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Tawjihiya Calculus, Algebra and Vectors Formulae

Flashcards on derivatives, integrals, logarithms, sequences, vectors and trigonometric identities for the Kuwait Tawjihiya mathematics paper. Original study material.

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Derivative of x^n

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Derivative of x^n

\frac{d}{d x} x^{n} = n x^{n - 1}

Derivative of e^x and ln x

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

;

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

Derivative of sin x and cos x

(sin x)^{'} = cos x

;

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

Product rule

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

Quotient rule

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

Chain rule

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

Stationary points

Solve

f^{'} (x) = 0

;

f^{''} > 0

gives a minimum and

f^{''} < 0

a maximum.

Integral of x^n

\int x^{n} d x = \frac{x ^{n + 1}}{n + 1} + C

for

n \neq = - 1

Integral of 1/x

\int \frac{1}{x} d x = ln ∣ x ∣ + C

Definite integral and area

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

, the area under the curve when

f \geq 0

Laws of logarithms

lo g (ab) = lo g a + lo g b

;

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

;

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

Change of base

lo g_{a} b = \frac{l n b}{l n a}

Quadratic formula

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

Discriminant

b^{2} - 4 a c > 0

: two real roots;

= 0

: one;

< 0

: none.

Arithmetic sequence

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

;

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

Geometric sequence

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

;

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

;

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

for

∣ r ∣ < 1

Dot product of vectors

a \cdot b = ∣ a ∣∣ b ∣ cos θ = a_{1} b_{1} + a_{2} b_{2}

; perpendicular if

0

Magnitude of a vector

∣ a ∣ = a_{1}^{2} + a_{2}^{2}

Pythagorean identity

sin^{2} θ + cos^{2} θ = 1

Double angle formulas

sin 2 θ = 2 sin θ cos θ

;

cos 2 θ = 1 - 2 sin^{2} θ

Combinations

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

Permutations

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

Binomial theorem

(a + b)^{n} = \sum_{k = 0}^{n}_{n} C_{k} a^{n - k} b^{k}

Distance between two points

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

Equation of a line through a point

y - y_{1} = m (x - x_{1})

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