# Analysis-and-Approaches-1-Page-Formula-Sheet-V1.3

```Mathematics: Analysis &amp; Approaches SL &amp; HL
1 Page Formula Sheet – First Examinations 2021 – Updated Version 1.3
Topic 3: Geometry and trigonometry – SL &amp; HL
Prior Learning SL &amp; HL
Area: Parallelogram
Area: Triangle
Area: Trapezoid
Area: Circle
Circumference: Circle
Volume: Cuboid
Volume: Cylinder
Volume: Prism
Area: Cylinder curve
π΄ = πβ , π = base, β = height
1
π΄ = 2 (πβ) , π = base, β = height
1
π΄ = 2 (π + π)β , π, π = parallel sides, β = height
π΄ = ππ 2 , π = radius
πΆ = 2ππ, π = radius
π = ππ€β , π = length, π€ = width, β = height
π = ππ 2 β , π = radius, β = height
π = π΄β , π΄ = cross-section area, β = height
π΄ = 2ππβ , π = radius, β = height
Sum of π terms of an
arithmetic sequence
The πth term of a
geometric sequence
Sum of π terms of a
finite geometric seq.
Compound interest
Area: Cone curve
Volume: Sphere
Surface area: Sphere
Sine rule
Complex numbers
Modulus-argument (polar)
&amp; Exponential (Euler) form
De Moivre’s theorem
Axis of symmetry of a
equation in the form
πππ + ππ + π = π
Discriminant
Exponential and
logarithmic functions
3
2
π΄ = 4ππ , π = radius
π
=
sinπ΄
π
sinπ΅
=
Area: Triangle
π’π = π’1 + (π − 1)π
π’π = π’1 π π−1
πΉπ is future value, ππ is present value, π is
the number of years, π is the number of
compounding periods per year, π% is the
nominal annual rate of interest
Pythagorean identity
log π π₯π¦ = log π π₯ + log π π¦
π₯
log π = log π π₯ − log π π¦
π¦
log π π₯ π = π log π π₯
logπ π₯
log π π₯ =
log π π
sinπΆ
Reciprocal trigonometric
identities
Pythagorean identities
Scalar product
ππ + (π1)ππ−1 π+. . . +(ππ)ππ−π ππ +. . . + ππ
π!
π
( ) = nC r =
π
π!(π−π)!
nC
r=
π!
π!(π−π)!
(π + π
)π
;
nP
=
r
π!
= (π−π)!
π(π − 1) π 2
π
( ) +. . . )
ππ (1 + π ( ) +
2!
π
π
π§ = π + ππ
π§ = π(cos π + π sin π) = ππ ππ = πcisπ
[π(cos π + π sin π )]π =
π(
π cos ππ + π sin ππ) = π π π πππ = π π cisππ
π¦ = ππ₯ + π ; ππ₯ + ππ¦ + π = 0 ;
π¦ − π¦1 = π(π₯ − π₯1 )
π¦2 − π¦1
π=
π₯2 − π₯1
π
π(π₯ ) = ππ₯ 2 + ππ₯ + π ⇒ π₯ = −
2π
π₯=
−π &plusmn; √π2 − 4ππ
,π ≠ 0
2π
β = π2 − 4ππ
π π₯ = π π₯ ln π ; log π π π₯ = π₯ = π logπ π₯
where π, π₯ &gt; 0 , π ≠ 1
π
∑ ππ π₯ π = 0
π=0
⇒ Sum is
(−1)π π0
−ππ−1
; product is
ππ
ππ
π΄ = π 2 π , π = radius, π = angle in radians
2
2
sin π
cos π
2
cos π + sin π = 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
sec π =
;
cos π
1
cosec π =
sin π
1 + tan2 π = sec 2 π ; 1 + cot 2 π = cosec 2 π
sin(π΄ &plusmn; π΅) = sin π΄ cos π΅ &plusmn; cos π΄ sin π΅
cos(π΄ &plusmn; π΅) = cos π΄ cos π΅ β sin π΄ sin π΅
tan π΄ &plusmn; tan π΅
tan(π΄ &plusmn; π΅) =
1 β tan π΄ tan π΅
tan 2π =
2 tan π
1 − tan2 π
|π| = √π£1 2 + π£2 2 + π£3 2
Variance ππ
π2 =
Linear transformation of
a single random variable
E(ππ + π) = πE(π) + π
Var(ππ + π) = π2 Var(π)
Variance of a discrete
random variable X
Variance of a continuous
random variable X
Derivative of ππ
