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Copyright©WeSolveThemLLC 1
TheUltimateCheatSheetforMath&PhysicsPreview
TableofContentsAlgebra..............................................................................................................................................2Arithmetic......................................................................................................................................................................................................2Exponents......................................................................................................................................................................................................3
Trigonometry.....................................................................................................................................4DoubleAngleFormulas...........................................................................................................................................................................4HalfAngleFormulas..................................................................................................................................................................................5SumandDifferenceFormulas..............................................................................................................................................................6
Precalculus.........................................................................................................................................7EquationofaLine......................................................................................................................................................................................7EquationofParabola................................................................................................................................................................................7EquationofCircle.......................................................................................................................................................................................7EquationofEllipse.....................................................................................................................................................................................7EquationofHyperbola.............................................................................................................................................................................7EquationofHyperbola.............................................................................................................................................................................7
Calculus..............................................................................................................................................8Tangentline..................................................................................................................................................................................................8Implicitdifferentiation............................................................................................................................................................................9
LinearAlgebra..................................................................................................................................10Rankofmatrixandpivots...................................................................................................................................................................10Lengthofavectorandtheunitvector...........................................................................................................................................11SolutionsofAugmentedMatrices....................................................................................................................................................12CoefficientMatrix....................................................................................................................................................................................12UniqueSolution.......................................................................................................................................................................................13InfiniteSolution.......................................................................................................................................................................................13NoSolution.................................................................................................................................................................................................13DifferentialEquations......................................................................................................................14First-OrderLinearNon-Homogeneous..........................................................................................................................................14OrderandLinearity................................................................................................................................................................................14ReductionofOrder.................................................................................................................................................................................15
Physics.............................................................................................................................................16Vectors.........................................................................................................................................................................................................16DotProduct................................................................................................................................................................................................16CrossProduct............................................................................................................................................................................................16MagnitudeorLengthofavector......................................................................................................................................................17ResultantVector......................................................................................................................................................................................17
QuickReference...............................................................................................................................19Arithmetic...................................................................................................................................................................................................19Exponential................................................................................................................................................................................................19Radicals.......................................................................................................................................................................................................19Fractions.....................................................................................................................................................................................................19Logarithmic................................................................................................................................................................................................19
Copyright©WeSolveThemLLC 2
QuadraticFormula..................................................................................................................................................................................20About:Thisbookcoversalltheformula,equationstipsandtricksanundergraduateSTEMmajorrequiresforAlgebra,Trigonometry,Precalculus,Calculus(alllevels/areas),LinearAlgebra,DifferentialEquations,andPhysics(Mechanics,E&M,Optics…)ThebookisapplicabletoanySTEMstudentatanypointoftheircareer.Itcanactasareview,aguidedassistantandortoolforstudentsoutsideofcollege.
AlgebraArithmetic
𝑎𝑏 ± 𝑎𝑐 = 𝑎 𝑏 ± 𝑐 = 𝑏 ± 𝑐 𝑎 10± 6 = 2 ∙ 5± 2 ∙ 3 = 2 5± 3 = 5± 3 2_________________________________________________________________________________________________________________
_
𝑎𝑏𝑐 =
𝑎𝑏𝑐
123 =
1231=12 ∙13 =
12 ∙ 3 =
16
__________________________________________________________________________________________________________________
𝑎𝑏 ±
𝑐𝑑 =
𝑎𝑑 ± 𝑏𝑐𝑏𝑑
12±
34 =
1 ∙ 4± 2 ∙ 32 ∙ 4 =
4± 68
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𝑎 − 𝑏𝑐 − 𝑑 =
𝑏 − 𝑎𝑑 − 𝑐
1− 23− 4 =
−(−1+ 2)−(−3+ 4) =
2− 14− 3
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𝑎𝑏 + 𝑎𝑐
𝑎 = 𝑏 + 𝑐,𝑎 ≠ 0 12± 16
4 =124 ±
164 = 3± 4
__________________________________________________________________________________________________________________
𝑎𝑏𝑐 =
𝑎𝑏𝑐
165 =
4 ∙ 45 = 4
45
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Copyright©WeSolveThemLLC 3
𝑎𝑏𝑐=
𝑎1 ∙
𝑐𝑏 =
𝑎𝑐𝑏
234
=2134
=21 ∙43 =
83
__________________________________________________________________________________________________________________
𝑎 ± 𝑏𝑐 =
𝑎𝑐 ±
𝑏𝑐
12± 165 =
125 ±
165
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𝑎𝑏𝑐𝑑
=𝑎𝑏 ∙𝑑𝑐 =
𝑎𝑑𝑏𝑐
1234
=12 ∙43 =
46 =
23
__________________________________________________________________________________________________________________
𝑖𝑓 𝑎 ± 𝑏 = 0 𝑡ℎ𝑒𝑛 𝑎 = ∓𝑏 𝑥 ± 2 = 0 ⇒ 𝑥 = ∓2
Exponents
𝑎! = 𝑎 2 = 2!_________________________________________________________________________________________________________________
_
𝑎! = 1 2! = 2!!! =2!
