Calculus formulas and rules cheat sheet PDF

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Applications & Interpretation - 1 Page Formula Sheet IB Mathematics SL & HL – First examinations 2021 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 = 𝑏ℎ , 𝑏 = base, ℎ = height

distance (𝑑) =

1 𝐴 = 2 (𝑏ℎ) , 𝑏 = base, ℎ = height

Distance between 2 points (𝒙𝟏 , 𝒚𝟏 , 𝒛𝟏 ) , (𝒙𝟐 , 𝒚𝟐 , 𝒛𝟐 ) √(𝑥 − 𝑥 )2 + (𝑦 − 𝑦 )2 + (𝑧 − 𝑧 )2 1 2 1 2 1 2

𝐴 = 𝜋𝑟 2 , 𝑟 = radius

Coordinates of the midpoint with endpoints (𝒙𝟏 , 𝒚𝟏 , 𝒛𝟏 ) , (𝒙𝟐 , 𝒚𝟐 , 𝒛𝟐 )

𝐴 = 2 (𝑎 + 𝑏)ℎ , 𝑎, 𝑏 = parallel sides, ℎ = height 1

𝐶 = 2𝜋𝑟, 𝑟 = radius

𝑉 = 𝑙𝑤ℎ , 𝑙 = length, 𝑤 = width, ℎ = height 𝑉 = 𝜋𝑟2 ℎ , 𝑟 = radius, ℎ = height

𝑉 = 𝐴ℎ , 𝐴 = cross-section area, ℎ = height 𝐴 = 2𝜋𝑟ℎ , 𝑟 = radius, ℎ = height

Coordinates of midpoint

(

,

𝑥 1 +𝑥 2 𝑦1 +𝑦2 2

2

Prior Learning HL only Solutions of a quadratic equation in the form 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄

𝑥=

2𝑎

Volume: Right cone Area: Cone curve

Surface area: Sphere

), for endpoints (𝑥1 , 𝑦1 ),(𝑥2 , 𝑦2 )

−𝑏 ± √𝑏2 − 4𝑎𝑐

Volume: Right-pyramid 𝑉 = 3 𝐴ℎ , 𝐴 = base area, ℎ = height

Volume: Sphere

Distance between two )2 ( )2 ( points (𝒙𝟏 , 𝒚𝟏 ) , (𝒙𝟐 , 𝒚𝟐 ) 𝑑 = √ 𝑥1 − 𝑥2 + 𝑦1 − 𝑦2

Sine rule

Cosine rule ,𝑎≠0 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

𝑢𝑛 = 𝑢1 + (𝑛 − 1)𝑑

Length of an arc

𝑛 𝑛 (2𝑢1 + (𝑛 − 1)𝑑) = (𝑢1 + 𝑢𝑛 ) 2 2

𝑠𝑛 =

𝑢𝑛 = 𝑢1 𝑟𝑛−1

𝑢1 (𝑟𝑛 − 1) 𝑢1 (1 − 𝑟 𝑛 ) ,𝑟 ≠ 1 = 1 −𝑟 𝑟−1

𝑠𝑛 =

𝑟 ) 𝐹𝑉 = 𝑃𝑉 × (1 + 100𝑘

𝑘𝑛

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

Area of a sector

Identities

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

∆ = 𝑏 2 − 4𝑎𝑐

𝑨 = (𝑎 𝑐

𝑏 ) → det 𝑨 = |𝑨| = 𝑎𝑑 − 𝑏𝑐 𝑑 1 𝑑 𝑏 ( ) → 𝑨−1 = det 𝑨 −𝑐 𝑑

−𝑏 ) 𝑎

𝑴𝑛 = 𝑷𝑫𝑛 𝑷−1 , where 𝑷 is the matrix of eigenvectors and 𝑫 is the diagonal matrix of eigenvalues

Magnitude of a vector Vector equ. of a line Parametric form of the equation of a line

Scalar product

Topic 2: Functions – SL & HL Equations of a straight line

𝑦 = 𝑚𝑥 + 𝑐 ; 𝑎𝑥 + 𝑏𝑦 + 𝑑 = 0 ; 𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1 ) 𝑦2 − 𝑦1 𝑥2 − 𝑥1

Gradient formula

𝑚=

Axis of symmetry of a quadratic function

𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 → 𝑥 = −

Topic 2: Functions – HL only Logistic function

𝑓(𝑥) =

𝐿

1 + 𝐶𝑒 −𝑘𝑥

, 𝐿, 𝑘, 𝐶 > 0

Angle between two vectors

𝑏

2𝑎

3

𝑎

sin𝐴

=

𝑏

sin𝐵

=

𝑐

sin𝐶

1 𝑎𝑏 sin 𝐶 2

cos2 𝜃 + sin2 𝜃 = 1 sin 𝜃 tan 𝜃 = cos 𝜃 cos 2𝜃 sin2𝜃 ( ) sin 2𝜃 −cos 2𝜃 reflection in the line 𝑦 = (tan 𝜃)𝑥

