Mathematics: Analysis & Approaches SL & HL
1 Page Formula Sheet – First Examinations 2021 – Updated Version 1.3
Prior Learning SL & HL
Area: Parallelogram
Topic 3: Geometry and trigonometry – SL & HL
Distance between 2 points
𝑑 = √(𝑥1 − 𝑥2 )2 + (𝑦1 − 𝑦2 )2 + (𝑧1 − 𝑧2 )2
(𝒙𝟏 , 𝒚𝟏 , 𝒛𝟏 ) , (𝒙𝟐 , 𝒚𝟐 , 𝒛𝟐 )
𝐴 = 𝑏ℎ , 𝑏 = base, ℎ = height
1
Area: Triangle
𝐴 = 2 (𝑏ℎ) , 𝑏 = base, ℎ = height
Area: Trapezoid
𝐴 = 2 (𝑎 + 𝑏)ℎ , 𝑎, 𝑏 = parallel sides, ℎ = height
Area: Circle
𝐴 = 𝜋𝑟 2 , 𝑟 = radius
1
Volume: Cuboid
𝑉 = 𝑙𝑤ℎ , 𝑙 = length, 𝑤 = width, ℎ = height
Volume: Cylinder
𝑉 = 𝜋𝑟 2 ℎ , 𝑟 = radius, ℎ = height
Volume: Prism
𝑉 = 𝐴ℎ , 𝐴 = cross-section area, ℎ = height
Area: Cylinder curve
𝐴 = 2𝜋𝑟ℎ , 𝑟 = radius, ℎ = height
,
2
2
), for endpoints (𝑥1 , 𝑦1), (𝑥2 , 𝑦2 )
Area: Cone curve
𝐴 = 𝜋𝑟𝑙 , 𝑟= radius, 𝑙 = slant height
Volume: Sphere
𝑉 = 3 𝜋𝑟 3 , 𝑟 = radius
Surface area: Sphere
𝐴 = 4𝜋𝑟 2 , 𝑟 = radius
Area: Triangle
Topic 1: Number and algebra - SL & HL
Length of an arc
4
𝑎
=
sin𝐴
𝑏
sin𝐵
=
𝑐
sin𝐶
𝑐 2 = 𝑎2 + 𝑏2 − 2𝑎𝑏 cos 𝐶
𝑎2 + 𝑏2 − 𝑐 2
cos 𝐶 =
2𝑎𝑏
1
𝐴 = 𝑎𝑏 sin 𝐶
2
𝑙 = 𝑟𝜃 , 𝑟 = radius, 𝜃 = angle in radians
The 𝒏th term of an
arithmetic sequence
𝑢𝑛 = 𝑢1 + (𝑛 − 1)𝑑
Area of a sector
𝐴 = 2 𝑟 2 𝜃 , 𝑟 = radius, 𝜃 = angle in radians
Sum of 𝒏 terms of an
arithmetic sequence
𝑛
𝑛
𝑠𝑛 = (2𝑢1 + (𝑛 − 1)𝑑) = (𝑢1 + 𝑢𝑛 )
2
2
Identity for 𝐭𝐚𝐧 𝜽
tan 𝜃 =
The 𝒏th term of a
geometric sequence
Pythagorean identity
cos 2 𝜃 + sin 𝜃 = 1
𝑢𝑛 = 𝑢1 𝑟 𝑛−1
Sum of 𝒏 terms of a
finite geometric seq.
𝑠𝑛 =
𝑢1 (𝑟 𝑛 − 1) 𝑢1 (1 − 𝑟 𝑛 )
=
,𝑟 ≠ 1
𝑟−1
1−𝑟
𝑟 𝑘𝑛
𝐹𝑉 = 𝑃𝑉 × (1 +
)
100𝑘
Double angle identities
sin 2𝜃 = 2 sin 𝜃 cos 𝜃
cos 2𝜃 = cos 2 𝜃 − sin2 𝜃
= 2 cos 2 𝜃 − 1
= 1 − 2 sin2 𝜃
𝐹𝑉 is future value, 𝑃𝑉 is present value, 𝑛 is
the number of years, 𝑘 is the number of
compounding periods per year, 𝑟% is the
nominal annual rate of interest
Reciprocal trigonometric
identities
Compound interest
𝑥
Exponents & logarithms 𝑎 = 𝑏 ⇔ 𝑥 = log 𝑎 𝑏 , 𝑎, 𝑏 > 0, 𝑎 ≠ 1
log 𝑎 𝑥𝑦 = log 𝑎 𝑥 + log 𝑎 𝑦
𝑥
log 𝑎 = log 𝑎 𝑥 − log 𝑎 𝑦
𝑦
log 𝑎 𝑥 𝑚 = 𝑚 log 𝑎 𝑥
log𝑏 𝑥
log 𝑎 𝑥 =
log 𝑏 𝑎
Exponents & logarithms
The sum of an infinite
geometric sequence
Binomial Theorem
for 𝑛 ∈ ℕ, (𝑎 + 𝑏)𝑛 =
Binomial coefficient
𝑠∞ =
𝑢1
, |𝑟| < 1
1−𝑟
𝑛!
