ 𝑒 𝑙 −𝑑 𝑖 𝑝 𝑚
𝑓 𝑗 𝑛
𝑔 𝑘 𝑜
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23
Table of Contents General Symbols and Notations ........................................................................................................................................................... 4 Types of numbers ....................................................................................................................................................................................... 5 Properties ...................................................................................................................................................................................................... 5 Meanings ........................................................................................................................................................................................................ 5 Complementation of sets ........................................................................................................................................................................ 5 Set Laws ......................................................................................................................................................................................................... 5 De Morgan’s Laws ...................................................................................................................................................................................... 6 Number of Elements in a Set ................................................................................................................................................................. 6 Axioms ............................................................................................................................................................................................................ 6 Arithmetic ...................................................................................................................................................................................................... 7 Exponents ...................................................................................................................................................................................................... 7 Radicals .......................................................................................................................................................................................................... 7 Complex Numbers ..................................................................................................................................................................................... 7 Adding and Subtracting Fractions ...................................................................................................................................................... 8 Logarithmic ................................................................................................................................................................................................... 8 Log “Base” Notation ................................................................................................................................................................................... 8 Log “Natural” Notation ............................................................................................................................................................................ 8 *Factoring ...................................................................................................................................................................................................... 9 Note: .................................................................................................................................................................................................................. 9 Long Division ............................................................................................................................................................................................... 9 Complete The Square ............................................................................................................................................................................... 9 Example 1: Solving for x (Formula 1) ............................................................................................................................................. 10 Example 2: Solving for x (Formula 2) ............................................................................................................................................. 10 Compositions ............................................................................................................................................................................................ 10 Functions .................................................................................................................................................................................................... 11 Vertical Line Test ..................................................................................................................................................................................... 11 Even/Odd Function ................................................................................................................................................................................. 11 Average Rate of Change ........................................................................................................................................................................ 11 Secant Line .................................................................................................................................................................................................. 11 Difference Quotient ................................................................................................................................................................................. 11 Distance Formula .................................................................................................................................................................................... 11 Midpoint Formula ................................................................................................................................................................................... 12 Quadratic Formula .................................................................................................................................................................................. 12 Proof: ............................................................................................................................................................................................................. 12 Discriminant: ............................................................................................................................................................................................. 12 Graphing a Line ........................................................................................................................................................................................ 12 Point Slope Form: ..................................................................................................................................................................................... 13 Slope Intercept Form: ............................................................................................................................................................................. 13 Standard or General Form ................................................................................................................................................................... 13 Parallel Line (equal slopes) ................................................................................................................................................................. 13 Perpendicular Line (product of slopes are -1) ............................................................................................................................. 14 *Domain Restrictions ............................................................................................................................................................................ 14 Polynomial .................................................................................................................................................................................................. 14 Fraction ........................................................................................................................................................................................................ 14 Radical, if n is even .................................................................................................................................................................................. 14 Radical, if n is odd .................................................................................................................................................................................... 14 Fraction with Radical in denominator ........................................................................................................................................... 14 Natural Log ................................................................................................................................................................................................. 14 Exponential ................................................................................................................................................................................................. 14 Inverse Functions .................................................................................................................................................................................... 14 Asymptotes, Holes and Graphs .......................................................................................................................................................... 14
2
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Hole in a Graph ......................................................................................................................................................................................... 15 Three General Cases for Horizontal Asymptotes ........................................................................................................................ 15 Ex. 1 Horizontal and Vertical .............................................................................................................................................................. 15 Ex. 2 Oblique ............................................................................................................................................................................................... 15 Ex. 3 Horizontal and Vertical .............................................................................................................................................................. 15 Inequalities ................................................................................................................................................................................................ 16 Interest Formulas .................................................................................................................................................................................... 16 Physics Formulas .................................................................................................................................................................................... 16 Symmetry ................................................................................................................................................................................................... 16 By Point ........................................................................................................................................................................................................ 16 Testing .......................................................................................................................................................................................................... 16 Variations (Proportionality) .............................................................................................................................................................. 16 Common Graphs and Formulas ........................................................................................................................................................ 17 Equation of a Line .................................................................................................................................................................................... 19 Equation of Parabola ............................................................................................................................................................................. 19 Equation of Circle ..................................................................................................................................................................................... 19 Equation of Ellipse ................................................................................................................................................................................... 19 Equation of Hyperbola (1) ................................................................................................................................................................... 19 Equation of Hyperbola (2) ................................................................................................................................................................... 19 Areas .............................................................................................................................................................................................................. 20 Surface Areas ............................................................................................................................................................................................. 20 Volumes ........................................................................................................................................................................................................ 20 Business Functions ................................................................................................................................................................................... 20 Average Rate of Change of f and Slope of Secant Line ............................................................................................................. 21 Difference Quotient ................................................................................................................................................................................. 21 Functions .................................................................................................................................................................................................... 21 Graph Shifts and Compressions ........................................................................................................................................................ 21 Systems of equations ............................................................................................................................................................................. 22 Rank of matrix and pivots ................................................................................................................................................................... 22 Determinate’s of a (2x2) matrix ....................................................................................................................................................... 23 Determinate of a (3x3) and higher matrices ............................................................................................................................... 23 Cofactor Expansion ................................................................................................................................................................................. 23
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3
General Symbols and Notations Symbol = ≠ ± ∓ 𝑖𝑓𝑓, ⇔ ⇒ < ≤ ≥ > × ∗ or ∙ … … … … … … … … ∞ 𝛥 ∑ 𝜃
…
…
Meaning Equal Not equal Plus or Minus Minus or Plus If and only if Implies Less than Less than equal Greater than equal Greater than Times Multiplication Multiplication Multiplication Multiplication Exponential Multiplication Infinity Displacement or change of Summation
𝑓 𝑥 𝑓 𝑥, 𝑦 ∈ ∀ ∴ ∵ ≡ , , ⊂ ⊆ , …,… ∪
Theta – reserved for angles Function of 𝑥 Function of 𝑥 and 𝑦 In or element of For all Therefore Because Equivalent Open interval Closed interval Proper Subset Subset (equal) Half open/closed Set of numbers Union
∩ ℝ 𝑃 𝑥! , 𝑦!