2! =22 = 1
__________________________________________________________________________________________________________________
𝑎!! =1𝑎! 2!! =
12! =
14
__________________________________________________________________________________________________________________
1𝑎!! = 𝑎!
12!! = 2! = 4
__________________________________________________________________________________________________________________
𝑎!𝑎! = 𝑎!!! 2!2! = 2!!! = 2!
__________________________________________________________________________________________________________________
Copyright©WeSolveThemLLC 4
𝑎!
𝑎! = 𝑎!!! 2!
2! = 2!!! = 2! = 2
__________________________________________________________________________________________________________________
𝑎𝑏
!=𝑎!
𝑏! 23
!
=2!
3! =49
__________________________________________________________________________________________________________________
𝑎𝑏
!!=𝑎!!
𝑏!! =𝑏!
𝑎! 12
!!
=1!!
2!! =2!
1 = 4
__________________________________________________________________________________________________________________
𝑎!!! = 𝑎
!! = 𝑎
!!
! 2!
!! = 2
!! = 2
!!
!
__________________________________________________________________________________________________________________
𝑎! ! = 𝑎!" = 𝑎!" = 𝑎! ! 2! ! = 2!∙! = 2 !∙! = 2! !
TrigonometryDoubleAngleFormulas
*Important
ThehalfangleanddoubleangleformulasalongwiththePythagoreanidentitiesareusedfrequentlythroughoutcalculus.Itisamustthatyoumemorizetheunderstandingandderivationsisfullycomprehended.
Foradetailedlistofallidentities,seethereferencesheetsinthebackofthebook.
Derivationforsin 2𝜃 = 2 sin𝜃 cos𝜃:
sin 2𝜃 = sin 𝜃 + 𝜃 = sin𝜃 cos𝜃 + sin𝜃 cos𝜃 = 2 sin𝜃 cos𝜃
__________________________________________________________________________________________________________________
Derivationforcos 2𝜃 = 1− 2 sin! 𝜃:
cos(2𝜃) = cos! 𝜃 − sin! 𝜃 = 2 cos! 𝜃 − 1 = 1− 2 sin! 𝜃
Copyright©WeSolveThemLLC 5
__________________________________________________________________________________________________________________
Asonecansee,theseformulasareallderivedfromthePythagoreanidentitiesandtherearemanywaystofindthem.Ifthiscanbeunderstoodproperlythenmemorizingthemisnotentirelynecessary.
OtherDerivations:
cos2𝜃 = cos(𝜃 + 𝜃) = cos𝜃 cos𝜃 − sin𝜃 sin𝜃 = cos! 𝜃 − sin! 𝜃
_________________________________________________________________________________________________________________
_
cos 2𝜃 = cos(𝜃 + 𝜃) = cos𝜃 cos𝜃 − sin𝜃 sin𝜃 = cos! 𝜃 − sin! 𝜃 = cos! 𝜃 − (1− cos! 𝜃)
= cos!−1+ cos! 𝜃 = 2 cos! 𝜃 − 1
__________________________________________________________________________________________________________________
cos2𝜃 = cos(𝜃 + 𝜃) = cos𝜃 cos𝜃 − sin𝜃 sin𝜃 = cos! 𝜃 − sin! 𝜃
= 1− sin! 𝜃 − sin! 𝜃 = 1− 2 sin! 𝜃
__________________________________________________________________________________________________________________
tan 2𝜃 = tan 𝜃 + 𝜃 =tan𝜃 + tan𝜃1− tan𝜃 tan𝜃 =
2 tan𝜃1− tan! 𝜃
HalfAngleFormulas
sin! 𝜃 =12 1− cos 2𝜃
Derivation:
sin! 𝜃 = 1− cos! 𝜃 = 1− cos𝜃 cos𝜃 = 1−12 cos 𝜃 − 𝜃 + cos 𝜃 + 𝜃
= 1−12 cos 0 + cos 2𝜃 = 1−
12 1 + cos2𝜃 = 1−
12−
12 cos2𝜃
=12−
12 cos2𝜃 =
12 [1− cos(2𝜃)]
Copyright©WeSolveThemLLC 6
__________________________________________________________________________________________________________________
cos! 𝜃 =12 [1+ 𝑐𝑜𝑠 2𝜃 ]
Derivation:
cos! 𝜃 = 1− sin! 𝜃 = 1− sin𝜃 sin𝜃 = 1−12 cos(𝜃 − 𝜃 − cos 𝜃 + 𝜃 ]
= 1−12 cos0− cos 2𝜃 = 1−
12 1 − cos2𝜃 = 1−
12+
12 cos2𝜃
=12+
12 cos2𝜃 =
12 1+ cos 2𝜃
__________________________________________________________________________________________________________________
tan! 𝜃 =1− cos(2𝜃)1+ cos(2𝜃)
Derivation:
tan! 𝜃 = sec! 𝜃 − 1 =1
cos𝜃
!
