Mathematics: Applications & Interpretation SL & HL
1 Page Formula Sheet – First Examinations 2021 – Updated Version 1.1
Topic 3: Geometry and trigonometry – SL & HL
Prior Learning SL & HL
Area: Parallelogram
Area: Triangle
Area: Trapezoid
Area: Circle
Circumference: Circle
Volume: Cuboid
𝐴 = 𝑏ℎ , 𝑏 = base, ℎ = height
1
𝐴 = 2 (𝑏ℎ) , 𝑏 = base, ℎ = height
1
𝐴 = 2 (𝑎 + 𝑏)ℎ , 𝑎, 𝑏 = parallel sides, ℎ = height
𝐴 = 𝜋𝑟 2 , 𝑟 = radius
𝐶 = 2𝜋𝑟, 𝑟 = radius
𝑉 = 𝑙𝑤ℎ , 𝑙 = length, 𝑤 = width, ℎ = height
Distance between 2 points
𝑑 = √(𝑥1 − 𝑥2 )2 + (𝑦1 − 𝑦2 )2 + (𝑧1 − 𝑧2 )2
(𝒙𝟏 , 𝒚𝟏 , 𝒛𝟏 ) , (𝒙𝟐 , 𝒚𝟐 , 𝒛𝟐 )
Coordinates of midpoint
of a line with endpoints
(𝒙𝟏 , 𝒚𝟏 , 𝒛𝟏 ) , (𝒙𝟐 , 𝒚𝟐 , 𝒛𝟐 )
Volume: Right cone
Area: Cone curve
Distance between two
𝑑 = √(𝑥1 − 𝑥2 )2 + (𝑦1 − 𝑦2 )2
points (𝒙𝟏 , 𝒚𝟏 ) , (𝒙𝟐 , 𝒚𝟐 )
𝐴 = 2𝜋𝑟ℎ , 𝑟 = radius, ℎ = height
Surface area: Sphere
Prior Learning HL only
Cosine rule
Volume: Prism
Area: Cylinder curve
Coordinates of midpoint
Solutions of a quadratic
equation in the form
𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄 = 𝟎
𝑉 = 𝐴ℎ , 𝐴 = cross-section area, ℎ = height
(
𝑥1 +𝑥2 𝑦1 +𝑦2
2
,
2
−𝑏 ± √𝑏2 − 4𝑎𝑐
𝑥=
,𝑎 ≠ 0
2𝑎
Area: Triangle
Topic 1: Number and algebra - SL & HL
The 𝒏th term of an
arithmetic sequence
Sum of 𝒏 terms of an
arithmetic sequence
The 𝒏th term of a
geometric sequence
Sum of 𝒏 terms of a
finite geometric seq.
Compound interest
Length of an arc
𝑢𝑛 = 𝑢1 + (𝑛 − 1)𝑑
𝑠𝑛 =
𝑛
𝑛
(2𝑢1 + (𝑛 − 1)𝑑) = (𝑢1 + 𝑢𝑛 )
2
2
𝑢𝑛 = 𝑢1 𝑟 𝑛−1
𝑠𝑛 =
Volume: Sphere
Sine rule
), for endpoints (𝑥1 , 𝑦1), (𝑥2 , 𝑦2 )
𝑢1 (𝑟 𝑛 − 1) 𝑢1 (1 − 𝑟 𝑛 )
=
,𝑟 ≠ 1
𝑟−1
1−𝑟
𝑟 𝑘𝑛
𝐹𝑉 = 𝑃𝑉 × (1 +
)
100𝑘
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
Area of a sector
Length of an arc
Area of a sector
Identities
for 𝑎, 𝑥, 𝑦 > 0
The sum of an infinite
geometric sequence
Complex numbers
Discriminant
