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Mathematics: Analysis & Approaches SL & HL
1 Page Formula Sheet – First Examinations 2021 – Updated Version 1.3
Topic 3: Geometry and trigonometry – SL & HL
Prior Learning SL & HL
Area: Parallelogram
Area: Triangle
Area: Trapezoid
Area: Circle
Circumference: Circle
Volume: Cuboid
Volume: Cylinder
Volume: Prism
Area: Cylinder curve
𝐴 = π‘β„Ž , 𝑏 = base, β„Ž = height
1
𝐴 = 2 (π‘β„Ž) , 𝑏 = base, β„Ž = height
1
𝐴 = 2 (π‘Ž + 𝑏)β„Ž , π‘Ž, 𝑏 = parallel sides, β„Ž = height
𝐴 = πœ‹π‘Ÿ 2 , π‘Ÿ = radius
𝐢 = 2πœ‹π‘Ÿ, π‘Ÿ = radius
𝑉 = πœ‹π‘Ÿ 2 β„Ž , π‘Ÿ = radius, β„Ž = height
𝑉 = π΄β„Ž , 𝐴 = cross-section area, β„Ž = height
𝐴 = 2πœ‹π‘Ÿβ„Ž , π‘Ÿ = radius, β„Ž = height
Area: Cone curve
Volume: Sphere
Surface area: Sphere
Sine rule
Area: Triangle
𝐹𝑉 is future value, 𝑃𝑉 is present value, 𝑛 is
the number of years, π‘˜ is the number of
compounding periods per year, π‘Ÿ% is the
nominal annual rate of interest
Pythagorean identity
Reciprocal trigonometric
identities
Pythagorean identities
Compound angle
identities
Exponents & logarithms
Double angle identity
for tan
log π‘Ž π‘₯𝑦 = log π‘Ž π‘₯ + log π‘Ž 𝑦
π‘₯
log π‘Ž = log π‘Ž π‘₯ − log π‘Ž 𝑦
𝑦
log π‘Ž π‘₯ π‘š = π‘š log π‘Ž π‘₯
log𝑏 π‘₯
log π‘Ž π‘₯ =
log 𝑏 π‘Ž
𝑠∞ =
Magnitude of a vector
𝑒1
, |π‘Ÿ| < 1
1−π‘Ÿ
Scalar product
π‘Žπ‘› + (𝑛1)π‘Žπ‘›−1 𝑏+. . . +(π‘›π‘Ÿ)π‘Žπ‘›−π‘Ÿ π‘π‘Ÿ +. . . + 𝑏𝑛
𝑛!
𝑛
( ) = nC r =
π‘Ÿ
π‘Ÿ!(𝑛−π‘Ÿ)!
Topic 1: Number and algebra - HL only
Combinations;
Permutations
nC
𝑛!
π‘Ÿ!(𝑛−π‘Ÿ)!
;
nP
r
𝑛!
= (𝑛−π‘Ÿ)!
𝑛(𝑛 − 1) 𝑏 2
𝑏
( ) +. . . )
π‘Žπ‘› (1 + 𝑛 ( ) +
2!