− 1 =1
cos𝜃 cos𝜃 − 1 =1
12 cos 𝜃 − 𝜃 + cos 𝜃 + 𝜃
− 1
=2
1+ cos 2𝜃 − 1 =2
1+ cos2𝜃 −1+ cos 2𝜃1+ cos 2𝜃 =
2− 1+ cos2𝜃1+ cos 2𝜃
=1− cos 2𝜃1+ cos 2𝜃
SumandDifferenceFormulas
sin 𝛼 ± 𝛽 = sin𝛼 cos𝛽 ± cos𝛼 sin𝛽
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cos(𝛼 ± 𝛽) = cos𝛼 cos𝛽 ∓ sin𝛼 cos𝛽_________________________________________________________________________________________________________________
_
tan 𝛼 ± 𝛽 =tan𝛼 ± tan𝛽1∓ tan𝛼 𝑡𝑎𝑛𝛽
Copyright©WeSolveThemLLC 7
Precalculus
EquationofaLine
𝑠𝑙𝑜𝑝𝑒 = 𝑚 =𝑦! − 𝑦!𝑥! − 𝑥!
𝑦 = 𝑚𝑥 + 𝑏
𝑦! − 𝑦! = 𝑚 𝑥! − 𝑥!
𝐴𝑥 + 𝐵𝑦 = 𝐶
EquationofParabola
Vertex: ℎ, 𝑘
𝑦 = 𝑎𝑥! + 𝑏𝑥 + 𝑐
𝑦 = 𝑎 𝑥 − ℎ ! + 𝑘
EquationofCircle
Center: ℎ, 𝑘 Radius:𝑟
𝑥 − ℎ ! + 𝑦 − 𝑘 ! = 𝑟!
EquationofEllipse
RightPoint: ℎ + 𝑎, 𝑘
LeftPoint: ℎ − 𝑎, 𝑘
TopPoint: ℎ, 𝑘 + 𝑏
BottomPoint: ℎ, 𝑘 − 𝑏
𝑥 − ℎ !
𝑎! +𝑦 − 𝑘 !
𝑏! = 1
EquationofHyperbola
Center: ℎ, 𝑘 Slope:± !
!
Asymptotes:𝑦 = ± !!𝑥 − ℎ + 𝑘
Vertices: ℎ + 𝑎, 𝑘 , ℎ − 𝑎, 𝑘
𝑥 − ℎ !
𝑎! −𝑦 − 𝑘 !
𝑏! = 1
EquationofHyperbola
Center: ℎ, 𝑘 Slope:± !
!
Asymptotes:𝑦 = ± !!𝑥 − ℎ + 𝑘
Vertices: ℎ, 𝑘 + 𝑏 , ℎ, 𝑘 − 𝑏
𝑦 − 𝑘 !
𝑎! −𝑥 − ℎ !
𝑏! = 1
Copyright©WeSolveThemLLC 8
CalculusTangentline
Findtheequationofthetangentlineat𝑥 = 3 for 𝑦 = 𝑥!