𝑛
( ) = nC r =
𝑟
𝑟!(𝑛−𝑟)!
Topic 1: Number and algebra - HL only
Combinations;
Permutations
nC
𝑛!
𝑟!(𝑛−𝑟)!
;
nP
𝑛!
r
= (𝑛−𝑟)!
(𝑎 + 𝑏)𝑛 =
Extension of Binomial
Theorem, 𝑛 ∈ ℚ
𝑏
𝑛(𝑛 − 1) 𝑏 2
𝑎𝑛 (1 + 𝑛 ( ) +
( ) +. . . )
𝑎
2!
𝑎
Complex numbers
𝑧 = 𝑎 + 𝑏𝑖
Modulus-argument (polar)
& Exponential (Euler) form
De Moivre’s theorem
r=
𝑧 = 𝑟(cos 𝜃 + 𝑖 sin 𝜃) = 𝑟𝑒
𝑖𝜃
= 𝑟cis𝜃
[𝑟(cos 𝜃 + 𝑖 sin 𝜃)]𝑛 =
𝑟
𝑛 (cos
𝑛 𝑖𝑛𝜃
𝑛𝜃 + 𝑖 sin 𝑛𝜃) = 𝑟 𝑒
𝑛
= 𝑟 cis𝑛𝜃
Topic 2: Functions – SL & HL
Equations of a
straight line
Gradient formula
𝑏
2𝑎
𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 ⇒ 𝑥 = −
Solutions of a quadratic
equation in the form
𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄 = 𝟎
𝑥=
Discriminant
∆ = 𝑏2 − 4𝑎𝑐
Exponential and
logarithmic functions
𝑎 𝑥 = 𝑒 𝑥 ln 𝑎 ; log 𝑎 𝑎 𝑥 = 𝑥 = 𝑎 log𝑎 𝑥
−𝑏 ± √𝑏2 − 4𝑎𝑐
,𝑎 ≠ 0
2𝑎
where 𝑎, 𝑥 > 0 , 𝑎 ≠ 1
Topic 2: Functions – HL only
𝑛
∑ 𝑎𝑟 𝑥 𝑟 = 0
𝑟=0
⇒ Sum is
Vector product
Area of a
parallelogram
−𝑎𝑛−1
(−1)𝑛 𝑎0
; product is
𝑎𝑛
𝑎𝑛
cosec 𝜃 =
1
sin 𝜃
1 + tan2 𝜃 = sec 2 𝜃 ; 1 + cot 2 𝜃 = cosec 2 𝜃
𝜎=√
Linear transformation of
a single random variable
E(𝑎𝑋 + 𝑏) = 𝑎E(𝑋) + 𝑏
Var(𝑎𝑋 + 𝑏) = 𝑎2 Var(𝑋)
Expected value: Continuous
random variable X
E(𝑋) = 𝜇 = ∫−∞ 𝑥𝑓(𝑥)d𝑥
1 − tan2 𝜃
𝒗 ∙ 𝒘 = 𝑣1 𝑤1 + 𝑣2 𝑤2 + 𝑣3 𝑤3
𝒗 ∙ 𝒘 = |𝒗||𝒘| cos 𝜃
where 𝜃 is the angle between 𝒗 and 𝒘
𝑣1 𝑤1 + 𝑣2 𝑤2 + 𝑣3 𝑤3
cos 𝜃 =
|𝒗||𝒘|
𝑥 = 𝑥0 + 𝜆𝑙, 𝑦 = 𝑦0 + 𝜆𝑚, 𝑧 = 𝑧0 + 𝜆𝑛
𝐴 = |𝒗 × 𝒘| , where 𝒗 and 𝒘 form two
adjacent sides of a parallelogram
𝑎𝑥 + 𝑏𝑦 + 𝑐𝑧 = 𝑑
Var(𝑋) = ∫−∞(𝑥 − 𝜇)2 𝑓(𝑥)d𝑥
∞
∞
= ∫−∞ 𝑥 2 𝑓(𝑥)d𝑥 − 𝜇2
Topic 5: Calculus - SL & HL
