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26 cardsMaturità / Esame di Stato (Italy)

Maturità Mathematics Key Formulas

26 flashcards with the main formulas and rules of calculus, algebra, geometry, probability and sequences. Study material written for practice, not the official exam.

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Derivative of

x^{n}

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Derivative of

x^{n}

n x^{n - 1}

(the power rule).

Product rule

(uv)^{'} = u^{'} v + u v^{'}

Quotient rule

(\frac{u}{v})^{'} = \frac{u ^{'} v - u v ^{'}}{v ^{2}}

for

v \neq = 0

Chain rule

(f (g (x)))^{'} = f^{'} (g (x)) \cdot g^{'} (x)

Derivative of

e^{x}

and

ln x

(e^{x})^{'} = e^{x}

and

(ln x)^{'} = \frac{1}{x}

for

x > 0

Derivatives of

sin x

and

cos x

(sin x)^{'} = cos x

and

(cos x)^{'} = - sin x

Discriminant of

a x^{2} + b x + c

Δ = b^{2} - 4 a c

: two real roots if

Δ > 0

, one if

Δ = 0

, none if

Δ < 0

Quadratic formula

x = \frac{- b \pm b ^{2} - 4 a c}{2 a}

Fundamental theorem of calculus

\int_{a}^{b} f (x) d x = F (b) - F (a)

, where

F^{'} = f

Integration by parts

\int u d v = uv - \int v d u

Standard limit with sine

lim_{x \to 0} \frac{s i n x}{x} = 1

Standard limit with the exponential

lim_{x \to 0} \frac{e ^{x} - 1}{x} = 1

Equation of a line through

(x_{0}, y_{0})

with slope

m

y - y_{0} = m (x - x_{0})

Distance between two points

d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}

Equation of a circle with centre

(a, b)

and radius

r

(x - a)^{2} + (y - b)^{2} = r^{2}

Number of combinations of

k

from

n

(k n) = \frac{n !}{k ! ( n - k )!}

Probability of independent events

A

and

B

both occurring

P (A \cap B) = P (A) \cdot P (B)

Conditional probability

P (A ∣ B) = \frac{P ( A \cap B )}{P ( B )}

for

P (B) > 0

n

-th term of an arithmetic sequence

a_{n} = a_{1} + (n - 1) d

, where

d

is the common difference.

n

-th term of a geometric sequence

a_{n} = a_{1} r^{n - 1}

, where

r

is the common ratio.

Sum of the first

n

terms of an arithmetic sequence

S_{n} = \frac{n ( a _{1} + a _{n} )}{2}

Volume of a cylinder

V = π r^{2} h

Volume of a sphere

V = \frac{4}{3} π r^{3}

Pythagorean identity

sin^{2} x + cos^{2} x = 1

Remainder theorem

The remainder of

p (x)

divided by

x - a

p (a)

Law of logarithms for a product

lo g_{b} (x y) = lo g_{b} x + lo g_{b} y

for

x, y > 0

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