Identify
𝑦 − 𝑓 𝑎 = 𝑓! 𝑎 𝑥 − 𝑎 , 𝑥! = 𝑎
𝑎 = 3
𝑓 𝑎 = 𝑓 3 = 3 ! = 9
𝑓! 𝑎 =𝑑𝑑𝑥 𝑥
! = 2𝑥
𝑓! 3 = 6Gobackandtakealookatthedifferencefromthelimitdefinitionprocessandthepowerruleprocess.
Nowplugeverythinginto𝑦 − 𝑦! = 𝑚 𝑥 − 𝑥!
𝑦 − 9 = 6 𝑥 − 3
∴ 𝑦 = 6𝑥 − 9
Graphingisalwaysgoodpractice
Copyright©WeSolveThemLLC 9
Implicitdifferentiation
Given𝑥𝑦 + 𝑦 = 𝑦! − 𝑥 find !"
!"
Simplytake !
!"ofthewholeequation
𝑑𝑑𝑥 𝑥𝑦 + 𝑦 = 𝑦! − 𝑥
⇒ 𝑑𝑑𝑥 𝑥𝑦 +
𝑑𝑑𝑥 𝑦 =
𝑑𝑑𝑥 𝑦
! −𝑑𝑑𝑥 𝑥
⇒ 𝑥𝑑𝑑𝑥 𝑦 + 𝑦
𝑑𝑑𝑥 𝑥 +
𝑑𝑦𝑑𝑥 = 2𝑦
𝑑𝑑𝑥 𝑦 − 1
⇒ 𝑥𝑑𝑦𝑑𝑥 + 𝑦 1 +
𝑑𝑦𝑑𝑥 = 2𝑦
𝑑𝑦𝑑𝑥 − 1
Feelfreetosubstitute𝑦! for !"
!"ifitistoomessy
⇒ 𝑥𝑦! + 𝑦 + 𝑦! = 2𝑦𝑦! − 1
⇒ 𝑥𝑦! + 𝑦! − 2𝑦𝑦! = −1− 𝑦
⇒ 𝑦! 𝑥 + 1− 2𝑦 = − 1+ 𝑦
⇒ 𝑦! =− 1+ 𝑦𝑥 + 1− 2𝑦 =
− 1+ 𝑦− 2𝑦 − 1− 𝑥 =
1+ 𝑦2𝑦 − 1− 𝑥
∴ 𝑑𝑦𝑑𝑥 =
𝑦 + 12𝑦 − 1− 𝑥
Copyright©WeSolveThemLLC 10
LinearAlgebraRankofmatrixandpivots
𝟏 1 , 𝑟𝑎𝑛𝑘 𝐴! = 1
𝟏 1 , 𝑟𝑎𝑛𝑘 𝐴! = 1
𝟏1 , 𝑟𝑎𝑛𝑘 𝐴! = 1
𝟏0 , 𝑟𝑎𝑛𝑘 𝐴! = 1
𝟏 1 1 , 𝑟𝑎𝑛𝑘 𝐴! = 1
𝟏 1 1 , 𝑟𝑎𝑛𝑘 𝐴!" = 1
𝟏11, 𝑟𝑎𝑛𝑘 𝐴! = 1
𝟏00, 𝑟𝑎𝑛𝑘 𝐴!! = 1
𝟏 00 𝟏 , 𝑟𝑎𝑛𝑘 𝐴! = 2
𝟏 1 11 1 11 1 1
, 𝑟𝑎𝑛𝑘 𝐴!" = 1
𝟏 0 00 𝟏 1 , 𝑟𝑎𝑛𝑘 𝐴! = 2
𝟏 1 11 1 −𝟏1 1 1
, 𝑟𝑎𝑛𝑘 𝐴!" = 2
𝟏 00 00 𝟏
, 𝑟𝑎𝑛𝑘 𝐴! = 2
𝟏 1 10 𝟏 10 0 𝟏
, 𝑟𝑎𝑛𝑘 𝐴!" = 3
Copyright©WeSolveThemLLC 11
Note:maxrankisthesmallerdimensionof𝑛×𝑚e.g.3×7 meansthat3isthehighestpossiblerank.Itgoeswiththetransposeaswelli.e.7×3stillhasahighestrankof3.