Modulus-argument
(polar) & Exponential
(Euler) form
Determinant of a
2×2 matrix
Inverse of a
2×2 matrix
Power formula for a
matrix
𝑠∞ =
Gradient formula
Axis of symmetry of a
quadratic function
𝑏
) ⇒ det 𝑨 = |𝑨| = 𝑎𝑑 − 𝑏𝑐
𝑑
1
𝑏
𝑑
(
) ⇒ 𝑨−1 =
𝑑
det 𝑨 −𝑐
−𝑏
)
𝑎
𝑴𝑛 = 𝑷𝑫𝑛 𝑷−1 , where 𝑷 is the matrix of
eigenvectors and 𝑫 is the diagonal matrix
of eigenvalues
𝑦 = 𝑚𝑥 + 𝑐 ; 𝑎𝑥 + 𝑏𝑦 + 𝑑 = 0 ;
𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1 )
𝑦2 − 𝑦1
𝑚=
𝑥2 − 𝑥1
𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 ⇒ 𝑥 = −
Topic 2: Functions – HL only
Logistic function
𝑓(𝑥) =
𝐿
1 + 𝐶𝑒 −𝑘𝑥
𝑎
sin𝐴
=
𝑏
sin𝐵
=
sin𝐶
𝑐 2 = 𝑎2 + 𝑏2 − 2𝑎𝑏 cos 𝐶
cos 𝐶 =
𝑎2 + 𝑏2 − 𝑐 2
2𝑎𝑏
1
𝐴 = 𝑎𝑏 sin 𝐶
2
𝜃
× 2𝜋𝑟
360
𝜃 = angle in degrees, 𝑟 = radius
𝑙=
𝜃
𝐴=
× 𝜋𝑟 2
360
𝜃 = angle in degrees, 𝑟 = radius
𝑙 = 𝑟𝜃
𝑟 = radius, 𝜃 = angle in radians
1
𝐴 = 𝑟2 𝜃
2
𝑟 = radius, 𝜃 = angle in radians
cos 2 𝜃 + sin2 𝜃 = 1
sin 𝜃
tan 𝜃 =
cos 𝜃
1 0
)
0 𝑘
˒ vertical stretch with scale factor of 𝑘
(
𝑘 0
) , centre (0,0)
0 𝑘
˒ enlargement with scale factor of 𝑘
(
𝑏
2𝑎
, 𝐿, 𝑘, 𝐶 > 0
cos 𝜃 sin 𝜃
) , clockwise rotation
−sin 𝜃 cos 𝜃
of angle 𝜃 about the origin (𝜃 > 0)
(
Magnitude of a vector
Vector equ. of a line
Parametric form of the
equation of a line
Scalar product
Angle between two
vectors
Vector product
Area of a
parallelogram
|𝒗| = √𝑣1 + 𝑣2 + 𝑣3
2
2
2
Complementary events
P(𝐴) + P(𝐴′ ) = 1
Combined events
Mutually exclusive events
Conditional probability
Binomial distribution
Mean ; Variance
𝒗 ∙ 𝒘 = |𝒗||𝒘| cos 𝜃
𝑣1 𝑤1 + 𝑣2 𝑤2 + 𝑣3 𝑤3
|𝒗||𝒘|
Linear combinations of 𝒏
independent random
variables, 𝑿𝟏 , 𝑿𝟐 , … . 𝑿𝒏
Unbiased estimate of
population variance
Poisson distribution
Mean ; Variance
Transition matrices
P(𝐴 ∩ 𝐵)
P(𝐵)
P(𝐴 ∩ 𝐵) = P(𝐴)P(𝐵)
E(𝑋) = ∑ 𝑥 P(𝑋 = 𝑥)
𝑋~B(𝑛, 𝑝)
E(𝑋) = 𝑛𝑝 ; Var(𝑋) = 𝑛𝑝(1 − 𝑝)
E(𝑎1 𝑋1 ± 𝑎2 𝑋2 ±. . . ±𝑎𝑛 𝑋𝑛 ) =