1 0 ( ) 0 𝑘 vertical stretch with scale factor of 𝑘 (𝑘 0) centre (0,0) 0 𝑘 enlargement with scale factor of 𝑘

Vector product

|𝒗| = √𝑣1 + 𝑣2 + 𝑣3 2

2

𝒗 ∙ 𝒘 = 𝑣1 𝑤1 + 𝑣2 𝑤2 + 𝑣3 𝑤3 𝒗 ∙ 𝒘 = |𝒗||𝒘| cos 𝜃

𝐴 = |𝒗 × 𝒘| , where 𝒗 and 𝒘 form two adjacent sides of a parallelogram

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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

Area enclosed by a curve and the 𝒙-axis

𝐴 = ∫ 𝑦 𝑑𝑥 ,

Integral of 𝒙

The trapezoidal rule where ℎ =

𝑏−𝑎 𝑛

𝑥 𝑛+1 + 𝐶 , 𝑛 ≠ −1 𝑛+1

∫ 𝑥 𝑛 𝑑𝑥 = 𝑏

𝑏

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

𝑓(𝑥 ) = cos 𝑥 → 𝑓′(𝑥 ) = − sin 𝑥 𝑓(𝑥 ) = 𝑒

𝑥

→ 𝑓′(𝑥 ) = 𝑒

𝑥

1

cos2 𝑥

1 𝑥 𝑑𝑦 𝑑𝑦 𝑑𝑢 = × 𝑦 = 𝑔(𝑢) , 𝑢 = 𝑓 (𝑥) → 𝑑𝑥 𝑑𝑢 𝑑𝑥 𝑑𝑣 𝑑𝑢 𝑑𝑦 =𝑢 +𝑣 𝑦 = 𝑢𝑣 → 𝑑𝑥 𝑑𝑥 𝑑𝑥 𝑑𝑣 𝑑𝑢 𝑣 𝑑𝑥 − 𝑢 𝑑𝑥 𝑑𝑦 𝑢 = 𝑦= → 𝑑𝑥 𝑣2 𝑣 1 ∫ 𝑑𝑥 = ln|𝑥| + 𝐶 𝑥 𝑓(𝑥 ) = ln 𝑥 → 𝑓′(𝑥 ) =

∫ sin 𝑥 𝑑𝑥 = − cos 𝑥 + 𝐶

Standard integrals

∫ cos 𝑥 𝑑𝑥 = sin 𝑥 + 𝐶

∫

𝑣1 𝑤1 + 𝑣2 𝑤2 + 𝑣3 𝑤3 cos 𝜃 = |𝒗||𝒘|

|𝒗 × 𝒘| = |𝒗||𝒘| sin 𝜃

E(𝑎1 𝑋1 ± 𝑎2 𝑋2 ±. . . ±𝑎𝑛 𝑋𝑛 ) = 𝑎1 E(𝑋1 ) ± 𝑎2 E(𝑋2 )± . . . ±𝑎𝑛 E(𝑋𝑛 )

𝑓(𝑥 ) = 𝑥 𝑛 → 𝑓′(𝑥 ) = 𝑛𝑥 𝑛− 1

𝒏

where 𝜃 is the angle between 𝒗 and 𝒘

𝑣2 𝑤3 − 𝑣3 𝑤2 𝒗 × 𝒘 = ( 𝑣3 𝑤1 − 𝑣1 𝑤3 ) 𝑣1 𝑤2 − 𝑣2 𝑤1

𝑋~B(𝑛, 𝑝) E(𝑋) = 𝑛𝑝 ; Var(𝑋) = 𝑛𝑝(1 − 𝑝)

Derivative of 𝒙𝒏

𝑥 = 𝑥0 + 𝜆𝑙, 𝑦 = 𝑦0 + 𝜆𝑚, 𝑧 = 𝑧0 + 𝜆𝑛

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E(𝑋) = ∑ 𝑥 P(𝑋 = 𝑥)

Topic 5: Calculus - SL & HL

𝒓 = 𝒂 + 𝜆𝒃

where 𝜃 is the angle between 𝒗 and 𝒘

Area of a parallelogram

P(𝐴 ∩ 𝐵 ) = P(𝐴)P(𝐵)

Linear transformation of a E(𝑎𝑋 + 𝑏) = 𝑎E(𝑋) + 𝑏 single random variable Var(𝑎𝑋 + 𝑏) = 𝑎 2 Var(𝑋)

Unbiased estimate of population variance

𝑙 = 𝑟𝜃 𝑟 = radius, 𝜃 = angle in radians 1 𝐴 = 𝑟2 𝜃 2 𝑟 = radius, 𝜃 = angle in radians

P (𝐴 ∩ 𝐵 ) P(𝐵)

Topic 4: Statistics and probability – HL only

𝜃 = angle in degrees, 𝑟 = radius

𝜃 × 𝜋𝑟 2 360

2

P(𝐴|𝐵) =

Binomial distribution Mean ; Variance

Linear combinations of 𝒏 independent random variables, 𝑿𝟏 , 𝑿𝟐 , … . 𝑿𝒏

𝐴=

𝑛 = ∑ 𝑘𝑖=1𝑓𝑖

P(𝐴 ∪ 𝐵 ) = P(𝐴) + P(𝐵)