IQR = 𝑄3 − 𝑄1
𝑥̅ =
∑𝑘
𝑖=1 𝑓𝑖 𝑥𝑖
𝑛
, where 𝑛
𝑓(𝑥) = 𝑥 𝑛 ⇒ 𝑓′(𝑥) = 𝑛𝑥 𝑛−1
𝑥 𝑛+1
∫ 𝑥 𝑛 𝑑𝑥 =
+ 𝐶 , 𝑛 ≠ −1
𝑛+1
𝑏
Area between curve
𝒚 = 𝒇(𝒙) & 𝒙-axis
𝐴 = ∫ 𝑦 𝑑𝑥 ,
Derivative of 𝐬𝐢𝐧 𝒙
𝑓(𝑥) = sin 𝑥 ⇒ 𝑓′(𝑥) = cos 𝑥
Derivative of 𝐜𝐨𝐬 𝒙
𝑓(𝑥) = cos 𝑥 ⇒ 𝑓′(𝑥) = − sin 𝑥
Derivative of 𝒆𝒙
𝑓(𝑥) = 𝑒 𝑥 ⇒ 𝑓′(𝑥) = 𝑒 𝑥
1
𝑓(𝑥) = ln 𝑥 ⇒ 𝑓′(𝑥) =
𝑥
𝑑𝑦
𝑑𝑦 𝑑𝑢
𝑦 = 𝑔(𝑢) , 𝑢 = 𝑓(𝑥) ⇒
=
×
𝑑𝑥
𝑑𝑢 𝑑𝑥
𝑑𝑦
𝑑𝑣
𝑑𝑢
𝑦 = 𝑢𝑣 ⇒
=𝑢
+𝑣
𝑑𝑥
𝑑𝑥
𝑑𝑥
𝑑𝑢
𝑑𝑣
𝑢
𝑑𝑦 𝑣 𝑑𝑥 − 𝑢 𝑑𝑥
𝑦=
⇒
=
𝑣
𝑑𝑥
𝑣2
d𝑣 d2 𝑠
𝑎=
= 2
d𝑡 d𝑡
Derivative of 𝐥𝐧 𝒙
Chain rule
Product rule
Acceleration
Distance; Displacement
travelled from 𝒕𝟏 to 𝒕𝟐
where 𝑓(𝑥) > 0
𝑎
𝑡2
𝑡2
dist = ∫ |𝑣(𝑡)| 𝑑𝑡 ;
disp = ∫ 𝑣(𝑡) 𝑑𝑡
𝑡1
𝑡1
1
∫ 𝑑𝑥 = ln|𝑥| + 𝐶
𝑥
∫ sin 𝑥 𝑑𝑥 = − cos 𝑥 + 𝐶
Standard integrals
∫ 𝑒 𝑥 𝑑𝑥 = 𝑒 𝑥 + 𝐶
Area enclosed by a
curve and 𝒙-axis
= ∑𝑘𝑖=1 𝑓𝑖
P(𝐴 ∪ 𝐵) = P(𝐴) + P(𝐵) − P(𝐴 ∩ 𝐵)
Mutually exclusive
events
P(𝐴 ∪ 𝐵) = P(𝐴) + P(𝐵)
𝑎
Derivative of 𝒇(𝒙)
from first principles
d𝑦
𝑓(𝑥 + ℎ) − 𝑓(𝑥)
= 𝑓′(𝑥) = lim (
)
ℎ→0
d𝑥
ℎ
Standard
derivatives
𝑓(𝑥) = tan 𝑥 ⇒ 𝑓 ′ (𝑥) = sec 2 𝑥
𝑓(𝑥) = sec 𝑥 ⇒ 𝑓 ′ (𝑥) = sec 𝑥 tan 𝑥
𝑓(𝑥) = cosec 𝑥 ⇒ 𝑓 ′ (𝑥) = −cosec 𝑥 cot 𝑥
𝑓(𝑥) = cot 𝑥 ⇒ 𝑓 ′ (𝑥) = −cosec 2 𝑥
𝑓(𝑥) = 𝑎 𝑥 ⇒ 𝑓 ′ (𝑥) = 𝑎 𝑥 (ln 𝑎)
1
𝑓(𝑥) = log𝑎 𝑥 ⇒ 𝑓 ′ (𝑥) =
𝑥 ln 𝑎
1
𝑓(𝑥) = arcsin 𝑥 ⇒ 𝑓 ′ (𝑥) =
√1 − 𝑥 2
1
𝑓(𝑥) = arccos 𝑥 ⇒ 𝑓 ′ (𝑥) = −
√1 − 𝑥 2
1
𝑓(𝑥) = arctan 𝑥 ⇒ 𝑓 ′ (𝑥) =
1 + 𝑥2
P(𝐴) =
Combined events
𝑏
𝐴 = ∫ |𝑦| 𝑑𝑥
Topic 5: Calculus – HL only
𝑛(𝐴)
𝑛(𝑢)
Complementary events P(𝐴) + P(𝐴′ ) = 1
1
P(𝐴|𝐵) =
P(𝐴 ∩ 𝐵)
P(𝐵)
Independent events
P(𝐴 ∩ 𝐵) = P(𝐴)P(𝐵)
Expected value: Discrete
random variable X
E(𝑋) = ∑ 𝑥 P(𝑋 = 𝑥)