𝐴 = 1 2−1 −2
1 11 1
1 11 1 𝑅1+ 𝑅2 ⇐ 𝑅2
~ 𝟏 20 0
1 1𝟐 2
1 12 2 ⇒ 𝑟𝑎𝑛𝑘 𝐴 = 2
𝐴𝑥 = 𝑏 ⇒3 2 31 3 33 2 1
131~𝟏 0 00 𝟏 00 0 𝟏
−37870
, 𝑟𝑎𝑛𝑘 𝐴 = 3 𝑖. 𝑒. 𝐴 = 𝑓𝑢𝑙𝑙 𝑟𝑎𝑛𝑘
Lengthofavectorandtheunitvector
Givenavector𝒙 = 𝑥 = 𝑥!, 𝑥!, 𝑥!,… , 𝑥! =
𝑥!𝑥!𝑥!⋮𝑥!
Thelengthofthevectoristhemagnitudeofthevector
𝒙 = 𝑥!! + 𝑥!! + 𝑥!! +⋯+ 𝑥!!
Ex:
Findthelengthof 1,2,3,4
1,2,3,4 =
1234
⇒ 1,2,3,4 = 1! + 2! + 3! + 4! = 1+ 4+ 9+ 16 = 30 units
Copyright©WeSolveThemLLC 12
Example:
Fromthevectorabove,finditsunitvector.
𝑣𝒗 =
𝒗𝒗
⇒ 𝑣𝒗
= 𝒗𝒗 = 1 units
𝒙𝒙 =
11+ 4+ 9+ 16
1234
=1,2,3,430
=130,230,330,430
𝑥𝑥 =
130
!
+230
!
+330
!
+430
!
=130+
430+
930+
1630 =
3030 = 1 units
SolutionsofAugmentedMatrices
Considerthebasicscenarioi.e.rememberfromalgebrawhenyouhave𝑎𝑥 + 𝑏𝑦 = 𝑐and𝑑𝑥 +𝑒𝑦 = 𝑓?Rememberthatthesetwolineseitherlyeoneachother,intersectornevertouch,andthismeanstheyhaveeitherauniquesolution,infinitesolutions,onnosolution.Thesamegoeswith
𝑎𝑥 + 𝑏𝑦 + 𝑐𝑧 = 𝑑,exceptthisisaplane.
Forℝ!,considerthefollowingsystemanditsthreepossiblesolutionsafterreduction:
CoefficientMatrix
𝑎𝑥 + 𝑏𝑦 + 𝑐𝑧 = 𝑑𝑒𝑥 + 𝑓𝑦 + 𝑔𝑦 = ℎ𝑖𝑥 + 𝑗𝑦 + 𝑘𝑧 = 𝑙
⇒ 𝑎 𝑏 𝑐𝑒 𝑓 𝑔𝑖 𝑗 𝑘
𝑥𝑦𝑧=
𝑑ℎ𝑙
⇒ 𝑎 𝑏 𝑐𝑒 𝑓 𝑔𝑖 𝑗 𝑘
𝑑ℎ𝑙
TheCoefficientMatrix=𝑎 𝑏 𝑐𝑒 𝑓 𝑔𝑖 𝑗 𝑘
Copyright©WeSolveThemLLC 13
UniqueSolution
~1 0 00 1 00 0 1
∗∗∗⇒
𝑥𝑦𝑧=
∗∗∗
In2𝐷/3𝐷hereisasinglepointofintersection
InfiniteSolution
~1 0 00 1 00 0 0
∗∗0⇒
𝑥𝑦𝑧=
∗∗0+ 𝑠
001
In3𝐷twoplaneslieontopofeachotherIn2𝐷twolineslieontopofeachother
NoSolution
~1 0 00 1 00 0 0
∗∗∗⇒
𝑥𝑦0=
∗∗∗
Twoplanes/linesnevertouch
Copyright©WeSolveThemLLC 14
DifferentialEquations
First-OrderLinearNon-Homogeneous
cos! 𝑥 sin 𝑥𝑑𝑦𝑑𝑥 + cos
! 𝑥 𝑦 = 1
Form
𝑦! + 𝑃 𝑥 𝑦 = 𝑄 𝑥 ⇒ 𝑦 =1𝐼 𝑥 ∫ 𝑄 𝑥 𝐼 𝑥 𝑑𝑥 + 𝐶 ⇔ 𝐼 𝑥 = 𝑒 ! ! !"