Intersection Real numbers Point
4
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Example 0 = 0 1 ≠ 0 𝑥 = ±𝑎 ⇒ 𝑥 = 𝑎 𝑜𝑟 𝑥 = −𝑎 𝑥 = ∓𝑎 ⇒ 𝑥 = −𝑎 𝑜𝑟 𝑥 = 𝑎 𝑝 ⇒ 𝑞 and 𝑞 ⇒ 𝑝 then 𝑝 ⇔ 𝑞 𝑝 ⇒ 𝑞 𝑥 − 𝑎 < 0 ⇒ 𝑥 < 𝑎 𝑥 − 𝑎 ≤ 0 ⇒ 𝑥 ≤ 𝑎 𝑥 − 𝑎 ≥ 0 ⇒ 𝑥 ≥ 𝑎 𝑥 − 𝑎 > 0 ⇒ 𝑥 > 𝑎 2×3 = 6 2 ∗ 3 = 6 or 2 ⋅ 3 = 6 2 3 = 6 2 3 = 6 2 3 = 6 2 3 !!! 3 − 2 = 6 ! 1 = 6 Never ends 𝛥𝑥 = 𝑥 − 𝑥! !
𝑎! 𝑥 ! = 𝑎! 𝑥 ! + 𝑎! 𝑥 ! + 𝑎! 𝑥 ! !!!
𝜋 = 45° 4 𝑓 𝑥 = 𝑥 ! + ⋯ 𝑓 𝑥, 𝑦 = 𝑥𝑦 ! + ⋯ 𝑥 ∈ 𝑎, 𝑏 means 𝑎 ≤ 𝑥 < 𝑏 ∀! (𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥) 𝑥 − 𝑎 = 0 ⇔ 𝑥 = 𝑎 ∴ 𝑥 = 𝑎 ∵ 𝑥 − 𝑎 = 0, 𝑥 = 𝑎 −2, 3 ≡ −2 < 𝑥 < 3 −2, 3 ≡ −2 < 𝑥 < 3 2, 3 ≡ 2 ≤ 𝑥 ≤ 3 𝐴 ⊂ 𝐵 ⇒ 𝐵 ⊄ 𝐴 𝐴 ⊆ 𝐵 ⇒ 𝐴 = 𝐵 1, 4 ≡ 1 ≤ 𝑥 < 4 1,3,5,7 𝐷 = −∞, 0 ∪ 0, ∞ 1,2,3 ∪ 3,4,5 = 1, 2, 3, 4, 5 1,2,3 ∩ 3,4,5 = 3 𝐷 = −∞, ∞ 1, 𝑓 1 𝜃=
Types of numbers
Integers … , −3, −2, −1,0, 1,2,3, …
Rational ! , 𝑏 ≠ 0 and ! 𝑎, 𝑏 are integers
Irrational A number that cannot be expressed as a fraction e.g. 𝜋
Complex 𝑥 = 𝑎 + 𝑏𝑖 where 𝑎 and 𝑏 are any number
Properties Reflexive 𝑎 = 𝑎
Symmetric 𝑎 = 𝑏 then 𝑏 = 𝑎
Transitive 𝑎 = 𝑏 and 𝑏 = 𝑐 then𝑎 = 𝑐
Substitution 𝑎 = 𝑏 then b can replace 𝑎
Meanings Both 𝐴 and 𝐵 have the same elements Subset: If every element of a set 𝐴 is in 𝐵 Proper Subset: If every element in A is also in B but 𝐴 ≠ 𝐵: Intersection: The elements that are both in 𝐴 and in 𝐵 Union: All elements from 𝐴 and 𝐵 are in 𝐴 union 𝐵 Compliment: If 𝐴 ⊂ 𝑈, and 𝑈 is the universal set
𝐴 = 𝐵 𝐴 ⊆ 𝐵 ⇒ 𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴 ⇔ 𝐴 = 𝐵 𝐴 ⊂ 𝐵 𝐴∩𝐵 = 𝑥 𝑥 ∈𝐴∧𝑥 ∈𝐵 𝐴∪𝐵 = 𝑥 𝑥 ∈𝐴∨𝑥 ∈𝐵∨𝑥 ∈𝐴∩𝐵 𝐴 = 𝐴! = 𝑥 𝑥 ∈ 𝑈 ∧ 𝑥 ∉ 𝐴
Complementation of sets a. 𝑈 ! = ∅
b. ∅! = 𝑈
c.
𝐴!
!
= 𝐴
d. 𝐴 ∪ 𝐴! = 𝑈
e. 𝐴 ∩ 𝐴! = ∅
Set Laws 𝐴 ∪ 𝐵 = 𝐵 ∪ 𝐴 𝐴 ∩ 𝐵 = 𝐵 ∩ 𝐴 𝐴∪ 𝐴∩ 𝐴∪ 𝐴∩
𝐵 ∪ 𝐶 = 𝐴 ∪ 𝐵 ∪ 𝐶 𝐵 ∩ 𝐶 = 𝐴 ∩ 𝐵 ∩ 𝐶 𝐵∩𝐶 = 𝐴∪𝐵 ∩ 𝐴∪𝐶 𝐵∪𝐶 = 𝐴∩𝐵 ∪ 𝐴∩𝐶
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Commutative law for union Commutative law for intersection Associative law for union Associative law for intersection Distributive law for union Distributive law for intersection
5
De Morgan’s Laws
i.
𝐴∪𝐵
!
= 𝐴! ∩ 𝐵!
ii.
𝐴∩𝐵
!
= 𝐴! ∪ 𝐵!