𝑎1 E(𝑋1 ) ± 𝑎2 E(𝑋2 )± . . . ±𝑎𝑛 E(𝑋𝑛 )
Var(𝑎1 𝑋1 ± 𝑎2 𝑋2 ±. . . ±𝑎𝑛 𝑋𝑛 ) =
𝑎1 2 Var(𝑋1 ) + 𝑎2 2 Var(𝑋2 ) + ⋯ +
𝑎𝑛 2 Var(𝑋𝑛 )
2
𝑠𝑛−1
=
𝑛
𝑠2
𝑛−1 𝑛
Sample statistics
𝑋~Po(𝑚)
E(𝑋) = 𝑚 ; Var(𝑋) = 𝑚
𝑻𝑛 𝒔0 = 𝒔𝑛 , where 𝒔0 is the initial state
𝑓(𝑥) = 𝑥 𝑛 ⇒ 𝑓′(𝑥) = 𝑛𝑥 𝑛−1
Area enclosed by a
curve and the 𝒙-axis
𝐴 = ∫ 𝑦 𝑑𝑥 ,
Integral of 𝒙𝒏
The trapezoidal rule
where ℎ =
𝑏−𝑎
𝑛
∫ 𝑥 𝑛 𝑑𝑥 =
𝑏
𝑏
𝑥 𝑛+1
+ 𝐶 , 𝑛 ≠ −1
𝑛+1
𝑎
∫ 𝑦 𝑑𝑥 ≈
where 𝑓(𝑥) > 0
𝑎
1
ℎ((𝑦0 + 𝑦𝑛 ) + 2(𝑦1 + 𝑦2 +. . . +𝑦𝑛−1 ))
2
Topic 5: Calculus – HL only
Derivative of 𝐬𝐢𝐧 𝒙
𝑓(𝑥) = sin 𝑥 ⇒ 𝑓′(𝑥) = cos 𝑥
Derivative of 𝐭𝐚𝐧 𝒙
𝑓(𝑥) = tan 𝑥 ⇒ 𝑓′(𝑥) =
Derivative of 𝐜𝐨𝐬 𝒙
Derivative of 𝒆𝒙
Derivative of 𝐥𝐧 𝒙
Chain rule
Product rule
Quotient rule
Standard integrals
𝑓(𝑥) = cos 𝑥 ⇒ 𝑓′(𝑥) = − sin 𝑥
𝑓(𝑥) = 𝑒 𝑥 ⇒ 𝑓′(𝑥) = 𝑒 𝑥
Volume of revolution
about 𝒙 or 𝒚-axes
Distance; Displacement
travelled from 𝒕𝟏 to 𝒕𝟐
Euler’s method
Euler’s method for
coupled systems
Exact solution for
coupled linear
differential equations
1
cos 2 𝑥
1
𝑥
𝑑𝑦
𝑑𝑦 𝑑𝑢
𝑦 = 𝑔(𝑢) , 𝑢 = 𝑓(𝑥) ⇒
=
×
𝑑𝑥
𝑑𝑢 𝑑𝑥
𝑑𝑦
𝑑𝑣
𝑑𝑢
𝑦 = 𝑢𝑣 ⇒
=𝑢
+𝑣
𝑑𝑥
𝑑𝑥
𝑑𝑥
𝑑𝑣
𝑑𝑢
𝑢
𝑑𝑦 𝑣 𝑑𝑥 − 𝑢 𝑑𝑥
𝑦=
⇒
=
𝑣2
𝑣
𝑑𝑥
1
∫ 𝑑𝑥 = ln|𝑥| + 𝐶
𝑥
𝑓(𝑥) = ln 𝑥 ⇒ 𝑓′(𝑥) =
∫ sin 𝑥 𝑑𝑥 = − cos 𝑥 + 𝐶
∫ cos 𝑥 𝑑𝑥 = sin 𝑥 + 𝐶
∫
Acceleration
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P(𝐴|𝐵) =
Derivative of 𝒙𝒏
where 𝜃 is the angle between 𝒗 and 𝒘
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P(𝐴 ∪ 𝐵) = P(𝐴) + P(𝐵)
Topic 5: Calculus - SL & HL
Area enclosed by a
curve and 𝒙 or 𝒚-axes
𝐴 = |𝒗 × 𝒘| , where 𝒗 and 𝒘 form two
adjacent sides of a parallelogram
P(𝐴 ∪ 𝐵) = P(𝐴) + P(𝐵) − P(𝐴 ∩ 𝐵)
Linear transformation of a E(𝑎𝑋 + 𝑏) = 𝑎E(𝑋) + 𝑏
single random variable
Var(𝑎𝑋 + 𝑏) = 𝑎2 Var(𝑋)
𝑣2 𝑤3 − 𝑣3 𝑤2
𝒗 × 𝒘 = (𝑣3 𝑤1 − 𝑣1 𝑤3 )
𝑣1 𝑤2 − 𝑣2 𝑤1
|𝒗 × 𝒘| = |𝒗||𝒘| sin 𝜃
= ∑𝑘𝑖=1 𝑓𝑖
, where 𝑛
Topic 4: Statistics and probability – HL only
where 𝜃 is the angle between 𝒗 and 𝒘