π‘Ž
π‘Ž
𝑧 = π‘Ž + 𝑏𝑖
Complex numbers
Modulus-argument (polar)
& Exponential (Euler) form
De Moivre’s theorem
r=
(π‘Ž + 𝑏)𝑛 =
Extension of Binomial
Theorem, 𝑛 ∈ β„š
𝑧 = π‘Ÿ(cos πœƒ + 𝑖 sin πœƒ) = π‘Ÿπ‘’
π‘Ÿ
[π‘Ÿ(cos πœƒ + 𝑖 sin πœƒ)]𝑛 =
𝑛 (cos
π‘–πœƒ
𝑛 π‘–π‘›πœƒ
π‘›πœƒ + 𝑖 sin π‘›πœƒ) = π‘Ÿ 𝑒
Topic 2: Functions – SL & HL
Equations of a
straight line
Gradient formula
Axis of symmetry of a
quadratic function
Solutions of a quadratic
equation in the form
π’‚π’™πŸ + 𝒃𝒙 + 𝒄 = 𝟎
Discriminant
Exponential and
logarithmic functions
= π‘Ÿcisπœƒ
𝑛
= π‘Ÿ cisπ‘›πœƒ
𝑦 = π‘šπ‘₯ + 𝑐 ; π‘Žπ‘₯ + 𝑏𝑦 + 𝑑 = 0 ;
𝑦 − 𝑦1 = π‘š(π‘₯ − π‘₯1 )
𝑦2 − 𝑦1
π‘š=
π‘₯2 − π‘₯1
𝑓(π‘₯) = π‘Žπ‘₯ 2 + 𝑏π‘₯ + 𝑐 ⇒ π‘₯ = −
−𝑏 ± √𝑏2 − 4π‘Žπ‘
π‘₯=
,π‘Ž ≠ 0
2π‘Ž
𝑏
2π‘Ž
βˆ† = 𝑏2 − 4π‘Žπ‘
π‘Ž π‘₯ = 𝑒 π‘₯ ln π‘Ž ; log π‘Ž π‘Ž π‘₯ = π‘₯ = π‘Ž logπ‘Ž π‘₯
where π‘Ž, π‘₯ > 0 , π‘Ž ≠ 1
Topic 2: Functions – HL only
Sum & product of the
roots of polynomial
equations of the form
π‘Ž
=
sin𝐴
𝑏
sin𝐡
=
𝑛
∑ π‘Žπ‘Ÿ π‘₯ π‘Ÿ = 0
π‘Ÿ=0
⇒ Sum is
(−1)𝑛 π‘Ž0
−π‘Žπ‘›−1
; product is
π‘Žπ‘›
π‘Žπ‘›
1
𝐴 = 2 π‘Ÿ 2 πœƒ , π‘Ÿ = radius, πœƒ = angle in radians
sin πœƒ
cos πœƒ
2
cos 2 πœƒ + sin πœƒ = 1
Angle between two
vectors
Vector equ. of a line
Parametric form of the
equation of a line
Cartesian equations of
a line
Vector product
Area of a
parallelogram
Vector equ. of a plane
Equation of a plane
Cartesian equ. of a plane
1
;
cos πœƒ
cosec πœƒ =
1
sin πœƒ
1 + tan2 πœƒ = sec 2 πœƒ ; 1 + cot 2 πœƒ = cosec 2 πœƒ
sin(𝐴 ± 𝐡) = sin 𝐴 cos 𝐡 ± cos 𝐴 sin 𝐡
cos(𝐴 ± 𝐡) = cos 𝐴 cos 𝐡 βˆ“ sin 𝐴 sin 𝐡
tan 𝐴 ± tan 𝐡
tan(𝐴 ± 𝐡) =