Number of Elements in a Set Note: 𝐴 ∧ 𝐵 are finite sets 𝑛 𝐴∪𝐵 =𝑛 𝐴 +𝑛 𝐵 −𝑛 𝐴∩𝐵 𝑛 𝐴∩𝐵 =𝑛 𝐴 +𝑛 𝐵 −𝑛 𝐴∪𝐵 𝑛 𝐴 ∪ 𝐵 ∪ 𝐶 = 𝑛 𝐴 + 𝑛 𝐵 + 𝑛 𝐶 − 𝑛 𝐴 ∩ 𝐵 − 𝑛 𝐴 ∩ 𝐶 − 𝑛 𝐵 ∩ 𝐶 + 𝑛(𝐴 ∩ 𝐵 ∩ 𝐶)
Axioms Substitution Principle Commutative – Addition Commutative – Multiplication Associativity – Addition Associativity – Multiplication Reflexive Symmetric Transitive Distribution Property Cancellation Property Identity – Addition Additive Inverse Identity – Multiplication Multiplicative Property – Zero Multiplicative Property for -1 Multiplicative Inverse 6
If 𝑎 = 𝑏, then 𝑎 can be substituted for 𝑏 𝑎 + 𝑏 = 𝑏 + 𝑎 𝑎𝑏 = 𝑏𝑎 𝑎 + 𝑏 + 𝑐 = 𝑎 + 𝑏 + 𝑐 𝑎 𝑏𝑐 = 𝑎𝑏 𝑐 𝑎 = 𝑎 If 𝑎 = 𝑏 then 𝑏 = 𝑎 If 𝑎 = 𝑏 and 𝑏 = 𝑐 then 𝑎 = 𝑐 𝑎 𝑏 + 𝑐 = 𝑎𝑏 + 𝑎𝑐 and 𝑎 + 𝑏 𝑐 = 𝑎𝑐 + 𝑏𝑐 − −𝑎 = 𝑎 𝑎 + 0 = 𝑎 and 0 + 𝑎 = 𝑎 𝑎 + −𝑎 = 0 and – 𝑎 + 𝑎 = 0 𝑎 1 = 𝑎 and 1 𝑎 = 𝑎 𝑎 0 = 0 and 0 𝑎 = 0 𝑎 −1 = −𝑎 and −1 𝑎 = −𝑎 𝑎 𝑎!! = 1 and 𝑎!! 𝑎 = 1
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Arithmetic
𝑎𝑏 ± 𝑎𝑐 = 𝑎 𝑏 ± 𝑐 = 𝑏 ± 𝑐 𝑎
𝑎 𝑏 = 𝑎 𝑐 𝑏𝑐
𝑎 𝑐 𝑎𝑑 ± 𝑏𝑐 ± = 𝑏 𝑑 𝑏𝑑
𝑎𝑏 + 𝑎𝑐 = 𝑏 + 𝑐, 𝑎 ≠ 0 𝑎 𝑎±𝑏 𝑎 𝑏 = ± 𝑐 𝑐 𝑐
𝑎−𝑏 𝑏−𝑎 = 𝑐−𝑑 𝑑−𝑐 𝑎 𝑎 𝑐 𝑎𝑐 = ∙ = 𝑏 1 𝑏 𝑏 𝑐
𝑎
𝑖𝑓 𝑎 ± 𝑏 = 0 𝑡ℎ𝑒𝑛 𝑎 = ∓𝑏
𝑏 𝑎𝑏 = 𝑐 𝑐
𝑎 𝑏 = 𝑎 ∙ 𝑑 = 𝑎𝑑 𝑐 𝑏 𝑐 𝑏𝑐 𝑑
Exponents 𝑎! = 𝑎
𝑎! = 1
1 = 𝑎! 𝑎!! 𝑎 ! 𝑎! = ! 𝑏 𝑏 𝑎! ! = 𝑎!" = 𝑎!" = 𝑎! !
! !
=𝑎
𝑎 𝑏
=
𝑎 𝑎
!!
1 𝑎!
𝑎!! = !!!
𝑎! = 𝑎!!! ! 𝑎
𝑎!! 𝑏 ! = 𝑏 !! 𝑎!
𝑎!
! !
! !
!
= 𝑎! = 𝑎!
Radicals
𝑎=
!
𝑎=
!
!
! !
𝑎! = 𝑎 !
!
𝑎=
!"
𝑎=
!
! 𝑎!"
𝑎 = 𝑏
!
𝑎
!
𝑏
!
=
𝑥! = 𝑥 ,
𝑎! ! 𝑏!
𝑎 = 𝑏
! !
!
!
!
𝑎! = 𝑎 !
𝑎! = 𝑎 , 𝑛 𝑖𝑠 𝑒𝑣𝑒𝑛
−∞ < 𝑥 < ∞
𝑎! = 𝑎, 𝑛 𝑖𝑠 𝑜𝑑𝑑
𝑥
!
= 𝑥,
𝑥 ≥ 0
Complex Numbers 𝑥 = 𝑎 ± 𝑖𝑏 𝑖 = −1
Conjugate 𝑥 = 𝑎 ∓ 𝑏𝑖 !
𝑎 + 𝑏𝑖 𝑐 + 𝑑𝑖 = 𝑎𝑐 − 𝑏𝑑 + 𝑎𝑑 + 𝑏𝑐 𝑖
𝑖 = −1
𝑥𝑥 = 𝑎! + 𝑏 !
−𝑎 = 𝑖 𝑎, 𝑎 ≥ 0
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7
Adding and Subtracting Fractions
𝑎 𝑐 𝑎𝑑 ± 𝑏𝑐 ± = 𝑏 𝑑 𝑏𝑑
𝑔 𝑥 ℎ 𝑥 𝑔 𝑥 𝑟 𝑥 ± 𝑓 𝑥 ℎ 𝑥 ± = 𝑓 𝑥 𝑟 𝑥 𝑓 𝑥 𝑟 𝑥
Logarithmic
Log “Base” Notation Note: log 𝑥 = log!" 𝑥 or it may be log 𝑥 = ln 𝑥 = log ! 𝑥; log x is the general notation for ln x but in some books or calculators log x = log!" x and vice-versa. ln 𝑏 𝑦 = log ! 𝑥 ⇒ 𝑥 = 𝑏 ! 𝑒 = 2.718281828 … = log ! 𝑏 ln 𝑎 log ! 𝑎 = 1 log ! 1 = 0 log ! 𝑎 ! = 𝑥 ! log ! 𝑥 = ln 𝑥 log ! 𝑥𝑦 = log ! 𝑥 + log ! 𝑦 log ! 𝑥 = 𝑏 log ! 𝑥 ∞ ∞ 𝑥 1 𝑎! 𝑡 ! log ! = log ! 𝑥 − log ! 𝑦 !" 𝑒= 𝑒 = 𝑦 𝑛! 𝑛! !!! !!! Log “Natural” Notation *It is unlikely that the notation involving “log” will be used throughout the course; you may see it in the beginning of the course, as a review of some sort but that should be about all you’ll see. The “ln 𝑢” notation will be the standard as it is easier to manipulate. ln 𝑏 𝑦 = ln 𝑥 ⇒ 𝑥 = 𝑒 ! 𝑦 = 𝑒 ! ⇒ 𝑥 = ln 𝑦 log ! 𝑏 = ln 𝑎 ! ln 𝑎 = undefined, 𝑎 ≤ 0 ln 1 = 0 1 𝑒= 𝑛! !!!
