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Thanawiya Amma Calculus, Algebra and Geometry Formulas

Flashcards with the key formulas of Thanawiya Amma pure mathematics: differentiation, integration, binomial theorem, sequences, vectors and trigonometry. Original study material.

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Derivative of

x^{n}

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Derivative of

x^{n}

n x^{n - 1}

Product rule

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

Quotient rule

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

Chain rule for

f (g (x))

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

Derivative of

e^{u}

and

ln u

u^{'} e^{u}

and

\frac{u ^{'}}{u}

Derivatives of

sin x

and

cos x

cos x

and

- sin x

Equation of the tangent at

x = a

y - f (a) = f^{'} (a) (x - a)

Condition for a local minimum

f^{'} (x) = 0

and

f^{''} (x) > 0

\int x^{n} d x

for

n \neq = - 1

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

\int \frac{1}{x} d x

and

\int e^{k x} d x

ln ∣ x ∣ + C

and

\frac{e ^{k x}}{k} + C

Area between

y = f (x)

and

y = g (x)

with

f \geq g

[a, b]

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

Volume of revolution about the

x

-axis

V = π \int_{a}^{b} y^{2} d x

General term of

(a + b)^{n}

T_{r + 1} = (r n) a^{n - r} b^{r}

Sum of the first

n

terms of an arithmetic sequence

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

Sum to infinity of a geometric series

\frac{a}{1 - r}

when

∣ r ∣ < 1

Dot product of two vectors

a \cdot b = ∣ a ∣∣ b ∣ cos θ

Condition for perpendicular vectors

a \cdot b = 0

Distance from

(x_{0}, y_{0}, z_{0})

to the plane

a x + b y + cz + d = 0

\frac{∣ a x _{0} + b y _{0} + c z _{0} + d ∣}{a ^{2} + b ^{2} + c ^{2}}

Sine rule and cosine rule

\frac{a}{s i n A} = \frac{b}{s i n B}

and

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

sin 2 θ

and

cos 2 θ

2 sin θ cos θ

and

cos^{2} θ - sin^{2} θ

Modulus of

z = a + bi

∣ z ∣ = a^{2} + b^{2}

Remainder theorem

The remainder of

f (x)

divided by

(x - a)

f (a)

Sum and product of the roots of

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

- \frac{b}{a}

and

\frac{c}{a}

Equation of a circle with centre

(a, b)

and radius

r

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

Determinant of a

2 \times 2

matrix

a d - b c

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