1 βˆ“ tan 𝐴 tan 𝐡
tan 2πœƒ =
2 tan πœƒ
1 − tan2 πœƒ
|𝒗| = √𝑣1 2 + 𝑣2 2 + 𝑣3 2
Μ… , of a set of
Mean, 𝒙
data
𝒓 = 𝒂 + πœ†π’ƒ + πœ‡π’„
Probability of an event A
P(𝐴) =
𝑛
𝑛(𝐴)
𝑛(𝑒)
, where 𝑛
= ∑π‘˜π‘–=1 𝑓𝑖
Mutually exclusive
events
Derivative of 𝒙𝒏
Area between curve
π’š = 𝒇(𝒙) & 𝒙-axis
Derivative of 𝐬𝐒𝐧 𝒙
Derivative of 𝐜𝐨𝐬 𝒙
Derivative of 𝒆𝒙
Derivative of π₯𝐧 𝒙
Chain rule
Product rule
Quotient rule
Acceleration
Conditional probability
Independent events
Expected value: Discrete
random variable X
Binomial distribution
Mean ; Variance
Standardized normal
variable
P(𝐴|𝐡) =
P(𝐴 ∩ 𝐡)
P(𝐡)
P(𝐴 ∩ 𝐡) = P(𝐴)P(𝐡)
E(𝑋) = ∑ π‘₯ P(𝑋 = π‘₯)
𝑋~B(𝑛, 𝑝)
E(𝑋) = 𝑛𝑝 ; Var(𝑋) = 𝑛𝑝(1 − 𝑝)
π‘₯−πœ‡
𝑧=
𝜎
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− πœ‡2
𝑛
Var(𝑋) = E[(𝑋 − πœ‡)2 ] = E(𝑋 2 ) − [E(𝑋)]2
Var(𝑋) = ∑(π‘₯ − πœ‡)2 P(𝑋 = π‘₯)
= ∑ π‘₯ 2 P(𝑋 = π‘₯) − πœ‡2
∞
Var(𝑋) = ∫−∞(π‘₯ − πœ‡)2 𝑓(π‘₯)dπ‘₯
∞
= ∫−∞ π‘₯ 2 𝑓(π‘₯)dπ‘₯ − πœ‡2
𝑓(π‘₯) = π‘₯ 𝑛 ⇒ 𝑓′(π‘₯) = 𝑛π‘₯ 𝑛−1
π‘₯ 𝑛+1
∫ π‘₯ 𝑛 𝑑π‘₯ =
+ 𝐢 , 𝑛 ≠ −1
𝑛+1
𝑏
𝐴 = ∫ 𝑦 𝑑π‘₯ ,
where 𝑓(π‘₯) > 0
π‘Ž
𝑓(π‘₯) = sin π‘₯ ⇒ 𝑓′(π‘₯) = cos π‘₯
𝑓(π‘₯) = cos π‘₯ ⇒ 𝑓′(π‘₯) = − sin π‘₯
𝑓(π‘₯) = 𝑒 π‘₯ ⇒ 𝑓′(π‘₯) = 𝑒 π‘₯
1
𝑓(π‘₯) = ln π‘₯ ⇒ 𝑓′(π‘₯) =
π‘₯
𝑑𝑦
𝑑𝑦 𝑑𝑒
𝑦 = 𝑔(𝑒) , 𝑒 = 𝑓(π‘₯) ⇒
=
×
𝑑π‘₯
𝑑𝑒 𝑑π‘₯
𝑑𝑦
𝑑𝑣
𝑑𝑒
𝑦 = 𝑒𝑣 ⇒
=𝑒
+𝑣
𝑑π‘₯
𝑑π‘₯
𝑑π‘₯
𝑑𝑣
𝑑𝑒
𝑒
𝑑𝑦 𝑣 𝑑π‘₯ − 𝑒 𝑑π‘₯
𝑦=
⇒
=
𝑣2
𝑣
𝑑π‘₯
d𝑣 d2 𝑠
π‘Ž=
= 2
d𝑑 d𝑑
𝑑2
𝑑2
dist = ∫ |𝑣(𝑑)| 𝑑𝑑 ;
disp = ∫ 𝑣(𝑑) 𝑑𝑑
𝑑1
𝑑1
1
∫ 𝑑π‘₯ = ln|π‘₯| + 𝐢
π‘₯
∫ sin π‘₯ 𝑑π‘₯ = − cos π‘₯ + 𝐢
∫ cos π‘₯ 𝑑π‘₯ = sin π‘₯ + 𝐢
∫ 𝑒 π‘₯ 𝑑π‘₯ = 𝑒 π‘₯ + 𝐢
𝑏
Area enclosed by a
curve and 𝒙-axis
𝐴 = ∫ |𝑦| 𝑑π‘₯