cos! 𝑥 sin 𝑥𝑑𝑦𝑑𝑥 + cos
! 𝑥 𝑦 = 1
⇒ 1
cos! 𝑥 sin 𝑥 cos! 𝑥 sin 𝑥𝑑𝑦𝑑𝑥 + cos
! 𝑥 𝑦 = 1 ⇒ 𝑦! + cot 𝑥 𝑦 = sec! 𝑥 csc 𝑥
⇒ 𝑃 𝑥 = cot 𝑥 ∧ 𝑄 𝑥 = sec! 𝑥 csc 𝑥 ∧ 𝐼 𝑥 = 𝑒 !"# ! !" = sin 𝑥
⇒ 𝑦 =1
sin 𝑥 sec! 𝑥 csc 𝑥 sin 𝑥 𝑑𝑥 + 𝐶 = csc 𝑥 sec! 𝑥 𝑑𝑥 + 𝐶 = csc 𝑥 tan 𝑥 + 𝐶
⇒ 𝑦 = csc 𝑥 tan 𝑥 + 𝐶 csc 𝑥 =1
sin 𝑥sin 𝑥cos 𝑥 + 𝐶 csc 𝑥 = sec 𝑥 + 𝐶 csc 𝑥
∴ cos! 𝑥 sin 𝑥𝑑𝑦𝑑𝑥 + cos
! 𝑥 𝑦 = 1 ⇔ 𝑦 = sec 𝑥 + 𝐶 csc 𝑥
OrderandLinearity
𝑦! + 𝑥𝑦!! − !!!
!!!= sin 𝑥𝑦 Sixth-Order-NonlinearandNonhomogeneous
𝑥𝑦!! − !!!
!!!= sin 𝑥 Sixth-Order-LinearandNonhomogeneous
𝑦′′+ 𝑦′+ 𝑦𝑥 = 0Second-Order-LinearandHomogeneous
Copyright©WeSolveThemLLC 15
𝑦′′+ 𝑦𝑦′ = 0Second-Order-NonlinearandHomogeneousNote:Althoughthepowerofyis1inthiscase,itisdependentupony’makingitnonlinear.
𝑦′′′+ 𝑦! + 𝑥𝑒! = 0Third-Order-NonlinearandHomogeneousReductionofOrderProcess:
GivenasecondorderlinearhomogeneousDEoftheform𝑦!! + 𝑃 𝑥 𝑦! + 𝑄 𝑥 = 0accompanied
with𝑦!(𝑥)
Solution
Sincethefirstsolutionisgiven,youmustfindthesecondsolution,whichis:
𝑦! 𝑥 = 𝑦! 𝑥𝑒! ! ! !"
𝑦! 𝑥 ! 𝑑𝑥, ∴ 𝑦 = 𝑐!𝑦! + 𝑐! 𝑦! 𝑥𝑒! ! ! !"
𝑦! 𝑥 ! 𝑑𝑥
Example:
𝑥!𝑦!! + 2𝑥𝑦! − 6𝑦 = 0, 𝑦! = 𝑥!
Find𝑃 𝑥
1𝑥! 𝑥!𝑦!! + 2𝑥𝑦! − 6𝑦 = 0 ⇒ 𝑦!! +
2𝑥 𝑦
! −6𝑥! 𝑦 = 0 ⇒ 𝑃 𝑥 =
2𝑥
∴ 𝑦! = 𝑥!𝑒!
!!!"
𝑥! ! 𝑑𝑥 = 𝑥!𝑒!! !" !
𝑥! 𝑑𝑥 = 𝑥!𝑒!" !!!
𝑥! 𝑑𝑥 = 𝑥!𝑥!!
𝑥! 𝑑𝑥 = 𝑥! 𝑥!! 𝑑𝑥
= 𝑥!1−5 𝑥
!! = −15 𝑥
!! ⇒ 𝑦! =1𝑥!
Theconstantcanbeignoredbecauseaconstanttimesaconstantisaconstant
∴ 𝑦 = 𝑐!𝑥! +
𝑐!𝑥!
Atthispointitshouldbecomeobviousthat𝑐! + 𝑐! +⋯+ 𝑐! = 𝐶,thisisalsotruefornumbersi.e.𝑐! + 5+ 𝑒 + ln 10 + 𝑒!! + 6𝑐! = 𝐶.Inotherwords:aconstantwithaconstantisaconstant.
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PhysicsVectors Notation
𝑎 = 𝑎!,𝑎! in2Dor𝑎 = 𝑎!,𝑎!,𝑎! in3D
Addition/Subtraction
𝑎 ± 𝑏 = 𝑎!,𝑎! ± 𝑏!, 𝑏! = 𝑎! ± 𝑏!,𝑎! ± 𝑏!