ln 𝑒 ! = 𝑥 ⇒ 𝑒 !" ! = 𝑥 ln 𝑥𝑦 = ln 𝑥 + ln 𝑦 Domains:
ln 𝑒 ! = 1 ⇒ 𝑒 !" ! = 1 𝑥 ln = ln 𝑥 − ln 𝑦 𝑦 ln 𝑥 , 𝐷 = 0, ∞
8
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ln 𝑥 ! = 𝑏 ln 𝑥 ln 𝑥 !! = ln
1 = − ln 𝑥 𝑥
ln 𝑥 , 𝐷 = 𝑥 𝑥 > 0, 𝑥 < 0
*Factoring
𝑥 ! + 𝑥 ! = 𝑥 ! 1 + 𝑥 !!! = 𝑥 ! 𝑥 !!! + 1 𝑥 ! + 2𝑎𝑥 + 𝑎! = 𝑥 + 𝑎 ! 𝑥 ! + 3𝑎𝑥 ! + 3𝑎! 𝑥 + 𝑎! = 𝑥 + 𝑎 ! 𝑥 ! + 𝑎! = (𝑥 + 𝑎)(𝑥 ! − 𝑎𝑥 + 𝑎! ) 𝑥 + 𝑎 ! = 𝑥 ! + 3𝑎𝑥 ! + 3𝑎! 𝑥 + 𝑎! Note: *Common mistake students make when solving for x:
𝑥! − 𝑥 = 0
𝑥 ! − 𝑎! = 𝑥 + 𝑎 𝑥 − 𝑎 𝑥 ! + 𝑎 + 𝑏 𝑥 + 𝑎𝑏 = (𝑥 + 𝑎)(𝑥 + 𝑏) 𝑥 ! − 3𝑎𝑥 ! + 3𝑎! 𝑥 − 𝑎! = 𝑥 − 𝑎 ! 𝑥 ! − 𝑎! = 𝑥 − 𝑎 𝑎! + 𝑎𝑥 + 𝑥 ! 𝑥 − 𝑎 ! = 𝑥 ! − 3𝑎𝑥 ! + 3𝑎! 𝑥 − 𝑎!
The solution of 𝑥 = 0 was lost, thus:
⇒
𝑥! = 𝑥
⇒
𝑥 = 1 ⇒
𝑥! − 𝑥 = 0 𝑥 𝑥 − 1 = 0 ⇔ 𝑥 = 0 𝑜𝑟 𝑥 = 1
Long Division (quotient)(divisor)+(remainder)=dividend P=Divisor
Q=Dividend
R=Quotient
Complete The Square
𝑦 = 𝑎𝑥 ! + 𝑏𝑥 + 𝑐
𝑏 𝑏 𝑏 ! 𝑏 ! = 𝑎 𝑥! + 𝑥 + 𝑐 = 𝑎 𝑥! + 𝑥 + − + 𝑐 𝑎 𝑎 2𝑎 2𝑎 𝑏 𝑏 ! 𝑏 ! 𝑏 ! 𝑏! = 𝑎 𝑥! + 𝑥 + −𝑎 +𝑐 =𝑎 𝑥+ −𝑎 + 𝑐 𝑎 2𝑎 2𝑎 2𝑎 4𝑎! 𝑏 ! 𝑏! =𝑎 𝑥+ − +𝑐 2𝑎 4𝑎 𝑏 ! 𝑏! ∴ 𝑦=𝑎 𝑥+ +𝑐− 2𝑎 4𝑎
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9
Example 1: Solving for x (Formula 1) 𝑎𝑥 ! + 𝑏𝑥 = 0 𝑏 0 𝑏 ⇒ 𝑥! + 𝑥 = ⇒ 𝑥 ! + 𝑥 + 0 = 0 𝑎 𝑎 𝑎 𝑏 𝑏 ! 𝑏 ! ! ⇒ 𝑥 + 𝑥+ − =0 ⇒ 𝑎 2𝑎 2𝑎 ⇒
𝑥+
𝑏 2𝑎
!
∴
𝑥=0
or
=
𝑏! 4𝑎!
⇒
𝑥+
𝑏 𝑏 𝑥 + 𝑥+ 𝑎 2𝑎
𝑏 𝑏! =± 2𝑎 4𝑎!
!
!