Derivative of 𝒇(𝒙)
from first principles
𝑓(π‘₯ + β„Ž) − 𝑓(π‘₯)
d𝑦
= 𝑓′(π‘₯) = lim (
)
β„Ž→0
β„Ž
dπ‘₯
π‘Ž
Topic 5: Calculus – HL only
Standard
derivatives
P(𝐴 ∪ 𝐡) = P(𝐴) + P(𝐡) − P(𝐴 ∩ 𝐡)
P(𝐴 ∪ 𝐡) = P(𝐴) + P(𝐡)
𝑛
Topic 5: Calculus - SL & HL
Complementary events P(𝐴) + P(𝐴′ ) = 1
Combined events
2
∑π‘˜
𝑖=1 𝑓𝑖 (π‘₯𝑖 −πœ‡)
=
∞
Variance of a continuous
random variable X
π‘Žπ‘₯ + 𝑏𝑦 + 𝑐𝑧 = 𝑑
∑π‘˜
𝑖=1 𝑓𝑖 π‘₯𝑖
𝑛
E(𝑋) = πœ‡ = ∫−∞ π‘₯𝑓(π‘₯)dπ‘₯
Variance of a discrete
random variable X
𝒓 βˆ™ 𝒏 = 𝒂 βˆ™ 𝒏 (using the normal vector)
π‘₯Μ… =
𝜎=√
Variance
π‘₯ − π‘₯0 𝑦 − 𝑦0 𝑧 − 𝑧0
=
=
𝑙
π‘š
𝑛
𝐴 = |𝒗 × π’˜| , where 𝒗 and π’˜ form two
adjacent sides of a parallelogram
2
∑π‘˜
𝑖=1 𝑓𝑖 π‘₯𝑖
E(π‘Žπ‘‹ + 𝑏) = π‘ŽE(𝑋) + 𝑏
Var(π‘Žπ‘‹ + 𝑏) = π‘Ž2 Var(𝑋)
Standard integrals
𝑣2 𝑀3 − 𝑣3 𝑀2
𝒗 × π’˜ = (𝑣3 𝑀1 − 𝑣1 𝑀3 )
𝑣1 𝑀2 − 𝑣2 𝑀1
|𝒗 × π’˜| = |𝒗||π’˜| sin πœƒ
where πœƒ is the angle between 𝒗 and π’˜
2
∑π‘˜
𝑖=1 𝑓𝑖 (π‘₯𝑖 −πœ‡)
Linear transformation of
a single random variable
π‘₯ = π‘₯0 + πœ†π‘™, 𝑦 = 𝑦0 + πœ†π‘š, 𝑧 = 𝑧0 + πœ†π‘›
IQR = 𝑄3 − 𝑄1
P(𝐡𝑖 )P(𝐴|𝐡𝑖 )
P(𝐡1 )P(𝐴|𝐡1 ) + P(𝐡2 )P(𝐴|𝐡2 ) + P(𝐡3 )P(𝐴|𝐡3 )
Standard Deviation 𝝈
Distance; Displacement
travelled from π’•πŸ to π’•πŸ
𝒓 = 𝒂 + πœ†π’ƒ
P(𝐡)P(𝐴|𝐡)
P(𝐡)P(𝐴|𝐡) + P(𝐡′ )P(𝐴|𝐡′ )
𝜎2 =
Topic 4: Statistics and probability - SL & HL
Interquartile range
P(𝐡𝑖 |𝐴) =
Variance 𝝈𝟐
𝒗 βˆ™ π’˜ = 𝑣1 𝑀1 + 𝑣2 𝑀2 + 𝑣3 𝑀3
𝒗 βˆ™ π’˜ = |𝒗||π’˜| cos πœƒ
where πœƒ is the angle between 𝒗 and π’˜
𝑣1 𝑀1 + 𝑣2 𝑀2 + 𝑣3 𝑀3
cos πœƒ =
|𝒗||π’˜|
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Bayes’ theorem
Integral of 𝒙𝒏
sin 2πœƒ = 2 sin πœƒ cos πœƒ
