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ESAT Mathematics Key Formulas and Rules

32 flashcards covering the algebra, trigonometry, sequences, calculus and geometry results used in ESAT Mathematics 1 and 2. Original study material for practice.

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Quadratic formula for

a x^{2} + b x + c = 0

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Quadratic formula for

a x^{2} + b x + c = 0

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

Discriminant of a quadratic and what it tells you

b^{2} - 4 a c

: positive gives two real roots, zero gives one repeated root, negative gives no real roots.

Completing the square for

x^{2} + b x + c

(x + \frac{b}{2})^{2} + c - \frac{b ^{2}}{4}

Difference of two squares

a^{2} - b^{2} = (a - b) (a + b)

Index laws for multiplying and dividing powers

a^{m} \times a^{n} = a^{m + n}

and

a^{m} \div a^{n} = a^{m - n}

Meaning of a negative and a fractional index

a^{- n} = \frac{1}{a ^{n}}

and

a^{m / n} = (n a)^{m}

Factor theorem

(x - a)

is a factor of

f (x)

if and only if

f (a) = 0

Remainder theorem

The remainder when

f (x)

is divided by

(x - a)

f (a)

Three laws of logarithms

lo g a + lo g b = lo g ab

lo g a - lo g b = lo g \frac{a}{b}

lo g a^{k} = k lo g a

Change of base for logarithms

lo g_{a} b = \frac{lo g _{c} b}{lo g _{c} a}

Relationship between

e^{x}

and

ln x

They are inverse functions:

e^{l n x} = x

and

ln e^{x} = x

n

th term and sum of an arithmetic sequence

u_{n} = a + (n - 1) d

and

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

n

th term and sum of a geometric sequence

u_{n} = a r^{n - 1}

and

S_{n} = \frac{a ( 1 - r ^{n} )}{1 - r}

Sum to infinity of a geometric series

S_{\infty} = \frac{a}{1 - r}

, valid only when

∣ r ∣ < 1

Binomial expansion of

(a + b)^{n}

for positive integer

n

\sum_{r = 0}^{n} (r n) a^{n - r} b^{r}

, where

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

Derivative of

x^{n}

e^{k x}

and

ln x

n x^{n - 1}

k e^{k x}

and

\frac{1}{x}

Derivatives of

sin x

and

cos x

(in radians)

\frac{d}{d x} sin x = cos x

and

\frac{d}{d x} cos x = - sin x

Chain rule

\frac{d y}{d x} = \frac{d y}{d u} \times \frac{d u}{d x}

Product rule

\frac{d}{d x} (uv) = u \frac{d v}{d x} + v \frac{d u}{d x}

How to classify a stationary point

Find where

f^{'} (x) = 0

; if

f^{''} (x) > 0

it is a minimum, if

f^{''} (x) < 0

it is a maximum.

Integral of

x^{n}

(

n \neq = - 1

\frac{1}{x}

and

e^{k x}

\frac{x ^{n + 1}}{n + 1} + c

ln ∣ x ∣ + c

and

\frac{1}{k} e^{k x} + c

Area under a curve between

x = a

and

x = b

\int_{a}^{b} y d x

, taking areas below the

x

-axis as negative.

Fundamental trigonometric identities

sin^{2} θ + cos^{2} θ = 1

and

tan θ = \frac{sin θ}{cos θ}

Double-angle formulae

sin 2 θ = 2 sin θ cos θ

and

cos 2 θ = cos^{2} θ - sin^{2} θ = 1 - 2 sin^{2} θ = 2 cos^{2} θ - 1

Sine rule and cosine rule

\frac{a}{sin A} = \frac{b}{sin B} = \frac{c}{sin C}

and

a^{2} = b^{2} + c^{2} - 2 b c cos A

Area of a triangle from two sides and the included angle

\frac{1}{2} ab sin C

Arc length and sector area (angle

θ

in radians)

Arc length

= r θ

; sector area

= \frac{1}{2} r^{2} θ

Equation of a circle with centre

(a, b)

and radius

r

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

Distance and midpoint between

(x_{1}, y_{1})

and

(x_{2}, y_{2})

Distance

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

; midpoint

= (\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2})

Gradients of parallel and perpendicular lines

Parallel lines have equal gradients; perpendicular gradients multiply to

- 1

Percentage change using multipliers

Increase by

r %

: multiply by

1 + \frac{r}{100}

; decrease by

r %

: multiply by

1 - \frac{r}{100}

Effect of enlargement with length scale factor

k

Areas scale by

k^{2}

and volumes by

k^{3}

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