⇒
𝑥=−
𝑏 ! = 2𝑎 𝑏 𝑏 ± 2𝑎 2𝑎
𝑏 𝑥=− 𝑎
Example 2: Solving for x (Formula 2) ! 𝑎𝑥 + 𝑏𝑥 + 𝑐 = 0 𝑏 𝑐 0 𝑏 𝑐 𝑏 ! 𝑏 ! ⇒ 𝑥! + 𝑥 + = ⇒ 𝑥! + 𝑥 + + − = 0 𝑎 𝑎 𝑎 𝑎 𝑎 2𝑎 2𝑎 𝑏 𝑏 ! 𝑏! 𝑐 𝑏 ! 𝑏 ! − 4𝑎𝑐 ! ⇒ 𝑥 + 𝑥+ = ! !− ⇒ 𝑥+ = 𝑎 2𝑎 2 𝑎 𝑎 2𝑎 4𝑎! 𝑏 𝑏 ! − 4𝑎𝑐 𝑏 𝑏 ! − 4𝑎𝑐 ⇒ 𝑥+ =± ⇒ 𝑥=− ± 2𝑎 2𝑎 2𝑎 2𝑎 −𝑏 ± 𝑏 ! − 4𝑎𝑐 ∴ 𝑥= 2𝑎
Compositions
𝑓∘𝑔 𝑥 =𝑓 𝑔 𝑥
𝑓∙𝑔 𝑥 =𝑓 𝑥 𝑔 𝑥
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𝑓±𝑔 𝑥 =𝑓 𝑥 ±𝑔 𝑥 𝑓 𝑓 𝑥 𝑥 = , 𝑔 𝑔 𝑥
𝑔 𝑥 ≠ 0
Functions
Vertical Line Test 𝑓 𝑥 is a function if it passes the vertical line test i.e. if you draw a vertical line anywhere on the graph, and the graph of 𝑓 only crosses it once. Even/Odd Function Even: 𝑓 −𝑥 = 𝑓 𝑥 Odd: 𝑓 −𝑥 = −𝑓 𝑥 (symmetric with respect to 𝑦-axis) (symmetric with respect to origin) Average Rate of Change 𝛥𝑦 𝑓 𝑥 − 𝑓 𝑥! = , 𝑥 ≠ 𝑥! 𝛥𝑥 𝑥 − 𝑥! Secant Line ! ! !! ! The slope of the secant line is the same as the average rate of change i.e. 𝑚 = !!! ! you then take one of the two points and plug the it into 𝑦 − 𝑦! = Difference Quotient 𝑚=
Distance Formula
! ! !! !! !!!!
!
𝑥 − 𝑥! and simplify.
𝑓 𝑥 + 𝛥𝑥 − 𝑓 𝑥 𝑓 𝑥+ℎ −𝑓 𝑥 = 𝛥𝑥 ℎ
Distance between two points on a number line 𝑃 𝑥! = 𝑃 𝑥! = 𝑃 𝑎 , 𝑄 = 𝑄 𝑥 = 𝑄 𝑥! = 𝑄 𝑏 𝑑 𝑃, 𝑄 = 𝑥 − 𝑥! ! = 𝑥 − 𝑥! = 𝑏−𝑎 ! = 𝑏−𝑎 = 𝑥! − 𝑥! ! = 𝑥! − 𝑥!
Distance between two points in a Cartesian coordinate system i.e. x vs. y graph 𝑃 𝑥! , 𝑦! , 𝑄 𝑥, 𝑦 𝑑 𝑃, 𝑄 = 𝑥 − 𝑥! ! + 𝑦 − 𝑦! !
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Midpoint Formula 𝑃 𝑥! , 𝑦!
&
𝑄 𝑥! , 𝑦!
𝑚 𝑃, 𝑄 =
Quadratic Formula !
𝑎𝑥 + 𝑏𝑥 + 𝑐 = 0
⇔
Proof: ! 𝑎𝑥 + 𝑏𝑥 + 𝑐 = 0 ⇒
⇒
𝑥! + 𝑥! 𝑦! + 𝑦! , 2 2
−𝑏 ± 𝑏 ! − 4𝑎𝑐 𝑥= 2𝑎
𝑏 𝑐 0 𝑏 𝑐 𝑏 ! 𝑏 ! ! ! 𝑥 + 𝑥+ = ⇒ 𝑥 + 𝑥+ + − = 0 𝑎 𝑎 𝑎 𝑎 𝑎 2𝑎 2𝑎 𝑏 𝑏 ! 𝑏! 𝑐 𝑏 ! 𝑏 ! − 4𝑎𝑐 ! 𝑥 + 𝑥+ = ! !− ⇒ 𝑥+ = 𝑎 2𝑎 2 𝑎 𝑎 2𝑎 4𝑎!
⇒ ∴
𝑥+
𝑏 𝑏 ! − 4𝑎𝑐 =± 2𝑎 2𝑎
⇒
𝑥=−
𝑏 𝑏 ! − 4𝑎𝑐 ± 2𝑎 2𝑎
−𝑏 ± 𝑏 ! − 4𝑎𝑐 𝑥= 2𝑎
Discriminant:
i) Two real solutions if 𝑏 ! − 4𝑎𝑐 > 0
ii) Repeated solutions if 𝑏 ! − 4𝑎𝑐 = 0 iii) Two complex solutions 𝑖𝑓𝑏 ! − 4𝑎𝑐 < 0
Graphing a Line
From the form 𝑦 = 𝑚𝑥 + 𝑏 you can easily graph a line by identifying two points and then connecting them. ! ! The equation will more generally appear as 𝑦 = ±! 𝑥 + 𝑏 where 𝑚 = ±!, 𝑐 is the rise and ± 𝑑 is the run (𝑐 always goes up and 𝑑 goes either left or right.)
The first point is 𝑃! 0, 𝑏 The second point is 𝑃! (±𝑑, 𝑏 + 𝑐)
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Plot these two points and connect a line through them. Point Slope Form: 𝑦 − 𝑦! = 𝑚 𝑥 − 𝑥! 𝛥𝑦 𝑚 = 𝑠𝑙𝑜𝑝𝑒 = 𝛥𝑥 𝛥𝑦 𝑦 − 𝑦! ⇒ 𝑚= = 𝛥𝑥 𝑥 − 𝑥! 𝑦 − 𝑦! ⇒ 𝑚= 𝑥 − 𝑥! ⇒ 𝑥 − 𝑥! 𝑚 = 𝑦 − 𝑦! ∴ 𝑦 − 𝑦! = 𝑚(𝑥 − 𝑥! ) Slope Intercept Form: 𝑦 = 𝑚𝑥 + 𝑏 𝛥𝑦 𝛥𝑦 𝑦 − 𝑦! 𝑚 = 𝑠𝑙𝑜𝑝𝑒 = ⇒ 𝑚= = 𝛥𝑥 𝛥𝑥 𝑥 − 𝑥! 𝑦 − 𝑦! ⇒ 𝑚= 𝑥 − 𝑥! ⇒ 𝑥 − 𝑥! 𝑚 = 𝑦 − 𝑦! ⇒ 𝑚𝑥 − 𝑚𝑥! = 𝑦 − 𝑦! ⇒ 𝑦 = 𝑚𝑥 − 𝑚𝑥! + 𝑦! ⇒ 𝑦 = 𝑚𝑥 + 𝑦! − 𝑚𝑥! ⇒ 𝑦 = 𝑚𝑥 + 𝑦! − 𝑚𝑥! , setting 𝑏 = 𝑦! − 𝑚𝑥! ∴ 𝑦 = 𝑚𝑥 + 𝑏 Standard or General Form 𝐴𝑥 + 𝐵𝑦 = 𝐶 Parallel Line (equal slopes) 𝑦! = 𝑚𝑥 + 𝑏! ∥ 𝑦! = 𝑚𝑥 + 𝑏!