cos 2πœƒ = cos 2 πœƒ − sin2 πœƒ
= 2 cos 2 πœƒ − 1
= 1 − 2 sin2 πœƒ
sec πœƒ =
P(𝐡|𝐴) =
Expected value: Continuous
random variable X
𝑐
sin𝐢
Topic 3: Geometry and trigonometry – HL only
Exponents & logarithms π‘Ž = 𝑏 ⇔ π‘₯ = log π‘Ž 𝑏 , π‘Ž, 𝑏 > 0, π‘Ž ≠ 1
Binomial coefficient
𝐴 = 4πœ‹π‘Ÿ 2 , π‘Ÿ = radius
Double angle identities
π‘₯
Binomial Theorem
for 𝑛 ∈ β„•, (π‘Ž + 𝑏)𝑛 =
4
𝑉 = 3 πœ‹π‘Ÿ 3 , π‘Ÿ = radius
𝑒1 (π‘Ÿ 𝑛 − 1) 𝑒1 (1 − π‘Ÿ 𝑛 )
=
,π‘Ÿ ≠ 1
π‘Ÿ−1
1−π‘Ÿ
π‘Ÿ π‘˜π‘›
𝐹𝑉 = 𝑃𝑉 × (1 +
)
100π‘˜
), for endpoints (π‘₯1 , 𝑦1), (π‘₯2 , 𝑦2 )
𝑠𝑛 =
The sum of an infinite
geometric sequence
𝐴 = πœ‹π‘Ÿπ‘™ , π‘Ÿ= radius, 𝑙 = slant height
tan πœƒ =
2
𝑒𝑛 = 𝑒1 π‘Ÿ 𝑛−1
Sum of 𝒏 terms of a
finite geometric seq.
1
𝑉 = 3 πœ‹π‘Ÿ 2 β„Ž , π‘Ÿ= radius, β„Ž = height
Identity for 𝐭𝐚𝐧 𝜽
,
Cosine rule
Length of an arc
The 𝒏th term of a
geometric sequence
1
Volume: Right-pyramid 𝑉 = 3 π΄β„Ž , 𝐴 = base area, β„Ž = height
Area of a sector
2
𝑒𝑛 = 𝑒1 + (𝑛 − 1)𝑑
Sum of 𝒏 terms of an
arithmetic sequence
π‘₯1 + π‘₯2 𝑦1 + 𝑦2 𝑧1 + 𝑧2
,
,
)
2
2
2
𝑛
𝑛
𝑠𝑛 = (2𝑒1 + (𝑛 − 1)𝑑) = (𝑒1 + 𝑒𝑛 )
2
2
(
The 𝒏th term of an
arithmetic sequence
(
𝑐 2 = π‘Ž2 + 𝑏2 − 2π‘Žπ‘ cos 𝐢
π‘Ž2 + 𝑏2 − 𝑐 2
cos 𝐢 =
2π‘Žπ‘
1
𝐴 = π‘Žπ‘ sin 𝐢
2
𝑙 = π‘Ÿπœƒ , π‘Ÿ = radius, πœƒ = angle in radians
π‘₯1 +π‘₯2 𝑦1 +𝑦2
Topic 1: Number and algebra - SL & HL
Compound interest
Coordinates of midpoint of
a line with endpoints
(π’™πŸ , π’šπŸ , π’›πŸ ) , (π’™πŸ , π’šπŸ , π’›πŸ )
Volume: Right cone
𝑉 = π‘™π‘€β„Ž , 𝑙 = length, 𝑀 = width, β„Ž = height
Distance between two
)2 (𝑦
)2
points (π’™πŸ , π’šπŸ ) , (π’™πŸ , π’šπŸ ) 𝑑 = √(π‘₯1 − π‘₯2 + 1 − 𝑦2