Conditional probability

𝜃 × 2𝜋𝑟 360 𝜃 = angle in degrees, 𝑟 = radius 𝑙=

, where

P(𝐴 ∪ 𝐵 ) = P(𝐴) + P(𝐵) − P(𝐴 ∩ 𝐵)

Mutually exclusive events

Expected value of a discrete random variable X

𝑐 2 = 𝑎 2 + 𝑏2 − 2𝑎𝑏 cos 𝐶 𝑎2 + 𝑏 2 − 𝑐 2 cos 𝐶 = 2𝑎𝑏 𝐴=

P(𝐴) + P(𝐴′ ) = 1

Combined events

Independent events

𝐴 = 4𝜋𝑟 2 , 𝑟 = radius

( cos 𝜃 sin𝜃) , clockwise rotation −sin 𝜃 cos 𝜃 of angle 𝜃 about the origin (𝜃 > 0)

𝑧 = 𝑟 (cos 𝜃 + 𝑖 sin 𝜃 ) = 𝑟𝑒 𝑖𝜃 = 𝑟cis𝜃 𝑎 𝑐

3

(cos 𝜃 −sin 𝜃) , anticlockwise rotation sin 𝜃 cos 𝜃 of angle 𝜃 about the origin (𝜃 > 0)

𝑧 = 𝑎 + 𝑏𝑖

𝑨=(

𝑉 = 𝜋𝑟 , 𝑟 = radius 4

𝑘 0 ( ) 0 1 horizontal stretch by scale factor of 𝑘

Transformation matrices

𝑢1 , |𝑟| < 1 1 −𝑟

𝐴 = 𝜋𝑟𝑙 , 𝑟= radius, 𝑙 = slant height

Complementary events

𝑛

𝑛(𝐴) P(𝐴) = 𝑛(𝑢)

Probability of an event 𝑨

Transition matrices

log𝑎

𝑠∞ =

1

3

𝑥 =

Length of an arc

Topic 1: Number and algebra - HL only

The sum of an infinite geometric sequence

𝑉 = 𝜋𝑟 2 ℎ , 𝑟= radius, ℎ = height

∑ 𝑘𝑖=1 𝑓𝑖 𝑥𝑖

Mean, 𝒙 , of a set of data

Topic 3: Geometry and trigonometry – HL only

𝑣𝐴 = approximate value, 𝑣𝐸 = exact value log𝑎 𝑥𝑦 = log 𝑎 𝑥 + log𝑎 𝑦 𝑥 = log𝑎 𝑥 − log𝑎 𝑦 𝑦 log𝑎 𝑥 𝑚 = 𝑚 log 𝑎 𝑥 for 𝑎, 𝑥, 𝑦 > 0

1

IQR = 𝑄3 − 𝑄1

Interquartile range

Poisson distribution Mean ; Variance

Exponents & logarithms 𝑎𝑥 = 𝑏 ↔ 𝑥 = log 𝑎 𝑏 , 𝑎, 𝑏 > 0, 𝑎 ≠ 1 𝑣𝐴 − 𝑣𝐸 | × 100% 𝜀=| 𝑣𝐸 Percentage error

Laws of logarithms

𝑥1 + 𝑥2 𝑦1 + 𝑦2 𝑧1 + 𝑧2 ) ( , , 2 2 2

Topic 4: Statistics and probability - SL & HL

1 𝑑𝑥 = tan 𝑥 + 𝐶 cos2 𝑥

∫ 𝑒 𝑥 𝑑𝑥 = 𝑒 𝑥 + 𝐶

Area enclosed by a curve and 𝒙 or 𝒚-axes Volume of revolution about 𝒙 or 𝒚-axes Acceleration Distance; Displacement travelled from 𝒕𝟏 to 𝒕𝟐

Euler’s method

Euler’s method for coupled systems Exact solution for coupled linear differential equations

𝑏

𝑏

𝐴 = ∫ |𝑦| 𝑑𝑥 or 𝐴 = ∫ |𝑥| 𝑑𝑦 𝑎

𝑎

𝑎

𝑎

𝑏

𝑏

𝑉 = ∫ 𝜋𝑦 2 𝑑𝑥 or 𝑉 = ∫ 𝜋𝑥 2 𝑑𝑦 𝑎=

d𝑣 d𝑣 d2 𝑠 =𝑣 = d𝑠 d𝑡 d𝑡 2

dist = ∫𝑡 2|𝑣(𝑡)| 𝑑𝑡 ; disp = ∫𝑡 2𝑣(𝑡) 𝑑𝑡 𝑡

1

𝑡

1

𝑦𝑛+1 = 𝑦𝑛 + ℎ × 𝑓(𝑥𝑛 , 𝑦𝑛 ); 𝑥𝑛+1 = 𝑥𝑛 + ℎ where ℎ is a constant (step length)

𝑥𝑛+1 = 𝑥 𝑛 + 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...