cos 𝜃 =
𝑛
𝑛(𝐴)
P(𝐴) =
𝑛(𝑢)
Probability of an event 𝑨
𝑥 = 𝑥0 + 𝜆𝑙, 𝑦 = 𝑦0 + 𝜆𝑚, 𝑧 = 𝑧0 + 𝜆𝑛
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𝑥̅ =
𝒓 = 𝒂 + 𝜆𝒃
𝒗 ∙ 𝒘 = 𝑣1 𝑤1 + 𝑣2 𝑤2 + 𝑣3 𝑤3
∑𝑘
𝑖=1 𝑓𝑖 𝑥𝑖
̅ , of a set of data
Mean, 𝒙
Expected value of a
discrete random variable X
𝑐
IQR = 𝑄3 − 𝑄1
Interquartile range
Independent events
𝐴 = 4𝜋𝑟 2 , 𝑟 = radius
cos 𝜃 − sin 𝜃
) , anticlockwise rotation
sin 𝜃 cos 𝜃
of angle 𝜃 about the origin (𝜃 > 0)
𝑧 = 𝑟(cos 𝜃 + 𝑖 sin 𝜃) = 𝑟𝑒 𝑖𝜃 = 𝑟cis𝜃
Topic 2: Functions – SL & HL
Equations of a
straight line
4
𝑉 = 3 𝜋𝑟 3 , 𝑟 = radius
(
∆ = 𝑏2 − 4𝑎𝑐
𝑎
𝑨=(
𝑐
𝐴 = 𝜋𝑟𝑙 , 𝑟= radius, 𝑙 = slant height
𝑘 0
)
0 1
˒ horizontal stretch by scale factor of 𝑘
𝑧 = 𝑎 + 𝑏𝑖
𝑎
𝑐
𝑉 = 3 𝜋𝑟 ℎ , 𝑟= radius, ℎ = height
(
𝑢1
, |𝑟| < 1
1−𝑟
𝑨=(
2
cos 2𝜃 sin 2𝜃
)
sin 2𝜃 −cos 2𝜃
˒ reflection in the line 𝑦 = (tan 𝜃)𝑥
Transformation
matrices
log 𝑎 𝑥 𝑚 = 𝑚 log 𝑎 𝑥
1
(
Topic 1: Number and algebra - HL only
log 𝑎 𝑥𝑦 = log 𝑎 𝑥 + log 𝑎 𝑦
𝑥
log 𝑎 = log 𝑎 𝑥 − log 𝑎 𝑦
𝑦
1
Topic 3: Geometry and trigonometry – HL only
Exponents & logarithms 𝑎 𝑥 = 𝑏 ⇔ 𝑥 = log 𝑎 𝑏 , 𝑎, 𝑏 > 0, 𝑎 ≠ 1
𝑣𝐴 − 𝑣𝐸
𝜀=|
| × 100%
𝑣𝐸
Percentage error
𝑣𝐴 = approximate value, 𝑣𝐸 = exact value
Laws of logarithms
𝑥1 + 𝑥2 𝑦1 + 𝑦2 𝑧1 + 𝑧2
,
,
)
2
2
2
Volume: Right-pyramid 𝑉 = 3 𝐴ℎ , 𝐴 = base area, ℎ = height
𝑉 = 𝜋𝑟 2 ℎ , 𝑟 = radius, ℎ = height
Volume: Cylinder
(
Topic 4: Statistics and probability - SL & HL
1
𝑑𝑥 = tan 𝑥 + 𝐶
cos 2 𝑥
∫ 𝑒 𝑥 𝑑𝑥 = 𝑒 𝑥 + 𝐶
𝑏
𝑏
𝐴 = ∫ |𝑦| 𝑑𝑥 or 𝐴 = ∫ |𝑥| 𝑑𝑦
𝑎
𝑏
𝑎
𝑏
𝑉 = ∫ 𝜋𝑦 2 𝑑𝑥 or 𝑉 = ∫ 𝜋𝑥 2 𝑑𝑦
𝑎=
𝑎
d𝑣 d2 𝑠
d𝑣
=
=𝑣
d𝑡 d𝑡 2
d𝑠
𝑎
𝑡
𝑡
dist = ∫𝑡 2|𝑣(𝑡)| 𝑑𝑡 ; disp = ∫𝑡 2 𝑣(𝑡) 𝑑𝑡
1
1
𝑦𝑛+1 = 𝑦𝑛 + ℎ × 𝑓(𝑥𝑛 , 𝑦𝑛 ); 𝑥𝑛+1 = 𝑥𝑛 + ℎ
where ℎ is a constant (step length)
𝑥𝑛+1 = 𝑥𝑛 + ℎ × 𝑓1 (𝑥𝑛 , 𝑦𝑛 , 𝑡𝑛 )
𝑦𝑛+1 = 𝑦𝑛 + ℎ × 𝑓2 (𝑥𝑛 , 𝑦𝑛 , 𝑡𝑛 )
𝑡𝑛+1 = 𝑡𝑛 + ℎ
where ℎ is a constant (step length)
𝒙 = 𝐴𝑒 𝜆1 𝑡 𝒑1 + 𝐵𝑒 𝜆2𝑡 𝒑2