Coordinates of midpoint
Distance between 2 points
𝑑 = √(π‘₯1 − π‘₯2 )2 + (𝑦1 − 𝑦2 )2 + (𝑧1 − 𝑧2 )2
(π’™πŸ , π’šπŸ , π’›πŸ ) , (π’™πŸ , π’šπŸ , π’›πŸ )
Topic 4: Statistics and probability – HL only
Standard
integrals
𝑓(π‘₯) = tan π‘₯ ⇒ 𝑓 ′ (π‘₯) = sec 2 π‘₯
𝑓(π‘₯) = sec π‘₯ ⇒ 𝑓 ′ (π‘₯) = sec π‘₯ tan π‘₯
𝑓(π‘₯) = cosec π‘₯ ⇒ 𝑓 ′ (π‘₯) = −cosec π‘₯ cot π‘₯
𝑓(π‘₯) = cot π‘₯ ⇒ 𝑓 ′ (π‘₯) = −cosec 2 π‘₯
𝑓(π‘₯) = π‘Ž π‘₯ ⇒ 𝑓 ′ (π‘₯) = π‘Ž π‘₯ (ln π‘Ž)
1
𝑓(π‘₯) = logπ‘Ž π‘₯ ⇒ 𝑓 ′ (π‘₯) =
π‘₯ ln π‘Ž
1
𝑓(π‘₯) = arcsin π‘₯ ⇒ 𝑓 ′ (π‘₯) =
√1 − π‘₯ 2
1
𝑓(π‘₯) = arccos π‘₯ ⇒ 𝑓 ′ (π‘₯) = −
√1 − π‘₯ 2
1
𝑓(π‘₯) = arctan π‘₯ ⇒ 𝑓 ′ (π‘₯) =
1 + π‘₯2
1
∫ π‘Ž π‘₯ dπ‘₯ = ln π‘Ž π‘Ž π‘₯ + 𝐢
1
1
π‘₯
∫ π‘Ž2 +π‘₯2 dπ‘₯ = π‘Ž arctan (π‘Ž) + 𝐢
1
π‘₯
Integration by parts
∫ √π‘Ž2 −π‘₯2 dπ‘₯ = arcsin (π‘Ž) + 𝐢 , |π‘₯| < π‘Ž
Area enclosed by a
curve and π’š-axis
𝐴 = ∫π‘Ž |π‘₯| 𝑑𝑦
Volume of revolution
about 𝒙 or π’š-axes
Euler’s method
Integrating factor for
π’š′ + 𝑷(𝒙)π’š = 𝑸(𝒙)
Maclaurin series
d𝑣
d𝑒
∫ 𝑒 dπ‘₯ 𝑑π‘₯ = 𝑒𝑣 − ∫ 𝑣 dπ‘₯ 𝑑π‘₯
𝑉=
𝑏
𝑏
∫π‘Ž πœ‹π‘¦ 2 𝑑π‘₯
𝑏
or 𝑉 = ∫π‘Ž πœ‹π‘₯ 2 𝑑𝑦
𝑦𝑛+1 = 𝑦𝑛 + β„Ž × π‘“(π‘₯𝑛 , 𝑦𝑛 ); π‘₯𝑛+1 = π‘₯𝑛 + β„Ž
where β„Ž is a constant (step length)
𝑒 ∫ 𝑃(π‘₯)dπ‘₯
𝑓(π‘₯) = 𝑓(0) + π‘₯𝑓 ′ (0) +
2
Maclaurin series for
special functions
π‘₯ 2 ′′
𝑓 (0)+ . ..
2!
2
3
βˆ™ 𝑒 π‘₯ = 1 + π‘₯ + π‘₯2! + ... βˆ™ ln(1 + π‘₯) = π‘₯ − π‘₯2 + π‘₯3 − ...
3
5
2
4
βˆ™ sin π‘₯ = π‘₯ − π‘₯3! + π‘₯5! − ... βˆ™ cos π‘₯ = 1 − π‘₯2! + π‘₯4! − ...
3
5
βˆ™ arctan π‘₯ = π‘₯ − π‘₯3 + π‘₯5 − ...
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