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Perpendicular Line (product of slopes are -1) 𝑦! = 𝑚𝑥 + 𝑏! ⊥ 𝑦! = −
*Domain Restrictions
1 𝑥 + 𝑏! 𝑚
For the following, 𝑓 𝑥 , 𝑔 𝑥 , ℎ 𝑥 are assumed to be continuous for all real numbers. 𝑥 = 𝑎! 𝑥 ! ± 𝑎! 𝑥 !!! ± 𝑎! 𝑥 !!! ± ⋯ ± 𝑎! 𝑥 !!! Polynomial No Restrictions 𝑓 𝑥 Fraction 𝒈 𝒙 ≠ 𝟎 ℎ 𝑥 = 𝑔 𝑥 ! Radical, if n is even 𝒈 𝒙 ≥ 𝟎 𝑓 𝑥 = 𝑔(𝑥) ! Radical, if n is odd No Restrictions 𝑓 𝑥 = 𝑔(𝑥) 𝑓 𝑥 Fraction with Radical in 𝐈𝐟 𝐧 𝐢𝐬 𝐞𝐯𝐞𝐧 𝒈 𝒙 ℎ 𝑥 =! denominator > 𝟎 𝐢𝐟 𝐧 𝐢𝐬 𝐨𝐝𝐝 𝒈 𝒙 𝑔 𝑥 ≠ 𝟎 Natural Log 𝑓 𝑥 = ln 𝑔 𝑥 𝒈 𝒙 > 𝟎 ! ! Exponential 𝐍𝐨 𝐑𝐞𝐬𝐭𝐫𝐢𝐜𝐭𝐢𝐨𝐧𝐬 ℎ(𝑥) = 𝑓 𝑥
Inverse Functions 𝑦=𝑓 𝑥 ⇒ 𝑥 = 𝑓 𝑦 !! = 𝑓 𝑓 !! 𝑥 If 𝑓 𝑥 is one-to-one it has an inverse The domain of 𝑓 𝑥 is the range of 𝑓 !! 𝑥 The range of 𝑓 𝑥 is the domain of 𝑓 !! 𝑥 𝑦 = 𝑓 𝑥 ⇒ 𝑥 = 𝑓 𝑦 ⇒ 𝑦 = 𝑓 !! 𝑥
Asymptotes, Holes and Graphs
An asymptote occurs where the function is getting infinitely close to a line on the graph but never touches the line. Horizontal asymptotes may cross the line from time-to-time; it is the end behavior we are concerned with. There are three types of asymptotes: Horizontal, Vertical and Oblique. Oblique asymptotes, will most likely, not be used in your calculus course but vertical and horizontal will be used frequently in order to graph functions. 14 copyright © wesolvethem.com | WESOLVETHEM LLC
Hole in a Graph
𝑥! − 4 𝑓 𝑥 = 𝑥−2
⇒
𝑥≠2
Three General Cases for Horizontal Asymptotes Since there are so many conditions and situations for asymptotes and the methods learned in algebra are so minimal to what is used in calculus, we will come back to this later. Case 1 𝑥 ! + 𝑥 !!! + ⋯ 𝑛 > 𝑚 ⇒ 𝐻𝐴: 𝑦 = 0 𝑓 𝑥 = ! !!! 𝑥 +𝑥 +⋯ Case 2 𝑥 ! + 𝑥 !!! + ⋯ 𝑛 < 𝑚 ⇒ 𝐻𝐴: 𝑛𝑜𝑛𝑒 𝑓 𝑥 = ! !!! 𝑥 +𝑥 +⋯ 𝑎 Case 3 a𝑥 ! + 𝑥 !!! + ⋯ 𝑛 = 𝑚 ⇒ 𝐻𝐴: 𝑦 = 𝑓 𝑥 = 𝑏 b𝑥 ! + 𝑥 !!! + ⋯ Ex. 1 Horizontal 𝑥! + 𝑥 + 1 𝑓 𝑥 = and Vertical 𝑥! + 𝑥! + 𝑥 + 1 𝐻𝐴: 𝑦 = 0, 𝑉𝐴: 𝑥 = −2
Ex. 2 Oblique
𝑓 𝑥 =
𝑥! + 𝑥! + 𝑥 + 1 𝑥! + 𝑥 + 1
𝑁𝑜 𝐻𝐴,
𝑂𝐴:
𝐻𝐴:
3 𝑦= , 2
𝑉𝐴:
𝑦 = 𝑥
3𝑥 ! + 𝑥 𝑓 𝑥 = ! 2𝑥 + 1
Ex. 3 Horizontal and Vertical
1 𝑥 =−! 2
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Inequalities
𝑓 𝑥 𝑓 𝑥
< 𝑎 ⇒ −𝑎 < 𝑓 𝑥 < 𝑎 or 𝑓 𝑥 < 𝑎 and 𝑓 𝑥 > −𝑎 ≤ 𝑎 ⇒ −𝑎 ≤ 𝑓 𝑥 ≤ 𝑎 or 𝑓 𝑥 ≤ 𝑎 and 𝑓 𝑥 ≥ −𝑎
Interest Formulas
𝐴 = 𝐴! 𝑒 !"
𝑃 = 𝑃!
𝑟 12
𝑟 1 − 1 + 12
!!