Area between curve
π = π(π) &amp; π-axis
Derivative of π¬π’π§ π
Derivative of ππ¨π¬ π
Derivative of ππ
Derivative of π₯π§ π
Chain rule
Product rule
Quotient rule
Standard integrals
Acceleration
π₯ − π₯0 π¦ − π¦0 π§ − π§0
=
=
π
π
π
π΄ = |π &times; π| , where π and π form two
π = π + ππ + ππ
Μ , of a set of
Mean, π
data
Area enclosed by a
curve and π-axis
π₯Μ =
π
, where π
π(π΄)
π(π’)
Complementary events P(π΄) + P(π΄′ ) = 1
Probability of an event A
Combined events
Mutually exclusive
events
Conditional probability
Independent events
Expected value: Discrete
random variable X
Binomial distribution
Mean ; Variance
Standardized normal
variable
P(π΄) =
= ∑ππ=1 ππ
Derivative of π(π)
from first principles
Standard
derivatives
P(π΄ ∪ π΅) = P(π΄) + P(π΅) − P(π΄ ∩ π΅)
P(π΄ ∪ π΅) = P(π΄) + P(π΅)
2
∑π
π=1 ππ π₯π
π
− π2
π
∞
E(π) = π = ∫−∞ π₯π(π₯ )dπ₯
Var(π) = ∑(π₯ − π)2 P(π = π₯ )
= ∑ π₯ 2 P (π = π₯ ) − π 2
∞
Var(π) = ∫−∞(π₯ − π)2 π(π₯ )dπ₯
∞
= ∫−∞ π₯ 2 π(π₯ )dπ₯ − π2
π(π₯ ) = π₯ π ⇒ π′(π₯ ) = ππ₯ π−1
π₯ π+1
∫ π₯ π ππ₯ =
+ πΆ , π ≠ −1
π+1
π
where π (π₯ ) &gt; 0
π΄ = ∫ π¦ ππ₯ ,
π
π(π₯ ) = sin π₯ ⇒ π′(π₯ ) = cos π₯
π(π₯ ) = cos π₯ ⇒ π′(π₯ ) = − sin π₯
π(π₯ ) = π π₯ ⇒ π′(π₯ ) = π π₯
1
π(π₯ ) = ln π₯ ⇒ π′(π₯ ) =
π₯
ππ¦
ππ¦ ππ’
π¦ = π (π’ ) , π’ = π (π₯ ) ⇒
=
&times;
ππ₯
ππ’ ππ₯
ππ¦
ππ£
ππ’
π¦ = π’π£ ⇒
=π’
+π£
ππ₯
ππ₯
ππ₯
ππ£
ππ’
π’
ππ¦ π£ ππ₯ − π’ ππ₯
π¦=
⇒
=
π£2
π£
ππ₯
2
dπ£ d π
π=
=
dπ‘ dπ‘ 2
π‘2
π‘2
dist = ∫ |π£(π‘)| ππ‘ ;
disp = ∫ π£(π‘) ππ‘
π‘1
π‘1
1
∫ ππ₯ = ln|π₯| + πΆ
π₯
∫ sin π₯ ππ₯ = − cos π₯ + πΆ
∫ π π₯ ππ₯ = π π₯ + πΆ
π
π΄ = ∫ |π¦| ππ₯
π
Topic 5: Calculus – HL only
ππ₯ + ππ¦ + ππ§ = π
∑π
π=1 ππ π₯π
2
∑π
π=1 ππ (π₯π −π)
=
∫ cos π₯ ππ₯ = sin π₯ + πΆ
π β π = π β π (using the normal vector)
IQR = π3 − π1
π=√
π
Topic 5: Calculus - SL &amp; HL
π₯ = π₯0 + ππ, π¦ = π¦0 + ππ, π§ = π§0 + ππ
π£2 π€3 − π£3 π€2
π &times; π = (π£3 π€1 − π£1 π€3 )
π£1 π€2 − π£2 π€1
|π &times; π| = |π||π| sin π
where π is the angle between π and π
2
∑π
π=1 ππ (π₯π −π)
Var(π) = E[(π − π)2 ] = E(π 2 ) − [E(π)]2
Variance
Topic 4: Statistics and probability - SL &amp; HL
Interquartile range
P(π΅π )P(π΄|π΅π )
P(π΅1 )P(π΄|π΅1 ) + P(π΅2 )P(π΄|π΅2 ) + P(π΅3 )P(π΄|π΅3 )
Standard Deviation π
Distance; Displacement
travelled from ππ to ππ
dπ¦
π(π₯ + β) − π(π₯)
= π′(π₯) = lim (
)
β→0
dπ₯
β
π(π₯ ) = tan π₯ ⇒ π ′ (π₯ ) = sec 2 π₯
π(π₯ ) = sec π₯ ⇒ π ′ (π₯ ) = sec π₯ tan π₯
π (π₯ ) = cosec π₯ ⇒ π ′ (π₯ ) = −cosec π₯ cot π₯
π(π₯ ) = cot π₯ ⇒ π ′ (π₯ ) = −cosec 2 π₯
π(π₯ ) = π π₯ ⇒ π ′ (π₯ ) = π π₯ (ln π)
1
π(π₯ ) = logπ π₯ ⇒ π ′ (π₯ ) =
π₯ ln π
1
π (π₯ ) = arcsin π₯ ⇒ π ′ (π₯ ) =
√1 − π₯ 2
1
π(π₯ ) = arccos π₯ ⇒ π ′ (π₯ ) = −
√1 − π₯ 2
1
π (π₯ ) = arctan π₯ ⇒ π ′ (π₯ ) =
1 + π₯2
1
∫ π π₯ dπ₯ = ln π π π₯ + πΆ
1
1
π₯
P (π΄ ∩ π΅ )
P(π΄|π΅) =
P(π΅)
P(π΄ ∩ π΅) = P(π΄)P(π΅)
Standard
integrals
Integration by parts
∫ √π2 −π₯2 dπ₯ = arcsin (π) + πΆ , |π₯| &lt; π
π~B(π, π)
E(π) = ππ ; Var(π) = ππ(1 − π)