𝑎 ± 𝑏 = 𝑎!,𝑎!,𝑎! ± 𝑏!, 𝑏!, 𝑏! = 𝑎! ± 𝑏!,𝑎! ± 𝑏!,𝑎! ± 𝑏! Visually
DotProduct
𝑎 ⋅ 𝑏 = 𝑎!,𝑎! ⋅ 𝑏!, 𝑏! = 𝑎!𝑏! + 𝑎!𝑏!
𝑎 ⋅ 𝑏 = 𝑎!,𝑎!,𝑎! ⋅ 𝑏!, 𝑏!, 𝑏! = 𝑎!𝑏! + 𝑎!𝑏! + 𝑎!𝑏!CrossProduct
𝑎×𝑏 = −𝑏×𝑎
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𝑎×𝑏 =𝚤 𝚥 𝑘𝑎! 𝑎! 𝑎!𝑏! 𝑏! 𝑏!
= 𝚤
𝑎! 𝑎!𝑏! 𝑏! − 𝚥
𝑎! 𝑎!𝑏! 𝑏! + 𝑘
𝑎! 𝑎!𝑏! 𝑏!
= 𝚤 𝑎! 𝑏! − 𝑎! 𝑏! − 𝚥 𝑎! 𝑏! − 𝑎! 𝑏! + 𝑘 𝑎! 𝑏! − 𝑎! 𝑏!
𝚤, 𝚥,and𝑘arecalledunitvectors.Aunitvector,isavectoroflength1
𝚤 = 1, 0, 0 , 𝚥 = 0, 1, 0 , 𝑘 = 0, 0, 1
= 𝑎! 𝑏! − 𝑎! 𝑏! , 0, 0 − 0, 𝑎! 𝑏! − 𝑎! 𝑏! , 0 + 0, 0, 𝑎! 𝑏! − 𝑎! 𝑏!
= 𝑎! 𝑏! − 𝑎! 𝑏! , 𝑎! 𝑏! − 𝑎! 𝑏! , 𝑎! 𝑏! − 𝑎! 𝑏!
MagnitudeorLengthofavectorAboldletterisavectori.e.𝑎 = 𝒂 = 𝑎!,𝑎!,𝑎!
2𝐷, 𝒂 = 𝑎 = 𝑎 = 𝑎!! + 𝑎!!
3𝐷, 𝒂 = 𝑎 = 𝑎 = 𝑎!! + 𝑎!! + 𝑎!!
UnitizingavectorTomakethevectorbeoflength1butpreservethedirection.
2𝐷, 𝑎 =𝑎𝑎 =
𝑎!,𝑎!𝑎!! + 𝑎!!
3𝐷, 𝑎 =𝑎𝑎 =
𝑎!,𝑎!,𝑎!𝑎!! + 𝑎!! + 𝑎!!
ResultantVector
𝑅 = 𝑎 + 𝑏Inphysicsyouwillbeusuallybegiventhevectore.g.(e.g.=forexample)𝑣(𝑣=velocity)
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Theresultantvector,𝑣wouldbeavectorthatcanbebrokenintoa𝑥and𝑦component.Anglewithrespecttox-axis Anglewithrespecttoy-axis
𝑣 = 𝑣 cos𝜃 , 𝑣 sin𝜃 𝑣 = 𝑣 sin𝜙 , 𝑣 cos𝜙
𝑣,𝑣! = 𝑣 cos𝜃 , 𝑣! = 𝑣 cos𝜃 , 0𝑣! = 𝑣 sin𝜃 , 𝑣! = 0, 𝑣 sin𝜃
𝑣,𝑣! = 𝑣 sin𝜙 , 𝑣! = 𝑣 sin𝜙 , 0𝑣! = 𝑣 cos𝜙 , 𝑣! = 0, 𝑣 cos𝜙
𝑅 = 𝑣 = 𝑣! + 𝑣! = 𝑣 cos𝜃 , 0 + 0, 𝑣 sin𝜃 = 𝑣 cos𝜃 , 𝑣 sin𝜃 𝑣 = 𝑣 cos𝜃 ! + 𝑣 sin𝜃 ! = 𝑣! cos! 𝜃 + 𝑣! sin! 𝜃 = 𝑣! cos! 𝜃 + sin! 𝜃 = 𝑣! 1 = 𝑣
Thismaybeslightlyconfusingwiththenotationbecauseofthevectorsbutinphysics,youwillbegivenanumberforthevectori.e.𝑣 = −25!