L=Loan 𝑃 = Monthly Payment 𝑟 = Interest rate for annual 𝑡 = Loan length in months
Physics Formulas
(rate)(time)=distance 𝑟𝑡 = 𝑑
Symmetry
By Point 𝑥-axis For every point 𝑥, 𝑦 there is a 𝑥, −𝑦 𝑦-axis For every point 𝑥, 𝑦 there is a −𝑥, 𝑦 origin For every point 𝑥, 𝑦 there is a −𝑥, −𝑦 Testing 𝑥-axis: Replace each 𝑦 with a – 𝑦, if the same equation results, it is symmetric. 𝑦-axis: Replace each 𝑥 with a – 𝑥, if the same equation results, it is symmetric. Origin: Replace each 𝑥, 𝑦 with a −𝑥, – 𝑦, if the same equation results, it is symmetric.
Variations (Proportionality) 𝑘 is the constant of proportionality
𝑦 is proportional to x: y= 𝑘𝑥
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𝑦 is inversely proportional to ! 𝑥: 𝑦 = !
Common Graphs and Formulas 𝑦 = 𝑥!
𝑦 = 𝑥!
_________________________________________________________________________________________________________ 1 𝑦= 𝑥 𝑦= 𝑥
_________________________________________________________________________________________________________ ! 𝑦=𝑒 𝑦 = ln 𝑥
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𝑥! + 𝑦! = 1
𝑥 ! − 𝑦 ! = 1
_________________________________________________________________________________________________________ 𝑦! − 𝑥! = 1 𝑦 = 𝑥! − 1
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Equation of a Line 𝑠𝑙𝑜𝑝𝑒 = 𝑚 =
𝑦 = 𝑚𝑥 + 𝑏 𝑦! − 𝑦! = 𝑚 𝑥! − 𝑥! 𝐴𝑥 + 𝐵𝑦 = 𝐶 ! 𝑦 = 𝑎𝑥 + 𝑏𝑥 + 𝑐 𝑦 = 𝑎 𝑥 − ℎ ! + 𝑘 ! ! 𝑥 + 𝑦 + 𝑎𝑥 + 𝑏𝑦 + 𝑐 = 0 ⇒ ! 𝑥 − ℎ + 𝑦 − 𝑘 ! = 𝑟!
𝑦! − 𝑦! 𝑥! − 𝑥!
Equation of Parabola !
!
Vertex: ℎ, 𝑘 = − !! , 𝑓 − !!
Equation of Circle Center: ℎ, 𝑘 Radius: 𝑟 Equation of Ellipse
𝑥−ℎ 𝑎!
!
Right Point: ℎ + 𝑎, 𝑘
𝑦−𝑘 + 𝑏!
!
= 1
Left Point: ℎ − 𝑎, 𝑘 Top Point: ℎ, 𝑘 + 𝑏
Bottom Point: ℎ, 𝑘 − 𝑏 Equation of Hyperbola (1) Center: ℎ, 𝑘 ! Slope: ± !
𝑥−ℎ 𝑎!
!
−
𝑦−𝑘 𝑏!
!
= 1
!
Asymptotes: 𝑦 = ± ! 𝑥 − ℎ + 𝑘 Vertices: ℎ + 𝑎, 𝑘 , ℎ − 𝑎, 𝑘 Equation of Hyperbola (2) Center: ℎ, 𝑘 ! Slope: ± !
𝑦−𝑘 𝑎!
!
𝑥−ℎ − 𝑏!
!
= 1
!
Asymptotes: 𝑦 = ± ! 𝑥 − ℎ + 𝑘 Vertices: ℎ, 𝑘 + 𝑏 , ℎ, 𝑘 − 𝑏
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Areas Square: 𝐴 = 𝐿! = 𝑊 ! Rectangle: 𝐴 = 𝐿 ∙ 𝑊 Circle: 𝐴 = 𝜋 ∙ 𝑟 ! ! ! Ellipse: 𝐴 = 𝜋 ∙ 𝑎𝑏 Triangle: 𝐴 = ! 𝑏 ∙ ℎ Trapezoid: 𝐴 = ! 𝑎 + 𝑏 ∙ ℎ !" Parallelogram: 𝑏 ∙ ℎ Rhombus: 𝐴 = ! , 𝑝 and 𝑞 are the diagonals Surface Areas Cube: 𝐴! = 6𝐿! = 6𝑊 ! Box: 𝐴! = 2(𝐿𝑊 + 𝑊𝐻 + 𝐻𝐿) Sphere: 𝐴! = 4𝜋𝑟 ! Cone: 𝐴! = 𝜋𝑟 𝑟 + ℎ! + 𝑟 ! Cylinder: 2𝜋𝑟ℎ + 2𝜋𝑟 ! Volumes ! Cube: 𝑉 = 𝐿! = 𝑊 ! Box: 𝑉 = 𝐿 ∙ 𝑊 ∙ 𝐻 Sphere: 𝑉 = ! 𝜋 ∙ 𝑟 ! ! ! Cone: 𝑉 = ! 𝜋 ∙ 𝑟 ! ℎ Ellipsoid: 𝑉 = ! 𝜋 ∙ 𝑎𝑏𝑐, 𝑎, 𝑏, 𝑐 are the radii Business Functions Cost Function 𝐶 𝑥 Revenue Function 𝑅 𝑥 Profit Function 𝑃 𝑥 = 𝑅 𝑥 − 𝐶 𝑥 Marginal Cost Function 𝐶 ! 𝑥 ! Marginal Revenue Function 𝑅 𝑥 Marginal Profit Function 𝑃! 𝑥 = 𝑅! 𝑥 = 𝐶! 𝑥 𝐶 𝑥 Average Cost Function 𝐶 𝑥 = 𝑥 ! ! ! ! ! ! !! ! Average Revenue Function 𝑅 𝑥 = ! Average Profit Function 𝑃 𝑥 = ! = !
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Average Rate of Change of 𝒇 and Slope of Secant Line
!" !"
=
! ! !! ! !!!
= 𝑚!"#$%& from 𝑃! (𝑎, 𝑓 𝑎 ) and 𝑃! 𝑏, 𝑓 𝑏
Difference Quotient ! !"!!! !! !! ! 𝑚!"#$%& = , 𝛥𝑥 = ℎ ⇒ 𝑚!"#$%& = !"
!!! !! ! !