Standardized normal
variable
∞
∫ cos 𝑥 𝑑𝑥 = sin 𝑥 + 𝐶
𝑣2 𝑤3 − 𝑣3 𝑤2
𝒗 × 𝒘 = (𝑣3 𝑤1 − 𝑣1 𝑤3 )
𝑣1 𝑤2 − 𝑣2 𝑤1
|𝒗 × 𝒘| = |𝒗||𝒘| sin 𝜃
where 𝜃 is the angle between 𝒗 and 𝒘
Cartesian equ. of a plane
Binomial distribution
Mean ; Variance
𝑛
Variance of a continuous
random variable X
𝑥 − 𝑥0 𝑦 − 𝑦0 𝑧 − 𝑧0
=
=
𝑙
𝑚
𝑛
𝒓 = 𝒂 + 𝜆𝒃 + 𝜇𝒄
Conditional probability
2
∑𝑘
𝑖=1 𝑓𝑖 (𝑥𝑖 −𝜇)
Var(𝑋) = ∑(𝑥 − 𝜇)2 P(𝑋 = 𝑥)
= ∑ 𝑥 2 P(𝑋 = 𝑥) − 𝜇2
𝒓 = 𝒂 + 𝜆𝒃
𝒓 ∙ 𝒏 = 𝒂 ∙ 𝒏 (using the normal vector)
Probability of an event A
− 𝜇2
Var(𝑋) = E[(𝑋 − 𝜇)2 ] = E(𝑋 2 ) − [E(𝑋)]2
Quotient rule
Equation of a plane
̅ , of a set of
Mean, 𝒙
data
𝑛
Variance of a discrete
random variable X
|𝒗| = √𝑣1 2 + 𝑣2 2 + 𝑣3 2
Vector equ. of a plane
Interquartile range
=
𝑛
Variance
2 tan 𝜃
𝑋~B(𝑛, 𝑝)
E(𝑋) = 𝑛𝑝 ; Var(𝑋) = 𝑛𝑝(1 − 𝑝)
𝑥−𝜇
𝑧=
𝜎
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Standard Deviation 𝝈
Topic 4: Statistics and probability - SL & HL
𝑦 = 𝑚𝑥 + 𝑐 ; 𝑎𝑥 + 𝑏𝑦 + 𝑑 = 0 ;
𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1 )
𝑦2 − 𝑦1
𝑚=
𝑥2 − 𝑥1
Axis of symmetry of a
quadratic function
Sum & product of the
roots of polynomial
equations of the form
1
;
cos 𝜃
sec 𝜃 =
tan 2𝜃 =
Vector equ. of a line
Parametric form of the
equation of a line
Cartesian equations of
a line
2
∑𝑘
𝑖=1 𝑓𝑖 𝑥𝑖
𝜎2 =
Integral of 𝒙𝒏
Double angle identity
for tan
Angle between two
vectors
2
∑𝑘
𝑖=1 𝑓𝑖 (𝑥𝑖 −𝜇)
Variance 𝝈𝟐
Derivative of 𝒙𝒏
Compound angle
identities
Scalar product
𝑎𝑛 + (𝑛1)𝑎𝑛−1 𝑏+. . . +(𝑛𝑟)𝑎𝑛−𝑟 𝑏𝑟 +. . . + 𝑏𝑛
sin 𝜃
cos 𝜃
2
sin(𝐴 ± 𝐵) = sin 𝐴 cos 𝐵 ± cos 𝐴 sin 𝐵