π₯−π
π§=
π
Area enclosed by a
curve and π-axis
π΄ = ∫π |π₯| ππ¦
E(π) = ∑ π₯ P(π = π₯)
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P(π΅π |π΄) =
π β π = π£1 π€1 + π£2 π€2 + π£3 π€3
π β π = |π||π| cos π
where π is the angle between π and π
π£1 π€1 + π£2 π€2 + π£3 π€3
cos π =
|π||π|
π = π + ππ
P(π΅)P(π΄|π΅)
P(π΅)P(π΄|π΅ ) + P(π΅′ )P(π΄|π΅′ )
Bayes’ theorem
Integral of ππ
sin 2π = 2 sin π cos π
cos 2π = cos 2 π − sin2 π
= 2 cos 2 π − 1
= 1 − 2 sin2 π
Topic 3: Geometry and trigonometry – HL only
Magnitude of a vector
π’1
π ∞ =
, |π| &lt; 1
1−π
1
P(π΅|π΄) =
Expected value: Continuous
random variable X
π
Double angle identities
Cosine rule
Length of an arc
Topic 2: Functions – HL only
Sum &amp; product of the
roots of polynomial
equations of the form
π = ππ , π = radius
π’1 (π π − 1) π’1 (1 − π π )
π π =
=
,π ≠ 1
π−1
1−π
π ππ
πΉπ = ππ &times; (1 +
)
100π
2
Topic 2: Functions – SL &amp; HL
Equations of a
straight line
3
tan π =
2
), for endpoints (π₯1 , π¦1), (π₯2 , π¦2 )
Topic 1: Number and algebra - HL only
Extension of Binomial
Theorem, π ∈ β
4
Identity for π­ππ§ π½
,
Double angle identity
for tan
Combinations;
Permutations
π΄ = πππ , π= radius, π = slant height
π
π
π π = (2π’1 + (π − 1)π ) = (π’1 + π’π )
2
2
π₯1 +π₯2 π¦1 +π¦2
Exponents &amp; logarithms
Binomial coefficient
1
π = 3 ππ 2 β , π= radius, β = height
Area of a sector
(
Compound angle
identities
Binomial Theorem
for π ∈ β, (π + π)π =
1
Volume: Right-pyramid π = 3 π΄β , π΄ = base area, β = height
Exponents &amp; logarithms π π₯ = π ⇔ π₯ = log π π , π, π &gt; 0, π ≠ 1
The sum of an infinite
geometric sequence
π₯1 + π₯2 π¦1 + π¦2 π§1 + π§2
(
,
,
)
2
2
2
π 2 = π2 + π2 − 2ππ cos πΆ
π2 + π2 − π 2
cos πΆ =
2ππ
1
π΄ = ππ sin πΆ
2
π = ππ , π = radius, π = angle in radians
Topic 1: Number and algebra - SL &amp; HL
The πth term of an
arithmetic sequence
Coordinates of midpoint of
a line with endpoints
(ππ , ππ , ππ ) , (ππ , ππ , ππ )
Volume: Right cone
Distance between two
π = √(π₯1 − π₯2 )2 + (π¦1 − π¦2 )2
points (ππ , ππ ) , (ππ , ππ )
Coordinates of midpoint
Distance between 2 points
π = √(π₯1 − π₯2 )2 + (π¦1 − π¦2 )2 + (π§1 − π§2 )2
(ππ , ππ , ππ ) , (ππ , ππ , ππ )
Topic 4: Statistics and probability – HL only
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Volume of revolution
Euler’s method
Integrating factor for
π′ + π·(π)π = πΈ(π)
Maclaurin series
∫ π2 +π₯2 dπ₯ = π arctan (π) + πΆ
1
dπ£
π₯
dπ’
∫ π’ dπ₯ ππ₯ = π’π£ − ∫ π£ dπ₯ ππ₯
π
π
π
π = ∫π ππ¦ 2 ππ₯ or π = ∫π ππ₯ 2 ππ¦
π¦π+1 = π¦π + β &times; π(π₯π , π¦π ); π₯π+1 = π₯π + β
where β is a constant (step length)
π ∫ π(π₯)dπ₯
π(π₯ ) = π(0) + π₯π ′ (0) +
2
Maclaurin series for
special functions
π₯ 2 ′′
π (0)+ . ..
2!
2
3
β π π₯ = 1 + π₯ + π₯2! + ... β ln(1 + π₯) = π₯ − π₯2 + π₯3 − ...
3
5
2
4
β sin π₯ = π₯ − π₯3! + π₯5! − ... β cos π₯ = 1 − π₯2! + π₯4! − ...
β arctan π₯ = π₯ −
π₯3
3
+
π₯5
5
− ...
```