, ℎ ≠ 0
Functions Constant Function Identity Function Square Function Cube Function Square Root Function Cube Root Function Reciprocal Function Absolute Value Function Greatest Integer Function Piecewise Function Power Function Ration Function
𝑦 = 𝑐 𝑦 = 𝑥 𝑦 = 𝑥! 𝑦 = 𝑥! 𝑦 = 𝑥 ! 𝑦 = 𝑥 1 𝑦= 𝑥 𝑦= 𝑥 𝑦 = int 𝑥 ∗ 𝑔 𝑥 , 𝑥 ∈ 𝐷! 𝑓 𝑥 = ℎ 𝑥 , 𝑥 ∈ 𝐷! 𝑦 = 𝑎𝑥 ! 𝑔 𝑥 𝑓 𝑥 = , ℎ 𝑥 ≠ 0 ℎ 𝑥
Graph Shifts and Compressions Vertically up for 𝑓 𝑥 Vertically down for 𝑓 𝑥 Horizontally left for 𝑓 𝑥 Horizontally right for 𝑓 𝑥 𝑎𝑓 𝑥 multiply each y-coordinate by a Vertically Compressed: 0 < 𝑎 < 1 Vertically Stretched: 𝑎 > 1 ! 𝑓 𝑎𝑥 Multiply each x-coordinate by ! Horizontal Compression: 𝑎 > 1 Horizontal Stretch: 0 < 𝑎 < 1 Reflection about x-axis Reflection about y-axis
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𝑓 𝑥 + 𝑘 𝑓 𝑥 − 𝑘 𝑓 𝑥+ℎ 𝑓 𝑥−ℎ 𝑥, 𝑎𝑦 1 𝑥, 𝑦 𝑎 −𝑓 𝑥 𝑓 −𝑥
21
Systems of equations
𝑎𝑥 + 𝑏𝑦 = 𝑒 ⇒ 𝑐𝑥 + 𝑑𝑦 = 𝑓
𝑎𝑥 + 𝑏𝑦 + 𝑐𝑧 = 𝑑 𝑒𝑥 + 𝑓𝑦 + 𝑔𝑦 = ℎ 𝑖𝑥 + 𝑗𝑦 + 𝑘𝑧 = 𝑙
⇒
𝑎 𝑐 𝑎 𝑒 𝑖
𝑒 𝑏 𝑥 = 𝑓 𝑦 𝑑 𝑏 𝑓 𝑗
𝑐 𝑥 𝑑 𝑔 𝑦 = ℎ 𝑘 𝑧 𝑙
𝑎 The Coefficient Matrix = 𝑒 𝑖
𝑎 𝑐
⇒
𝑏𝑒 𝑑𝑓
⇒ 𝑏 𝑓 𝑗
𝑎 𝑒 𝑖
𝑏 𝑓 𝑗
𝑐 𝑑 𝑔ℎ 𝑘 𝑙
𝑐 𝑔 𝑘
Rank of matrix and pivots
𝟏 , 1
𝟏 0
𝟏 0 0
𝟏 , 0
𝟏
0 , 1
0 0 , 𝟏
𝑟𝑎𝑛𝑘 𝐴! = 2 𝑟𝑎𝑛𝑘 𝐴! = 2 𝑟𝑎𝑛𝑘 𝐴! = 2
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𝟏 1 1 𝟏 1 1 𝟏 0 0
1 1,
𝑟𝑎𝑛𝑘 𝐴!" = 1
𝑟𝑎𝑛𝑘 𝐴!! = 1
1 1 1 1 , 1 1 1 1 1 −𝟏 , 1 1 1 𝟏 0
𝑟𝑎𝑛𝑘 𝐴! = 1
𝑟𝑎𝑛𝑘 𝐴! = 1
𝟏 0 , 0
𝑟𝑎𝑛𝑘 𝐴! = 1
0 , 𝟏
0 𝟏
𝑟𝑎𝑛𝑘 𝐴! = 1
1,
𝟏
𝟏 1 , 1 𝟏 0
𝑟𝑎𝑛𝑘 𝐴! = 1
𝑟𝑎𝑛𝑘 𝐴! = 1
1 1,
𝟏
1,
𝟏
1 1 , 𝟏
𝑟𝑎𝑛𝑘 𝐴!" = 1
𝑟𝑎𝑛𝑘 𝐴!" = 2
𝑟𝑎𝑛𝑘 𝐴!" = 3
Determinate’s of a (2x2) matrix Various ways to check determinant (2x2): 𝑎 𝐴= 𝑐
𝑏 𝑑
𝑎 ⇒ det 𝐴 = 𝐴 = 𝑐
𝐴=
𝑎 0
𝑏 𝑐
⇒ det 𝐴 = 𝐴 =
𝑎 0
𝑏 = 𝑎 𝑐 𝑐
𝐴=
𝑎 0
0 𝑏
⇒ det 𝐴 = 𝐴 =
𝑎 0
0 = 𝑎 𝑏 𝑏
𝑏 = 𝑎 𝑑 − 𝑏 𝑐 𝑑
Determinate of a (3x3) and higher matrices Cofactor Expansion Note:
𝑎 𝑑 𝑔
𝑎 𝑐
𝑎 𝑏 𝑒 𝑓 𝑖 𝑗 𝑚 𝑛
𝑐 𝑔 𝑘 𝑜
𝑏 𝑒 ℎ
𝑑 𝑓 ℎ = +𝑎 𝑗 𝑙 𝑛 𝑝
𝑏 = 𝑎 𝑑 − 𝑏 𝑐 𝑑
𝑐 𝑓 = +𝑎 𝑒 ℎ 𝑖 𝑔 𝑘 𝑜
𝑑 𝑓 −𝑏 𝑔 𝑖
ℎ 𝑒 𝑙 −𝑏 𝑖 𝑝 𝑚
𝑔 𝑘 0
𝑓 𝑑 +𝑐 𝑔 𝑖
ℎ 𝑒 𝑙 +𝑐 𝑖 𝑝 𝑚
𝑓 𝑗 𝑛
𝑒 ℎ ℎ 𝑒 𝑙 −𝑑 𝑖 𝑝 𝑚
𝑓 𝑗 𝑛
𝑔 𝑘 𝑜
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