cos(𝐴 ± 𝐵) = cos 𝐴 cos 𝐵 ∓ sin 𝐴 sin 𝐵
tan 𝐴 ± tan 𝐵
tan(𝐴 ± 𝐵) =
1 ∓ tan 𝐴 tan 𝐵
Magnitude of a vector
Bayes’ theorem
1
Topic 3: Geometry and trigonometry – HL only
Pythagorean identities
P(𝐵)P(𝐴|𝐵)
P(𝐵)P(𝐴|𝐵) + P(𝐵′ )P(𝐴|𝐵′ )
P(𝐵𝑖 |𝐴) =
1
Volume: Right cone
Cosine rule
P(𝐵|𝐴) =
P(𝐵𝑖 )P(𝐴|𝐵𝑖 )
P(𝐵1 )P(𝐴|𝐵1 ) + P(𝐵2 )P(𝐴|𝐵2 ) + P(𝐵3 )P(𝐴|𝐵3 )
𝑉 = 3 𝜋𝑟 2 ℎ , 𝑟= radius, ℎ = height
Sine rule
Distance between two
)2 (𝑦
)2
points (𝒙𝟏 , 𝒚𝟏 ) , (𝒙𝟐 , 𝒚𝟐 ) 𝑑 = √(𝑥1 − 𝑥2 + 1 − 𝑦2
(
𝑥1 + 𝑥2 𝑦1 + 𝑦2 𝑧1 + 𝑧2
,
,
)
2
2
2
1
𝐶 = 2𝜋𝑟, 𝑟 = radius
Coordinates of midpoint
(
Volume: Right-pyramid 𝑉 = 3 𝐴ℎ , 𝐴 = base area, ℎ = height
Circumference: Circle
𝑥1 +𝑥2 𝑦1 +𝑦2
Coordinates of midpoint of
a line with endpoints
(𝒙𝟏 , 𝒚𝟏 , 𝒛𝟏 ) , (𝒙𝟐 , 𝒚𝟐 , 𝒛𝟐 )
Topic 4: Statistics and probability – HL only
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∫ 𝑎 𝑥 d𝑥 = ln 𝑎 𝑎 𝑥 + 𝐶
Standard
integrals
1
1
𝑥
∫ 𝑎2 +𝑥2 d𝑥 = 𝑎 arctan (𝑎) + 𝐶
1
𝑥
d𝑣
d𝑢
∫ √𝑎2 −𝑥2 d𝑥 = arcsin (𝑎) + 𝐶 , |𝑥| < 𝑎
Integration by parts
∫ 𝑢 d𝑥 𝑑𝑥 = 𝑢𝑣 − ∫ 𝑣 d𝑥 𝑑𝑥
Area enclosed by a
curve and 𝒚-axis
𝐴 = ∫𝑎 |𝑥| 𝑑𝑦
Volume of revolution
about 𝒙 or 𝒚-axes
𝑉 = ∫𝑎 𝜋𝑦 2 𝑑𝑥 or 𝑉 = ∫𝑎 𝜋𝑥 2 𝑑𝑦
Euler’s method
𝑏
𝑏
𝑏
𝑦𝑛+1 = 𝑦𝑛 + ℎ × 𝑓(𝑥𝑛 , 𝑦𝑛 ); 𝑥𝑛+1 = 𝑥𝑛 + ℎ
where ℎ is a constant (step length)
Integrating factor for
𝒚′ + 𝑷(𝒙)𝒚 = 𝑸(𝒙)
𝑒 ∫ 𝑃(𝑥)d𝑥
Maclaurin series
𝑓(𝑥) = 𝑓(0) + 𝑥𝑓 ′ (0) +
2
Maclaurin series for
special functions
𝑥 2 ′′
𝑓 (0)+ . ..
2!
2
3
∙ 𝑒 𝑥 = 1 + 𝑥 + 𝑥2! + ... ∙ ln(1 + 𝑥) = 𝑥 − 𝑥2 + 𝑥3 − ...
3
5
2
4
∙ sin 𝑥 = 𝑥 − 𝑥3! + 𝑥5! − ... ∙ cos 𝑥 = 1 − 𝑥2! + 𝑥4! − ...
3
5
∙ arctan 𝑥 = 𝑥 − 𝑥3 + 𝑥5 − ...