!,𝜃 = 25°(avectorhasmagnitudeanddirection,
whichmeansitcanbe𝑣 = −25!!cos 25° , −25!
!sin 25° orformagnitude 𝑣 = 𝑣 = 25!
!.
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QuickReferenceArithmetic
𝑎𝑏 ± 𝑎𝑐 = 𝑎 𝑏 ± 𝑐 = 𝑏 ± 𝑐 𝑎 𝑎𝑏𝑐 =
𝑎𝑏𝑐
𝑎𝑏 ±
𝑐𝑑 =
𝑎𝑑 ± 𝑏𝑐𝑏𝑑
𝑎 − 𝑏𝑐 − 𝑑 =
𝑏 − 𝑎𝑑 − 𝑐
𝑎𝑏 + 𝑎𝑐𝑎 = 𝑏 + 𝑐,𝑎 ≠ 0 𝑎
𝑏𝑐 =
𝑎𝑏𝑐
𝑎𝑏𝑐=
𝑎1 ∙
𝑐𝑏 =
𝑎𝑐𝑏
𝑎 ± 𝑏𝑐 =
𝑎𝑐 ±
𝑏𝑐
𝑎𝑏𝑐𝑑
=𝑎𝑏 ∙𝑑𝑐 =
𝑎𝑑𝑏𝑐
Exponential
𝑎! = 𝑎 𝑎! = 1 𝑎!! =1𝑎!
1𝑎!! = 𝑎! 𝑎!𝑎! = 𝑎!!!
𝑎!
𝑎! = 𝑎!!! 𝑎𝑏
!=𝑎!
𝑏! 𝑎𝑏
!!=𝑏!
𝑎! 𝑎!!! = 𝑎
!!
! 𝑎! ! = 𝑎! !
Radicals
𝑎!!= 𝑎!" = 𝑎
!!" 𝑎!! = 𝑎,𝑛 𝑖𝑠 𝑜𝑑𝑑 𝑎!! = 𝑎 ,𝑛 𝑖𝑠 𝑒𝑣𝑒𝑛
𝑎 = 𝑎! = 𝑎!! = 𝑎!! 𝑎!! = 𝑎
!!
𝑎𝑏
!=
𝑎!
𝑏! =𝑎!!
𝑏!!=
𝑎𝑏
!!
Fractions
𝑎𝑏 ±
𝑐𝑑 =
𝑎𝑑 ± 𝑏𝑐𝑏𝑑
𝑔 𝑥𝑓 𝑥 ±
ℎ 𝑥𝑟 𝑥 =
𝑔 𝑥 𝑟 𝑥 ± 𝑓 𝑥 ℎ 𝑥𝑓 𝑥 𝑟 𝑥
Logarithmicln 𝑏ln 𝑎 = log! 𝑏 𝑦 = log! 𝑥⇔ 𝑥 = 𝑏! 𝑒 ≈ 2.72 log! 𝑎 = 1
log! 1 = 0 log! 𝑎! = 𝑢 log! 𝑢 = ln𝑢 log! 𝑢! = 𝑏 log! 𝑢
log! 𝑢𝑣 = log! 𝑢 + log! 𝑣 log!𝑢𝑣 = log! 𝑢 − log! 𝑣 log! 𝑏 =
ln 𝑏ln 𝑎
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𝑣 = ln𝑢 ⇒ 𝑢 = 𝑒! 𝑣 = 𝑒! ⇒ 𝑢 = ln 𝑣 𝑒 =1𝑛!
!
!!!
ln𝑎 = undefined,𝑎 ≤ 0 ln 1 = 0 ln 𝑒! = 𝑢 ⇒ 𝑒!"! = 𝑢ln 𝑒! = 1 ⇒ 𝑒!" ! = 1 ln𝑢! = 𝑏 ln𝑢 ln𝑢𝑣 = ln𝑢 + ln 𝑣 ln
𝑢𝑣 = ln𝑢 − ln 𝑣
QuadraticFormula
𝑎𝑥! + 𝑏𝑥 + 𝑐 = 0 ⇒ 𝑥 =−𝑏 ± 𝑏! − 4𝑎𝑐
2𝑎